Image structure analysis for seismic interpretation · Image structure analysis for seismic...

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Image structure analysis for seismic interpretation Proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema, voorzitter van het College voor Promoties, in het openbaar te verdedigen op dinsdag 4 juni 2002 om 13.30 uur door Peter BAKKER doctorandus in de natuurkunde geboren te Linz Oostenrijk

Transcript of Image structure analysis for seismic interpretation · Image structure analysis for seismic...

Page 1: Image structure analysis for seismic interpretation · Image structure analysis for seismic interpretation Proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit

Image structure analysis for seismic interpretation

Proefschrift

ter verkrijging van de graad van doctoraan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema,voorzitter van het College voor Promoties,

in het openbaar te verdedigen op dinsdag 4 juni 2002 om 13.30 uurdoor

Peter BAKKER

doctorandus in de natuurkundegeboren te Linz Oostenrijk

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Dit proefschrift is goedgekeurd door de promotor:Prof. dr. ir. L.J. van Vliet

Samenstelling promotiecommissie:

Rector Magnificus, voorzitterprof.dr.ir. L.J. van Vliet, Technische Universiteit Delft, promotordr. P.W. Verbeek, Technische Universiteit Delft, toegevoegd promotorprof.dr.ir. A. Gisolf, Technische Universiteit Delftprof.dr.ir. R.L. Lagendijk, Technische Universiteit Delftprof.dr.ir. F.A. Gerritsen, Philips Medical Systemsdr. G.C. Fehmers, Shell International Exploration and Production B.V.dr. W.J. Niessen, University Hospital Utrechtprof.dr. I.T. Young, Technische Universiteit Delft, reserve lid

This project was financially supported by the Netherlands Ministry of Economic affairs,within the framework of the Innovation Oriented Research Programme (IOP Beeldverw-erking, project number IBV97005).

Advanced School for Computing and Imaging

This work was carried out in graduate school ASCI.ASCI dissertation series number 78.

ISBN: 90-75691-08-4c© 2002, Peter Bakker, all rights reserved.

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Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Traditional interpretation of 3-D seismic data . . . . . . . . . . . . . . . . 21.2 Improving the efficiency of the interpretation process . . . . . . . . . . . . 4

1.2.1 Structure enhancement for horizon tracking . . . . . . . . . . . . . 41.2.2 Seismic attributes for detection . . . . . . . . . . . . . . . . . . . . 5

1.3 Image processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.1 Adaptive filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2. Linear structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 The representation of linear structures . . . . . . . . . . . . . . . . . . . . 12

2.1.1 Orientation representation . . . . . . . . . . . . . . . . . . . . . . . 132.1.2 Gradient structure tensor . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Orientation adaptive filtering . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Edge preserving filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.1 Generalized Kuwahara filtering . . . . . . . . . . . . . . . . . . . . 222.3.2 Improving orientation estimation near borders . . . . . . . . . . . . 262.3.3 Edge preserving orientation adaptive filtering . . . . . . . . . . . . 29

2.4 Application: Automatic fault detection . . . . . . . . . . . . . . . . . . . . 322.5 Application: Structure enhancement . . . . . . . . . . . . . . . . . . . . . 37

3. Curvilinear structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.1 The GST for 3D plane-like curvilinear structures . . . . . . . . . . . . . . . 42

3.1.1 The quadratic surface approximation . . . . . . . . . . . . . . . . . 423.1.2 The quadratic GST for 3D surfaces . . . . . . . . . . . . . . . . . . 433.1.3 Experimental tests and results . . . . . . . . . . . . . . . . . . . . . 45

3.2 The GST for 2D curvilinear structures . . . . . . . . . . . . . . . . . . . . 513.3 Curvature adaptive filtering . . . . . . . . . . . . . . . . . . . . . . . . . . 523.4 Appendix A: Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.5 Appendix B: Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 56

4. Line-like curvilinear structures . . . . . . . . . . . . . . . . . . . . . . . . . 614.1 The GST for 3D line-like curvilinear structures . . . . . . . . . . . . . . . . 61

4.1.1 The quadratic curve approximation . . . . . . . . . . . . . . . . . . 614.1.2 The quadratic GST for space curves . . . . . . . . . . . . . . . . . . 63

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iv Contents

4.2 Experimental tests and results . . . . . . . . . . . . . . . . . . . . . . . . . 664.2.1 Circle image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.2.2 Helix image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.2.3 Ellipse image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.4 Application: Channel detection . . . . . . . . . . . . . . . . . . . . . . . . 744.5 Appendix A: Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.6 Appendix B: Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5. Structural analysis using a non-parametric description . . . . . . . . . . 815.1 The tracking of line-like curvilinear structures . . . . . . . . . . . . . . . . 82

5.1.1 Application to the tracking of growth rings . . . . . . . . . . . . . . 835.1.2 Application to the tracking of sedimentary structures . . . . . . . . 84

5.2 Non-parametric adaptive filtering . . . . . . . . . . . . . . . . . . . . . . . 885.3 Non-parametric confidence estimation . . . . . . . . . . . . . . . . . . . . . 92

5.3.1 Application to channel detection . . . . . . . . . . . . . . . . . . . . 93

6. Coherency estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.1 Coherency based on the eigenstructure of the covariance matrix . . . . . . 986.2 Coherency estimation using the GST . . . . . . . . . . . . . . . . . . . . . 1006.3 The presence of structural dip . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.3.1 Dip estimation by sampling the dip dependency . . . . . . . . . . . 1036.3.2 Comparison between the dip estimates of the GST and dip search . 104

6.4 An experimental comparison for fault detection . . . . . . . . . . . . . . . 108

7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137.1 Image processing approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137.2 Application to seismic interpretation . . . . . . . . . . . . . . . . . . . . . 114

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Samenvatting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Dankwoord . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Curriculum vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

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1. Introduction

The growing global population and living standards cause an increasing demand for energy.Despite the many efforts made to exploit ‘new’ energy sources such as solar energy andbio-mass, oil and gas continue to be the primary sources of energy. The total amount ofoil and gas produced each year is still increasing, and is very likely to continue to do so forat least 30 years. The oil industry is searching for new reservoirs on land and offshore, inincreasingly difficult environments. Furthermore, the trend in the oil industry has changedfrom producing ‘at any cost’ in the seventies to a cost efficient and environmentally awareproduction today.

Since the first seismic surveys, in the 1920’s, the seismic reflection method has played animportant role in the exploration of oil and gas. The seismic method is a powerful remotesensing technique that can image the subsurface over depths from several meters to severalkilometers. The basic idea is to first generate an acoustic wave field by a localized source.This field travels down the subsurface and partly reflects at locations where the acousticrock properties change. The reflected wave field is measured by an array of localizedreceivers on the surface.

The seismic method can be divided into three parts. It starts with the acquisition thatconsists of collecting raw data directly from the receivers. Usually, several different ‘shots’are recorded of the same location. These different shot-records are stacked, or averaged,to reduce the influence of noise. Next, all the acquired data is processed to isolate thesignal that corresponds to the travel-time of the reflected wave field from the surface tothe reflectors. Signals due to diffraction and multiples should be suppressed. Multiples arewave fields that have been reflected more than once. Migration techniques are used to geta sharp, ‘focused’ image of the subsurface. Finally, the identification of possible oil andgas reserves is done by interpretation of the processed image using geological models andinformation from well measurements such as logs and bore-hole images.

The first 3D seismic survey was shot over a field in Texas in 1967. Since then, therehas been an increasingly rapid expansion in the application of this technology. A 3Dseismic image I(x, y, t) has two spatial coordinates (x, y) parallel to the surface and oneperpendicular time coordinate (t). The time usually corresponds to the two-way travel-time, i.e. the time it takes for a wave field to travel from the surface to a reflector and back.The typical spatial sample spacing is 12.5 m and the time resolution varies from 25 m inthe shallow part of the data to 100 m in deeper parts. Although many geological featuresare still below seismic resolution, seismic images can in favorable circumstances reveal the

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2 Chapter 1 Introduction

internal structure of a reservoir. A 2D cross-section of a 3D seismic image is shown infigure 1.1. The 3D image is the shallow part of a processed 3D seismic survey, showing thesea-floor. The total time interval of the image is 0.75 s, assuming that the acoustic velocityis 2000 m/s, this interval corresponds to 750 m. The traditional 2D surveys gave only afew cross-sections of the subsurface, but the 3D surveys give a full representation of thesubsurface. This offers great advantages for both the processing and the interpretation ofthe seismic data. The development of 3D migration techniques has significantly improvedthe quality of seismic images. Densely sampled horizontal or time slices made it possiblefor the interpreter to map the lateral changes in the subsurface with a higher accuracy.

Although much progress has been made in the interpretation process since the first seismicsurveys, the current seismic interpretation methods still do not exploit the data to its fullpotential. Subtle geological features are often difficult to recognize in a seismic image forthe human expert. Furthermore, the total amount of data that has to be interpreted growsmore rapidly than the total number of interpreters, demanding an increasing efficiency.Image processing techniques can be used to enhance certain features in the data, and theycan provide the basis for the automation of certain interpretation tasks. In this thesiswe will present image processing techniques that can contribute to the automation of theinterpretation of seismic data.

21.3 km

t

x

Sea

0.75s

Figure 1.1: A vertical 2D cross-section of a 3D seismic image. Three horizons have been trackedand they are displayed as white lines and pointed to by white arrows.

1.1 Traditional interpretation of 3-D seismic data

The introduction of 3D seismic images caused a radical change in the interpretation. In-stead of having a dozen of paper sections, interpreters now have densely sampled 3D imagesand powerful workstations at their disposal. One of the direct advantages is the ability tofreely choose the view angle, and it was soon discovered that a horizontal view or a timeslice is very useful for the interpretation of depositional structures. Furthermore, the 24-bitcolor displays of modern workstations provide intuitive color maps with a large dynamicrange.

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1.1 Traditional interpretation of 3-D seismic data 3

Prior to the description of the traditional interpretation, we will introduce some terminol-ogy with the help of a simple geological model of the subsurface, in which we limit ourselvesto sedimentary rocks. The formation of sedimentary rocks begins with the physical depo-sition of sediments on the surface. The physical properties such as the grain size, and thechemical properties of the sediments depend on the environmental conditions. During thecontinuation of this physical deposition process, the older sediments are buried deeper intothe subsurface. The increasing of the pressure and the temperature with depth causes theconversion of the sediments into sedimentary rock. The sedimentary layers can be tilted,bend of fractured by the plate tectonic forces in the earth’s crust. A sequence of sedimen-tary layers formed in one type of sedimentological environment is called a stratigraphicsequence. A change of the sedimentological environment, for example caused by a sea-levelchange, means the beginning of a new stratigraphic sequence.

Seismic interpretation begins with mapping the large scale structure of the area. Thisstructural interpretation mainly consists of creating horizons and fault planes. Horizonsare surfaces that are created by the interpreter by selecting a reflector and following it overthe volume. As an example three horizons are shown as white lines in figure 1.1. Thereare several possible reasons why a reflector is selected to be interpreted as a horizon. Thesimplest reason is that the reflector is outstandingly clear and strong, making it easy totrack. Sequence boundaries are important horizons to distinguish between the differentgeological periods. Another example of an important horizon is the top of a reservoir. Afracture in the subsurface rock caused by tectonic forces is called a fault. Faults causediscontinuities in the layered structure that make the creation of horizons more difficult.To be able to continue a horizon over a fault, it is necessary to know the amount ofvertical displacement between both sides of the fault. It may be possible that one reflectorseamlessly continues over a fault into a different reflector. To avoid that these two reflectorsare incorrectly interpreted as belonging to one horizon, the entire fault-surface should beknown.

A common next step in the interpretation process is to map the structure inside a sedimen-tary layer. The horizons created in the structural interpretation can be used to approximatethe depositional surfaces within the time interval where the reflectors are approximatelyparallel to the horizons. With a depositional surface we mean the earth’s surface in a pre-vious geological time where paleodeposits were made upon. The sedimentary structuresmanifest themselves as lateral changes in the seismic response1 along the horizon, and canbe recognized by their morphology. A channel for example, a sedimentary structure formedby flowing water, has a very distinctive meandering morphology.

1 In essence the shape of the seismic wavelet. The amplitude and the phase of the wavelet are the mostcommonly used descriptors for this shape.

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1.2 Improving the efficiency of the interpretation

process

The volume of the seismic datasets has increased tremendously over the last three decades,the area covered by the surveys has grown and the sample density has become larger. Thelast few years a lot of research is done on time lapse seismic. The goal of this technique is tomonitor during the production, the changes inside a reservoir over time. Currently, thesedatasets contain only a few time samples, but installing a permanent array of receiverson top of the reservoir could lead to full 4D seismic images. Another development thatpotentially adds an extra dimension is the sampling of the amplitude versus the offset(AVO). This offset is the lateral distance between the start point of the acoustic wave andthe point of reflection. These developments have already contributed to an increase of atleast a factor 1000 of the total amount of data per survey. The demand for more datapoints will continue in the future. Seismic dataset currently require a couple of Terabytesof computer storage space.

It is clear that the analysis of this increasing amount of data, requires an increase in theefficiency of the interpretation process. One way forward is to improve the visualizationof the data. Interpreting 3D data by the visual inspection of 2D cross-sections is notoptimal. Improved 3D visualizations are provided by immersive virtual reality rooms. Thepossibility to literally walk through the data also presents a new form of human-computerinteraction. Another possibility to improve the visualization is to make a 2D animation ofthe 3D data by stepping through the consecutive slices at a high frame rate.

Although an improved 3D visualization can considerably speedup the manual interpreta-tion, it falls outside the scope of this thesis. The approach chosen in this thesis is to developimage processing tools for the automation of interpretation tasks. In the remainder of thissection we will indicate the application domain of the tools presented in this thesis.

1.2.1 Structure enhancement for horizon tracking

Three dimensional seismic data made it possible to delineate faults and sedimentary struc-tures laterally. Time slices, however, can cut through different depositional surfaces, show-ing only parts of these structures. This makes the interpretation of time slices difficult.The traditional solution is to approximate the depositional surface by picking a reflectorand follow it over the seismic image. This process is commonly referred to as horizontracking. Since it is very time consuming to manually mark every individual point of thehorizon, algorithms have been developed to automate this task. A poor data quality orcomplex faulting can hamper the tracking considerably, and the tracking software oftenfails at many points on the horizon.

The success of the automatic horizon tracker could be improved by enhancing the structural

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1.2 Improving the efficiency of the interpretation process 5

information in the seismic image. We will present an adaptive filter method that suppressesrandom fluctuations and small deviations. This filter should be flexible enough to avoidfiltering across faults, thereby preventing the merging of different reflectors from both sidesof the fault.

1.2.2 Seismic attributes for detection

Geological features in the data that are difficult to estimate by the human visual system,can be made explicit by the computation of seismic attributes. In the field of imageprocessing and pattern recognition attributes are referred to as features. Examples ofconventional seismic attributes are the complex seismic trace attributes: instantaneousamplitude, phase, frequency, and bandwidth [TKS79]. A seismic trace is a vertical line ofdata in a seismic image I(x = x0, y = y0, t).

Sedimentary structures are usually found by computing a seismic attribute in a certaintime window around a horizon. Computing the variance of the seismic response withina time window, for example, projects all lateral changes in a seismic sequence onto atwo dimensional surface. This allows the interpreter to check a volume of data for theoccurrence of sedimentary structures by inspecting only one 2D display. The computationof the 3D attributes dip and azimuth along a horizon are very useful for the structuralinterpretation [DGS+89].

Since a few years computers have become powerful enough to compute 3D seismic attributesover an entire data volume, creating attribute volumes. In 1995 Bahorich and Farmerintroduced a 3D seismic attribute they called the coherency cube [BF95]. Their algorithmcomputes the optimally lagged cross-correlation between neighboring traces. This wasthe first time a coherency measure was presented as a seismic attribute in the literature.Three dimensional seismic attributes like coherency, dip-magnitude, and dip-azimuth, canbe used to highlight lateral changes. A time slice of a 3D attribute volume is much easierto interpret than a time slice of a seismic volume. Because each point in the attributevolume is computed using a 3D neighborhood, the detection of geological features is lessdependent on the exact location, and therefore less dependent on carefully picked horizons.

The main use of seismic attributes is to create an intuitive display that allows an interpre-tation task to be performed both more efficiently and more effectively. In this thesis wewant to go one step further. We want to develop seismic attributes or features that allowthe automatic detection of certain geological features. We will focus on the automaticdetection of faults and the automatic detection of channels.

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1.3 Image processing

Now that we have sketched our application domain, it is time to introduce the imageprocessing approach. Although seismic interpretation is both the motivation and the ap-plication of this work, our goal is to develop image processing techniques that are generalenough to be applicable to other natural images as well. We will describe the requiredproperties for this more general class of images, and we will give some practical exam-ples. Nevertheless, seismic images will continue to form the thread that runs through thisintroduction.

Seismic images often show patterns with a layered structure due to the depositional natureof the subsurface, see for example figures 1.1 and 1.2a. In image processing a patternwith a certain regularity or structure is called a texture. The description of the ‘layered’textures in seismic images can be split up in two parts. One part is the geometricaldescription of the structure, the other part is the description of the signal perpendicularto the layered structure. Examples of geometrical properties are the orientation and thecurvature of the layered structure. An example of a property of the perpendicular signalis its characteristic frequency. In the case of a seismic image, the perpendicular signal isdetermined by the change in the acoustic impedance of the subsurface rock, convolved withthe seismic wavelet. This convolved signal is usually described by using a time-frequencyrepresentation [TKS79, Ste97, Des97]. The main subject of this thesis is the geometricaldescription of the structure of layered textures.

(a) seismic (b) wood (c) finger print (d) interference

Figure 1.2: Examples of natural images containing ‘layered’ textures.

The geometrical description is insensitive to the properties of the perpendicular signal2,and is therefore applicable to all images with a layered structure. Some practical examplesof images belonging to that class are shown in figure 1.2. Besides a seismic image (a), a2D cross-section of a CT image of a tree-trunk is shown (b). This image can be used forthe counting of growth rings. Furthermore, a finger print image (c) and an image of aninterference pattern of a vibrating plate (d) are shown.

2 The only demand on the perpendicular signal is that it is not constant. In other words, its spectrumshould contain frequencies other the zero.

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1.3 Image processing 7

The individual layers of a layered texture in an image can locally be approximated withisophotes. Isophotes are curves and surfaces with a constant intensity value. For thegeometrical description of these isophotes, we do not analyze the image intensities directly.We analyze their spatial relationship to characterize the shape of the isophotes. The firsttask for the geometrical description is to define properties that can describe this shape. Therelevant properties depend, however, on the scale of the analysis. At the smallest scale, thelevel of the individual points of the image, only the intensity of the image point itself can bemeasured. The definition of geometrical properties at this scale is therefore not possible. Atthe largest or global scale, the description of the isophotes in general becomes very complex.The geometrical properties should therefore be defined and estimated locally, which meansat some intermediate scale. The scale of the geometric analysis is not directly related tothe frequency content of the data, but merely determines the spatial extent of the datathat has to be modeled. In general, the optimal scale for the estimation of some featuremay vary over the image. The feature may also be present at several scales simultaneously.Both cases require the estimation of the feature at multiple scales.

Parametric description

The parametric description of the local geometry is based on the heuristic that decreasingthe analysis scale decreases the geometric complexity. In other words, at a small scale alayered structure can be described using a simple geometric model. This complexity canbe compared with the number of polynomial terms necessary to approximate an arbitraryfunction around a certain point. The larger the area around the point that needs to bemodeled correctly the more terms are needed. Since we want to describe the differentialgeometry, we first compute the gradient at each point, then we apply a parametric modelto the gradient information. In chapter 2 we start the analysis with the simplest gradientmodel, namely a straight model. A straight structure is fully determined by its orientation,and the representation and estimation of orientation is therefore discussed in detail inchapter 2.

Orientation representation and estimation is an intensively studied subject in image pro-cessing [Knu89, SF96, Mar97, GVV97, KHRV99, CST00]. An important property of anorientation representation is its modality. Imagine two straight layered textures3 at dif-ferent angles as drawn in figure 1.3. This configuration can occur in seismic images atsequence boundaries. The textures inside the windows 1 and 2, can be described using asingle orientation, thus using a uni-modal representation. The description of the textureinside window 3, requires two orientations, and therefore a multi-modal orientation rep-resentation. Uni-modal representations are in general more computationally efficient, andwe will therefore avoid multi-modal orientation estimation if possible.

Examining the images in figure 1.2, one can conclude that modeling these images as locally

3 Straight layered textures are more commonly referred to as oriented textures.

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8 Chapter 1 Introduction

3

1

2

Figure 1.3: Schematic display of two touching layered textures. The textures are analyzed insidea circular window at three different locations indicated by circles.

straight is feasible. It is also clear from these images that the curvature of the structurebecomes more important for larger scales. In chapters 3 and 4 we will extend the localstraight models by incorporating curvature.

Non-parametric description

A parametric description of the geometry at an even larger scale requires more parameters.The estimation of these higher order parameters increases the computational demand, andthe applicability of these parameters is limited. In chapter 5 we therefore turn to a non-parametric description. We will use the orientation and curvature estimates for a piece-wisedescription of layered structures.

Detection

Faults manifest themselves in seismic images as discontinuities in the layered structure, ascan be seen in figure 1.2a. Modeling this seismic image as locally straight gives an accuratedescription of the largest part of the data, but fails on the faults. In other words, if wedefine a resemblance measure between a planar reflector and the local data, then we canuse this resemblance measure for the detection of faults. A fault detection method basedon this idea is presented at the end of chapter 2.

Closely related to this approach is the estimation of seismic coherency. Different measuresfor the estimation of coherency are recently published in [MKF98, GM99, MK00], andsome of the techniques presented in these papers will be discussed in chapter 6. Thesearticles show that the seismic coherency is lower at both faults and channel banks. For thedelineation of faults one should use a large temporal window and a small spatial window,while the delineation of channels and other depositional structures requires a large spatialwindow size and a small temporal window. This is due to the fact that faults can continueacross several different geological time periods, but depositional structures are restrictedto a single time period. The typical vertical extent of a fault is 100-1000m and for adepositional structure 10-100m. The image processing techniques developed in chapters 4and 5 are applied to the detection of channels.

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1.3 Image processing 9

1.3.1 Adaptive filtering

Once we have a description of the structure of a texture, we can use this description forthe enhancement of this structure. We now define the description as the model of the dataand any deviation of the data from this supposed model as the noise in the data. Usingthese definitions, we can approach structure enhancement as a noise reduction problem.Noise in images is traditionally suppressed by low-pass filtering. This assumes that thesignal at some point changes slowly compared to the noise, i.e. the signal and the noiseare spectrally separable. In the case of layered textures this is only partly true. The signalchanges slowly inside the layers, and low-pass filtering across the layers suppresses boththe signal and the noise. The suppression of the signal can be avoided by locally adaptingthe shape of the low-pass filter window in such a way that the window ‘fits’ inside of asingle layer. The local shape of the layers is estimated using the parameter estimates ofthe local geometric model.

.......

Outputimage

Adaptive

filter

σ κφ

Parameters

Inputimage

Figure 1.4: Generic adaptive filtering scheme. Parameters (σ, φ, κ, . . . ) are locally estimated onthe input image to locally control an adaptive filter.

The processing scheme of adaptive filtering is depicted in figure 1.4. First the controlparameters are estimated. Candidate parameters are, in the case of structure enhancement,scale, orientation and curvature (σ, φ, κ). Next, the parameters are used to adapt the shapeof the filter window. In the case of the straight model, this results in disc-shaped windowssteered to the local orientation.

This kind of adaptive filtering can also be used as a method for providing the analysiswindow that yields an optimal property estimate. As we have seen above, the optimalwindows for the detection of faults and channels have an anisotropic shape. The faultdetection window is elongated along one dimension that should be steered perpendicularto the horizons. The window for the detection of channels, on the other hand, is elongatedin two dimensions and this disc-shaped window should be steered parallel to the horizonof interest.

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2. Linear structures

Classical image processing filters for smoothing and edge detection, are designed for verysimple local neighborhoods: homogeneous regions separated by discontinuities. However,a local neighborhood can also contain a pattern. If this pattern has some form of orderedstructure, then it is called a texture. Typical examples of textures are the patterns ondifferent types of cloth, in wood, or brick walls. In this chapter we are going to study aspecific type of texture, namely oriented textures. An oriented texture is a pattern with alinear structure. We start the analysis of oriented textures by giving a definition of linearstructure.

(a) (b) (c)

Figure 2.1: Images that consist of multiple domains. In the first image (a) the domains arecharacterized by the intensity. The domains in images (b) and (c) contain orientedtextures, which are characterized by the dominant orientation.

definition A linear structure in N-D space is shift invariant in at least one orientation,but not in all orientations.

For the mathematical definition of N-D linear structures, we describe a neighborhood as afunction f(x). A neighborhood f(x) is shift invariant along x1, if

f(x) ≡ f(x1, x2, · · · , xn) = f(x1 + d, x2, · · · , xn) (2.1)

for all values of the scalar d. Since the neighborhood f does not change as a function ofx1, we can write if as a function with a reduced dimensionality

f(x) = f(x2, x3, · · · , xn). (2.2)

Thus we also can define linear structures by the reduction of dimensionality. A neighbor-hood f has a linear structure if there exists a basis β, such that

fls(a) ≡ fls(a1, a2, · · · , an) = f(a1+i, a2+i, · · · , an) , i ∈ {1, 2, · · · , n − 1} (2.3)

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12 Chapter 2 Linear structures

with a = [x]β, the coordinate vector in basis β. We call the axes of the coordinate system(a1, a2, · · · , an), the principal axes of the linear structure.

In the two-dimensional case, there are two linear independent orientations, allowing onlyone possible linear structure. Namely, with one shift invariant orientation, and the otherorientations not shift invariant. Two examples of two-dimensional oriented textures aregiven in figure 2.1b and c. In n-dimensional data, there are n− 1 types of linear structuresthat differ by the number of independent shift invariant orientations (i). In this thesiswe differentiate between two types of linear structures: line-like and plane-like. A line-like structure is shift invariant along only one orientation (i = 1), and a plane-like linearstructure is shift invariant along two orientations (i = 2). The simplest examples of line-and plane-like linear structures are a single line and a single plane respectively. A line-likeoriented texture can also be regarded as a collection of individual lines, and therefore bereferred to as a line-bundle. Note that plane-like linear structures are only meaningful ifthe dimensionality n > 2.

Granlund and Knutsson [GK95] defined the simple or one-dimensional neighborhood as thebuilding block of their analysis. In a simple neighborhood the intensity fs only changesalong one orientation and can be mathematically represented by

fs(x) = f(x · n). (2.4)

We do not use this structure definition directly, because it has no clear geometrical meaning.Simple neighborhoods are identical to line-like linear structures in 2D, and identical toplane-like linear structures in 3D. The geometrical interpretation of line-like and plane-likelinear structures is the same for arbitrary dimensional data. The geometrical interpretationof a simple neighborhood on the other hand, is different for each dimensionality.

2.1 The representation of linear structures

In this section we address the problem of finding a representation for a neighborhoodwith a linear structure, that is suitable for further processing. A linear structure can becharacterized by the orientation of its principal axes and the number of shift invariantorientations (i). A property of linear structures is that they are invariant under point-inversion

fls(x) = fls(−x). (2.5)

The representation should have this property as well. In two dimensions this propertycorresponds to the fact that 180 degrees rotation of a line amounts to no change at all.This structure should therefore be represented by orientation, which is defined moduloπ (180◦) as opposed to direction which is defined modulo 2π (360◦).

An orientation estimate is a valuable feature that allows the segmentation of images withseveral oriented textures. If we want to detect the borders between the regions in the

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2.1 The representation of linear structures 13

images in figure 2.1, then we can use traditional edge detectors for the borders between theconstant intensity domains. The detection of borders between different oriented texturesrequires the estimation of orientation. Later on in this thesis we will use orientation to steeran adaptive filter. Another example of using the output of an orientation estimator forfurther processing, is to measure the change in orientation, i.e. the curvature [GWVV99].Stabilizing the orientation estimate by averaging is often required for these tasks. Therepresentation should therefore be continuous.

2.1.1 Orientation representation

We start by studying the two dimensional case. The most obvious representation of orien-tation is a scalar quantity. For example the rotation angle φ, corresponding to the rotationthat aligns the principal axes of the structure with the coordinate system. While it iseasy to restrict the scalar quantity to the interval [0, π), it is impossible to make it con-tinuous at the same time. For each arbitrary interval there is a ’jump’ between the twoend-points of that interval, i.e. it cannot be made cyclic. Think of visualizing orientationwith grey-values; at some angle there is always a jump from black to white, see figure 2.2.

Figure 2.2: Image of an oriented texture and its local orientation. Visualizing orientation withgrey-values suffers from two unavoidable black to white jumps.

Another possibility is to represent orientation by a vector x, e.g. the gradient vector.Although this vector gives a continuous representation, it is defined mod 2π. The vectorsx and −x do not map to the same value. Limiting the vectors to a two dimensionalhalf plane makes the representation discontinuous. The correct procedure is to double theangle.

(

r cos φr sin φ

)

→(

r cos 2φr sin 2φ

)

(2.6)

This vector representation was introduced by Granlund [Gra78].

A representation of 3D orientation was given by Knutsson in[Knu85]. He addressed theproblem by stating that a suitable representation should meet three basic requirements:

1. ’uniqueness’: the vectors x and −x should map to the same value,

2. ’uniform stretch’: preservation of the angle metric of the original space.

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14 Chapter 2 Linear structures

3. ’polar separability’: while the angle may change, the magnitude of the mapped vectorshould only be a function of the magnitude of the original vector.

The mapping he found was a complicated mapping to a 5D vector. Later Knutsson [Knu89]found a mapping M that maps the vector x onto the tensor T defined by

M : T ≡ xxT

||x|| . (2.7)

This tensor mapping also meets all three criteria and is therefore suitable for furtherprocessing. The mapping consists of applying the dyadic product and normalizing theresulting tensor with the magnitude of the vector. To be able to compare the mapping Mwith the mapping in eq. 2.6, we show the two-dimensional case in polar coordinates.

r

(

cos φsin φ

)

→ r

2

(

1 + cos 2φ sin 2φsin 2φ 1 − cos 2φ

)

(2.8)

The major advantage of the tensor mapping is that it also holds for higher dimensionalimages. Another interesting property of the tensor mapping is its inverse. The spacespanned by the tensors T that are created by mapped vectors according to eq. 2.7, is onlya sub-space of the entire tensor space. The least squares approximation Tls to a tensoroutside of this sub-space T′ is given by

Tls = λ1e1eT1 , (2.9)

where λ1 is the largest eigenvalue of T′ and e1 the corresponding eigenvector.

2.1.2 Gradient structure tensor

The structure tensor is defined as

T ≡(

xxT

||x||n)

, (2.10)

where ( ) indicates some weighted local average. The computation the structure tensorconsists of two steps. First the orientation is estimated for each point in the image usingorientation selective filters. Next, the filter outputs are mapped to the tensor representationand averaged. The normalization of the tensors with ||x||n, determines how the orientationaveraging is weighted with the corresponding intensity contrast. The structure tensor canbe used as a tool for the local analysis of linear structures. Several different implementationsand applications of the structure tensor have been reported in the literature. The earliestpublications are [KW87, RS91, Hag92], and the list of applications has steadily grown overthe last decade.

An efficient implementation of the structure tensor is the gradient structure tensor (GST).The first step is to estimate the gradient g = ∇I at scale σg. We compute the gradients by

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2.1 The representation of linear structures 15

convolving the image with the first order derivatives of a Gaussian. For an N-dimensionalimage the components of the gradients are

gi = I(x) ⊗ ∂

∂xiG(x; σg) , i ∈ {1, . . . , N} (2.11)

The second step consists of mapping the gradient to the tensor using the dyadic productand average the tensor components Tij at scale σT . The gradient structure tensor is definedby

T ≡ ggT , (2.12)

and the mapping used is equivalent to eq. 2.10, with n = 0. This means that the elementsof the GST can be interpreted as gradient energies. We compute the local average, orspatial integration, by convolving the tensor components with a Gaussian kernel.

Tij = Tij ⊗ G(x; σT ) (2.13)

The tensor scale is usually chosen three to ten times the gradient scale: 3σg < σT < 10σg.

Averaging the tensor has a number of advantages. Rapid changes in the orientation dueto noise on the gradient vector are suppressed, yielding a smooth orientation estimate.Furthermore, the GST not only contains information about the gradient energy in theorientation of the maximum gradient, but also in all perpendicular orientations. Thisextra information allows for the differentiation between different types of local structures,as we will see below.

As an example, we show the GST for a 3D local neighborhood f(x, y, z) using the derivativenotation by indexes.

g =

fx

fy

fz

T =

f 2x fxfy fxfz

fxfy f 2y fyfz

fxfz fyfz f 2z

(2.14)

The computation of the GST in 3D requires nine convolutions, three for the gradient andsix for the tensor smoothing. Since the tensor is symmetric only N(N + 1)/2 componentshave to be processed, where N is the dimensionality of the image.

The relevant information contained by the GST is extracted by computing its eigenvaluesand eigenvectors. Due to the way it is constructed the GST is symmetric and semi-positivedefinite. This means that the eigenvalues are real and positive. By diagonalizing the tensor,the eigenvalue analysis finds the the coordinate system that is aligned with the principalaxes. This aligned coordinate system is given by the set of eigenvectors, and we call it thegauge-coordinate system. The gradient energies along the principal axes are given by theeigenvalues, and the largest eigenvalue corresponds to the dominant orientation. For highdimensional cases (N > 3), the eigenvalue problem must be solved numerically, but in the2D and 3D cases the eigenvalues can be found analytically.

The structure tensor can also be based on other orientation filters than the gradient filterswe use. Knutsson [Knu89] used angular separable quadrature filters to estimate orientation.

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16 Chapter 2 Linear structures

The advantage of quadrature filters over derivative filters is that they are phase invariant,i.e. give a response on both even and odd structures. Derivative filters only give a responseon odd structures, e.g. on the edges and not on the ridge of a line. This limited support nolonger constitutes a problem since the tensor averaging combines the gradient informationfrom both slopes of a line, without the cancellation of opposite vectors. The minimumnumber of convolutions needed to estimate 3D orientation using the quadrature filters in3D is 12, while the computation of the gradient only requires 3 convolutions. Furthermore,mapping of the gradient to the tensor is much less complicated than the mapping of thequadrature filters.

The structure tensor analysis assumes a uni-modal orientation distribution. The dominantorientation corresponds to the weighted average orientation. In the case of multiple maximain orientation histogram, the dominant orientation will, in general, not coincide with amaximum.

Interpretation of the eigenvalues

The richness of the structure tensor analysis becomes apparent if we study the eigenvalues.Throughout the thesis we assume that the eigenvalues are ordered, i.e. λi > λi+1. Intwo dimensions we can distinguish three different cases, corresponding to different typesof local neighborhoods. They are given in the table below:

λ1 λ2 description0 0 Both eigenvalues are zero. No gradient energy, no contrast.

Constant intensity with no measurable structure.> 0 0 One eigenvalue is zero. Linear structure.> 0 > 0 Both eigenvalues are greater than zero. The underlying struc-

ture deviates from the linear structure model, e.g. due to noiseor curvature or multiple orientations. If λ1 = λ2, then thestructure is isotropic.

In practice, the eigenvalues should not be checked against zero, but against a thresholdvalue that is determined by the noise and the signal energy in the image. Without aquantitative estimate of the noise level, it will not be possible to distinguish between noiseand isotropic structures. The total gradient energy is given by the trace of the GST

Eg = Tr(T) =∑

i

λi. (2.15)

Contrast independent measures can be constructed dividing the eigenvalues by the totalenergy Eg. In this thesis we use the following contrast independent confidence measure

Can =λ1 − λ2

λ1 + λ2

. (2.16)

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2.1 The representation of linear structures 17

It takes values between 0 and 1 and indicates how much the local data resembles a linearstructure. The next table sums up the behavior of Can.

isotropic: λ1 ≈ λ2 Can ≈ 0Linear structure:(anisotropic)

λ1 � λ2 Can ≈ 1

The more isotropic a structure becomes (Can → 0), the more difficult it becomes to estimatethe orientation of that structure. Therefore we use Can as the confidence measure of theorientation estimation. Since a linear structure can also be viewed of as an anisotropicstructure, Can is also referred to as the anisotropy.

The analysis of the eigenvalues of the structure tensor in 3D is similar to the analysis in2D. The extra dimension introduces the plane-like linear structures and the different casesare given in the table below:

λ1 λ2 λ3 description0 0 0 all eigenvalues are zero. No gradient energy, no contrast. Con-

stant intensity with no measurable structure.> 0 0 0 Two eigenvalues are zero. Plane-like linear structure.> 0 > 0 0 One eigenvalue is zero. Line-like linear structure. If λ1 = λ2,

then the cross-section of the line-like structure is isotropic, e.g.cylindrical.

> 0 > 0 > 0 All eigenvalues are greater than zero. The underlying structuredeviates from the linear structure model. If λ1 = λ2 = λ3, thenthe structure is isotropic.

Again we like to have contrast independent measures. With three eigenvalues we are ableto construct two mutually independent confidence measures [KBV+99]

Cplane =λ1 − λ2

λ1 + λ2

, Cline =λ2 − λ3

λ2 + λ3

, (2.17)

and they both take values between 0 and 1. The confidence measure Cplane can be inter-preted as how much the neighborhood resembles a plane-like structure, and Cline indicatesthe resemblance to line-like structures. These measures can differentiate between the fol-lowing local structures.

isotropic: λ1 ≈ λ2 ≈ λ3 Cplane ≈ 0 Cline ≈ 0Plane-like: λ1 � λ2 ≈ λ3 Cplane ≈ 1 Cline ≈ 0Line-like: λ1 ≈ λ2 � λ3 Cplane ≈ 0 Cline ≈ 1

Application of the GST to seismic data

The seismic method is a remote sensing technique for the inspection of the outer layer ofthe earth’s crust with a maximum depth in the order of kilometers. This relatively thin

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18 Chapter 2 Linear structures

outer layer mainly consists of sedimentary rock which has a stratified structure. If there issufficient contrast in the acoustic impedance of the different layers that were deposited ontop of each other, then the stratification will be visible in the seismic image. Depositionalsurfaces, or more specifically the corresponding reflectors and seismic sequences, can locallybe modeled as a plane-like linear structure. The GST is an efficient image processing toolthat incorporates this model. We have seen that the GST yields parameter estimates ofthe local linear model and confidence measures for the resemblance of the model to theactual data.

Figure 2.3a contains a two dimensional seismic image of a vertical cross-section of ananticline. In a two dimensional vertical cross-section, the planar structure of the reflectorsreduces to a line-like structure, and can therefore be analyzed with the 2D GST. We haveprocessed this image with the 2D GST (σg = 1, σT = 4) and the resulting estimates fororientation, gradient energy Eg, and confidence Can are shown in figure 2.3b-d. The GSTyields a smooth and continuous estimate of the slowly varying structural orientation dueto the curvature of the anticline. The strong reflectors are clearly highlighted in the energyestimate and the confidence estimate gives lower values if the reflectors are discontinuous.

2.2 Orientation adaptive filtering

Random measurement noise, which is present in every real world image, hampers manualinterpretation by human experts as well as automatic segmentation and analysis by com-puters. Therefore many image processing techniques are developed to reduce noise. Thesemethods are either based on spatial correlation or a spectral analysis. Additive uncorre-lated noise on signals that only contain low frequency components, can be reduced in astraight forward manner by low-pass filtering. If the spectrum of the noise-free image aswell as the spectrum of the noise are known, then the best linear filter to separate the twosignals is the Wiener filter [Wie49, Pra72]. These spectra are in general not available. Inthe absence of a priori knowledge, the noise is usually assumed to be uncorrelated.

Edges or transitions, which are imported for the analysis of images, often form lines orsurfaces that can be locally modeled as a linear structure. In this linear structure modelwe have local a priori knowledge of the signal, namely shift invariance along one or moreorientations. A linear filter such as the Wiener filter is not able to exploit this local in-formation. The reduction of noise in an oriented texture domain or along an individualcontour needs an anisotropic smoothing operator that adapts to the local orientation. Can-didate approaches are elongated steerable filters [Hag92] [Fre92], and anisotropic diffusion[Wei98].

For this thesis we used a very flexible adaptive filter model. The adaptive behavior isobtained by local transformation of the image data. The transformation is formalized as acoordinate transformation based on locally estimated parameters. The transformed data

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2.2 Orientation adaptive filtering 19

(a) (b)

(c) (d)

Figure 2.3: Analyzing a seismic image of a vertical cross-section of an anticline (a), using the2D GST. Resulting in estimates of local orientation (b), gradient energy (c), andconfidence or anisotropy (d).

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20 Chapter 2 Linear structures

is computed by interpolation of the image data. Many different control parameters can beincorporated by simply adjusting the coordinate transform. Since this adaptive mechanismdoes not depend on the filter type, it can be used for non-linear filters as well, e.g. medianor variance filters. Steering an elongated filter window requires a robust and continuousrepresentation of orientation. Haglund [Hag92] showed that the structure tensor is suitablefor controlling an orientation adaptive filter. Our orientation adaptive filter consists of anelongated Gaussian filter steered by the eigenvectors of the GST.

(a)

2 2

12

σσ

(b) (c)

Figure 2.4: (a) Creating an elongated filter by combining isotropic filters. (b) The parametersσ1 and σ2 of a 2D elongated filter. (c) Steering an elongated filter to match thelocal structure.

An elongated filter can be created efficiently by combing several isotropic Gaussian filterresponses, see figure 2.4. The responses are sampled at regular intervals along a straightline. If the intervals are smaller than two times the scale of the isotropic filter (∆ < 2σ),the resulting mask is essentially flat. A practical implementation of this filter consistsof two stages. First an isotropic Gaussian filter is applied by simple convolution, settingthe filter widths σ1 . . . σn equal to σ. This filter step can be interpreted as selecting theappropriate scale. A steered filter completes the task by elongating the filter along the shiftinvariant orientations. The steered filter is one-dimensional (σ1) for line-like structures andtwo-dimensional (σ1,σ2) for plane-like structures. When relevant information is present atthe finest scale, the first of stage of the filter may be omitted. This is only allowed ifthe image is band-limited and sampled according to the sampling theory [Sha49]. Thesampling or Nyguist theory requires that the the sampling frequency is higher than twotimes the maximum frequency of the analog signal fsample > 2fmax.

To demonstrate the capabilities of an orientation adaptive filter, we applied it to a 2Dimage of a circular oriented texture, shown in figure 2.5. The parameter settings we usedfor the GST are σg = 1, σT = 5, and for the steered filter σ1 = 4. For comparison we alsoapplied isotropic Gaussian filters at four different scales σ = {1, 2, 3, 4}.Due to the circular shape of the oriented texture in the test image, we can immediatelyverify from the result in figure 2.5c that the adaptive filter is rotation invariant. Further-more, a limitation of the filter comes to light in the high curvature area. The elongatedfilter is no longer able to match the shape of the local texture in this area, and the filtererror increases with increasing curvature.

An efficient way to implement steerable filters is proposed in [FA91, Per95]. The presentedtechnique allows a convolution based filter to be steered to an arbitrary orientation by

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2.3 Edge preserving filtering 21

(a) (b) (c)

Figure 2.5: A demonstration of the need for anisotropic smoothing filters to improve the SNRof a noisy image of an oriented texture (a). (b) Isotropic Gaussian filters appliedto (a) at four different scales. (c) The result of orientation adaptive filtering. Thesizes of the filter are indicated by white drawings.

writing it as a linear combination of a set of linear basis filters. The basic idea is to samplethe angular response of the filter and to interpolate between these samples. A draw-backof this method is that a narrow elongated filter (σ1 � σ2) needs a lot of basis filters, sinceit has a high angular resolution. Furthermore, this technique does not work for non-linearfilters, and the extension of this technique to curvature adaptive filtering is not trivial.

2.3 Edge preserving filtering

The goal of filtering is to get a better description of the local image properties by replacingthe values of the individual points by a function of their neighborhood. The influenceof random fluctuations in the image, for example, can be reduced by integration over aneighborhood. This integration assumes that the image properties of the individual pointsinside the neighborhood, are approximately constant. The assumption of constant imageproperties is not met when the neighborhood overlaps a border between two regions withdifferent properties. Estimation of the images properties near such a border will result ina weighted average of the properties from the two different regions. This should obviouslybe avoided. The traditional example of this problem is low-pass filtering of an image withconstant gray-value regions. Although it will reduce the noise, it will also blur sharp edges.A filter that avoids overlapping a border, is therefore said to be edge preserving.

There are two principal mechanisms for filter windows to avoid a border. One of them isto make the filter window smaller near a border, i.e. scale adaptive filtering. The border isinterpreted as a fine scale image feature and the scale of the filter should be adapted to thissmaller scale. The second mechanism avoids borders by allowing decentralized windowsthat are ”repelled” by the border. A schematic visualization of the two mechanisms avoid-ing an orientation border is given in figure 2.6. Both mechanisms require an estimation of

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22 Chapter 2 Linear structures

(a) (b) (c)

Figure 2.6: Two principal mechanisms to avoid a border. The behavior of a normal filter (a).Avoiding border by adapting the scale of the windows (b), and by using decentralizedwindows (c).

the border locations. The central issue is to transform this border estimate into the controlparameter scale or displacement. An example of a filter based on decentralized windowsthat solves this issue is the generalized Kuwahara filter [BVV99].

2.3.1 Generalized Kuwahara filtering

A traditional filter for edge preserving smoothing of images containing constant grey-valueregions, is the Kuwahara filter [Kea76]. Kuwahara divided a square symmetric neighbor-hood into four slightly overlapping windows, each containing the central pixel, see fig.2.7a.In each window the mean mi and the variance si of the intensity is computed. The edgepreserving estimate of the mean mep is obtained by replacing the central pixel by the meanvalue of that window that has the smallest intensity variance.

mep = mr , r = arg{mini

si} , i ∈ 1, 2, 3, 4 (2.18)

The mean operation (uniform filter) reduces the noise and the variance is used to selectthe most homogeneous window. The proposition on which the filter is based is that thevariance in a window that overlaps an edge s2

edge, is greater than the variance in a windowin a homogeneous area s2

homogenous,

s2

edge > s2

homogenous. (2.19)

The window with the smallest variance, therefore avoids edges. This filter has been furtherdeveloped in [NM79] by increasing the number of windows to eight and changing the shapeof the windows to pentagons and hexagons.

The Kuwahara filter is an implementation of an edge preserving smoothing filter thatuses decentralized windows. The crucial part of this class of filters is finding the optimaldisplacement. The generalized Kuwahara filter [BVV99] finds the optimal displacement

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2.3 Edge preserving filtering 23

1

3 4

2

(a) (b)

Figure 2.7: a) traditional Kuwahara filter, b) a realization of the generalized Kuwahara filterwith circular windows. The dashed lines bound the neighborhood and the solid linesthe windows.

by locating the highest confidence value. Since the confidence of the parameter estimationis lower near borders, the filter avoids them. By evaluating all decentralized windowsthat still contain the current pixel, the filter is made rotation invariant and maintainsconnectivity. Each decentralized window has a parameter and a confidence estimate (pi, ci),the parameter from the window with the highest confidence value yields the edge preservingestimate pep.

pep = pr , r = arg{maxi

ci} (2.20)

Making a parameter estimate edge-preserved using the generalized Kuwahara filter, there-fore, requires the estimation the parameter and a corresponding confidence value.

A realization of the generalized Kuwahara filter with round windows is depicted in figure2.7b. In principal, there are no restrictions to the shape or size of the windows, exceptthat they should both be fixed. Note that the shape of the window determines the shapeof the neighborhood. To demonstrate the generalized Kuwahara filter we applied it toan image with three grey-value domains. We added uncorrelated noise to a SNR of 3dB. The parameter to be estimated is the Gaussian mean and the confidence estimateminus the gradient magnitude (−|∇I|) is the confidence value, both computed at scaleσ = 5. As can be seen from the results in figure 2.8, the Gaussian filter blurs the edgesand the generalized Kuwahara filter preserves sharp edges. The sharp edges created by thegeneralized Kuwahara filter do not per definition coincide with the original edges of thenoise-free image, since the certainty measure, on which the new edge location is based, isestimated on the noisy image.

A mostly undesired side effect of the Kuwahara filter is a blemished result in regions withoutclear edges and the creation of false contours. These artifacts are due to the fact that theKuwahara filter always selects. In homogeneous regions, however, the differences betweenthe confidence values are due to noise, and the selection should not take place. A solutionto this problem is to control the selection process by a function of the confidence valuef(c). We used a linear mixing of the normal parameter estimate and the edge preservedversion

pnew = p(1 − f(c)) + pepf(c), 0 < f(c) < 1. (2.21)

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24 Chapter 2 Linear structures

(a) Input image (b) −|∇I | (c) Gauss

(d) Generalized Kuwahara (e) Controlled selection (f) PM Diffusion λ = 0.06

(g) PM Diffusion λ = 0.08 (h) PM Diffusion λ = 0.1

Figure 2.8: Reducing the noise in an image with grey-value domains at 3 dB. The Gaussianfilter (c) blurs the edges, but the Generalized Kuwahara filter (d) preserves them.Controlled selection using the confidence values (b) is shown in (e). For comparisonthe result of Perona and Malik diffusion [PM90] is shown in (f,g,h). 100 iterationsare used in all cases.

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2.3 Edge preserving filtering 25

The task of the function f is to suppress false border detections due to noise, and enhancetrue border detections. Clipping the confidence between the 5 and the 95 global percentilegives visually good results, see figure 2.8e. The lower clipping bound is based on theassumption that in 5 percent of the data, the contrast of the noise is larger than thecontrast of the signal.

Related methods

Another approach to finding the optimal displacement of decentralized windows, is pre-sented in [FS99]. In this paper Fischl and Schwartz split up the computation of thedisplacement vector field v(x) in three parts

v(x) = m(x)d(x)o(x), (2.22)

where m is the magnitude, d is the direction or sign, and o is the orientation of thedisplacement vectors. They state that the orientation should be perpendicular to theborder, and the direction should be such that the decentralized window moves away fromthe border without crossing it. The magnitude or amount of displacement should movethe window until it is inside a homogeneous region or until the influence of another borderbecomes dominant. They worked out this algorithm for the filtering of constant grey-valueregions. The use the orientation of the gradient and the direction is found by locating theposition of the maximum gradient magnitude. This yields their initial estimate vi of thedisplacement. The magnitude is found extending the initial vector along its direction untilthe inner product between the initial vector at ”head” and ”tail” becomes non-positive.

In the case of a straight symmetric border, the displacement found by this algorithm isidentical to the displacement of the generalized Kuwahara filter that uses the gradientmagnitude as the confidence value. The complexity of the generalized Kuwahara filteris lower, since it is a one step algorithm and it does not explicitly enforce a displacementperpendicular to the border. Fischl and Schwartz compared their displaced window methodto Perona and Malik diffusion [PM90]. Treating the image intensity as a conserved quantitybeing allowed to diffuse over time, allows image enhancement to be performed by solvingthe diffusion equation. Perona and Malik diffusion is a nonlinear version of diffusion thatis controlled by the geometry. The magnitude of intensity gradient determines the amountof diffusion that is allowed in a certain direction. The corresponding diffusion equation isgiven by

∂I

∂t= div(g(|∇I|2)∇I) (2.23)

with

g(s2) =1

1 + s2/λ2(λ > 0) (2.24)

For a visual comparison we have applied the Perona and Malik diffusion filter to the testimage in figure 2.8, for three values of the parameter λ (0.06,0.08,0.1). The Fischl and

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26 Chapter 2 Linear structures

Schwartz algorithm is slightly more complex than our generalized Kuwahara filter, but oneto two orders of magnitude faster than 50 iterations of the Perona and Malik diffusion.Fischl and Schwartz have also shown that their algorithm gives results that are visuallycomparable to the results of Perona and Malik.

2.3.2 Improving orientation estimation near borders

The structure analysis using the GST is based on the assumption that there is locallyone dominant orientation, i.e. a uni-modal representation. The orientation estimation ofthe GST fails when there is locally more than one orientation present, for example at theborder of two oriented textures. The resulting orientation estimation near that border isa weighted average of the two dominant orientations from both sides.

A powerful solution for allowing multiple orientations in one neighborhood in the image, isto add orientation as a new dimension [Wal87, CH89, GVV97]. This orientation dimensionneeds to be sampled by applying a bank of directionally selective filters. The number offilters needed depends on the angular resolution. The drawbacks of this approach are thehigher computational complexity and the demand for more computer memory, especiallyin 3D. Orientation in 2D can be represented by one angle, thus adding orientation as adimension f(x, y) → f(x, y, φ) adds one dimension. The representation of orientation in3D requires two angles. Adding orientation to a 3D image f(x, y, z) → f(x, y, z, φ, θ)amounts to a total of five dimensions.

An image with multiple oriented textures can be correctly filtered without a full multi-modal representation of the data, as long as the textures do not overlap. A correct orien-tation estimation near border can be obtained by applying the generalized Kuwahara filterto the output of the GST. We will now demonstrate this combination by example. Weapply the GST to the three domains image and the orientation and confidence estimatesare shown resp. in figure 2.9c and b. It can be verified from this result that the confidenceestimate is lower near the borders. Applying the generalized Kuwahara filter with theGST orientation as the parameter estimate and Can as the confidence value, causes theGST windows to avoid the borders. The resulting improved orientation estimate is shownin figure 2.9d. A comparison of this result with the true orientation, shows that at somelocations there is a small shift in the position of the border.

Limitations of the GST confidence as orientation border detector

The success of the method described above, highly depends on the ability of GST confidenceestimate to locate the borders. We will now analyze a neighborhood around an orientationborder point. First, the neighborhood is split up in two parts A and B, in such a way thatneither of the these parts contain the border, see figure 2.10. Then the structure tensor iscomputed on both parts, according to

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2.3 Edge preserving filtering 27

(a) (b)

(c) (d) (e)

Figure 2.9: The orientation (c) and confidence estimate (b) of the GST applied to the three-domains image (a) at 9 dB. An improved orientation estimation near borders usingthe generalized Kuwahara filter with (c) as parameter and (b) as confidence. Thetrue orientation (e) is given for comparison.

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28 Chapter 2 Linear structures

A B

(a) (b)

1ee1Be1A

a bφ φφ

(c)

Figure 2.10: Orientation estimation at a border point indicated by the small circle in (a). Theneighborhood around this point is divided into parts A and B (a). The combinationof the structure tensors from these parts is shown in (b). The tensors are depictedby the ellipses spanned by their eigenvectors, and the dashed line denotes thecombined tensor.

Ti =

x∈i

w(x)g(x)g(x)Tdx , i ∈ {A, B}. (2.25)

We assume for the confidence estimates Ci of these tensors

Ci =λ1i − λ2i

λ1i + λ2i≈ 1 , i ∈ {A, B}, (2.26)

i.e. the patterns in sub-neighborhoods A and B have approximately a one-dimensionalstructure. This means that A and B can be characterized by a single dominant orientationλ1e1, and the orientation border can be described by one angle φ.

φ = arccose1A · e1B

||e1A|| ||e1B||(2.27)

The ability of the GST confidence to detect the orientation border is determined by theangular dependence of the confidence estimate C(φ), of the combined tensor T = TA+TB.An expression for C(φ) can be found by analyzing the eigenvalues of T. The tensor T isbrought to its diagonal form D, by writing it in the basis β spanned by the eigenvectorsof T.

D = [T]β = [TA]β + [TB]β = Q−1(φa)DAQ(φa) + Q(φb)DBQ−1(φb) (2.28)

where Q is a rotation in the two-dimensional sub-space P spanned by {e1A, e1B}, andφ = φa + φb as shown in figure 2.10. If

λ2i = λ3i = · · · = λNi , i ∈ {A, B},we can chose e2A and e2B such that they are elements of sub-space P. This gives thefollowing equations

λ1 = λ1A cos2 φa + λ1B cos2 φb + λ2A sin2 φa + λ2B sin2 φb

λ2 = λ1A sin2 φa + λ1B sin2 φb + λ2A cos2 φa + λ2B cos2 φb

(2.29)

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2.3 Edge preserving filtering 29

We can use these eigenvalues to construct C(φa, φb).

C(φA, φb) =λ1 − λ2

λ1 + λ2

=(λ1A − λ2A) cos 2φa + (λ1B − λ2B) cos 2φb

λ1A + λ2A + λ1B + λ2B

= CA cos(2φa) + CB cos(2φb)

(2.30)

The angles (φa, φb) can be found by solving

0 = (λ1A − λ2A) sin(2φa) − (λ1B − λ2B) sin(2φb)

φ = φa + φb.(2.31)

In the case that

λ1A = λ1B ≡ λ′

1 , λ2A = λ2B ≡ λ′

2 → φa = φb =φ

2,

we get

C(φ) =λ′

1 − λ′

2

λ′

1 + λ′

2

cos(φ). (2.32)

The angle φ is difference between the dominant orientations of the oriented textures atboth sides of the border. As a consequence, at a border with a small angular difference,the confidence estimate C(φ) will only decrease by a few percent. For example, if φ = π

6,

then the confidence will only decrease 10%. In the presence of noise it will be difficult todetect such a low-angle border.

2.3.3 Edge preserving orientation adaptive filtering

Earlier we have shown that noise in oriented texture domains can be reduced by orientationadaptive filtering. This filter fails near borders between oriented textures. The previoussection shows that it is possible to improve the orientation estimate near borders. Usingthis improved orientation estimate, we are able to correctly steer an elongated windowover images with several touching oriented textures. Still, we have to make sure that theadaptive filter does not overlap borders, see figure 2.11a. This is in fact the same problemwe encountered during the orientation estimation. The difference is that now we can onlyallow displacement along the shift invariant orientations, as depicted in figure 2.11b.

A steered version of the generalized Kuwahara filter immediately suggests itself. Thegeneralized Kuwahara filter should have the same dimensionality as the steered filter. One-dimensional for line-like structures and two-dimensional for plane-like structures. However,an efficient estimation of the parameter and confidence value in the steered displacedwindows is not possible, since the orientation can be different between each point in theimage. To keep the computation cost low, we are forced to reduce the number of displacedwindows. By varying this number, we experimentally verified the trade-off between ”visual

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30 Chapter 2 Linear structures

(a) (b)

Figure 2.11: Reducing noise near a border between two oriented textures, a) orientation adap-tive filter, b) edge preserving orientation adaptive filter.

quality” and computation cost. With visual quality we mean the judgment of the humanvisual system. A reasonable compromise is to use 5 (1D) or 9 (2D) displaced windowsevenly spread over the neighborhood.

We have applied the steered Kuwahara filter to reduce the noise in the two-dimensionalthree-domains test image. The signal-to-noise ratio in the image is 9 dB, and the resultsare shown in figure 2.12d. In each displaced 21*1 pixel window, we measured the Gaussianweighted mean with σ = 5 as the parameter estimate and the variance as the confidencevalue. The filter is steered using the improved orientation estimate of the GST withσg = 1.4 and σT = 5. The result after controlling the selection process according to eq.2.21, is shown in figure 2.12e. For comparison we have also shown the result of the normalorientation adaptive filter of the same size, and steered with both the GST orientation andthe improved orientation estimate. From figure 2.12 is it clear that the steered Kuwaharafilter gives the best result near borders. The created false contours inside the domainsare reduced by controlling the selection of the Kuwahara filter on the basis of the GSTconfidence.

A significant speedup could be obtained by using the GST confidence as the confidencevalue and computing the optimal displacements using a steered version of generalizedKuwahara filter. Applying these displacements to the result of an orientation adaptivefilter leads to edge preservation, see figure 2.12f. However, since the confidence valuewas not computed in the same window as the parameter estimate, a small error in theorientation estimate could cause large errors in filter result. Another remark we shouldmake is that we composed the three-domains image using oriented textures of differentscales, to show that the steered filters are not very sensitive to the scale selection. Sincethe scale separation between the domains in this test image is so clear, a multi-scale analysiswould in this case certainly lead to better results.

For comparison we have applied the coherence enhancing diffusion filter as described byWeickert in [Wei98], to the three-domains test image as well. This filter is based on theanisotropic diffusion equation

∂I

∂t= div(D · ∇I). (2.33)

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2.3 Edge preserving filtering 31

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 2.12: Noise reduction using an orientation adaptive filter steered by the GST (b), andthe improved orientation estimate (c). The result of the steered Kuwahara filter(d), and a space-variant mix of (c) and (d) on the basis of GST confidence (e).The faster steered generalized Kuwahara based on the GST confidence and (c).Coherence enhancing diffusion with 5,10, and 20 iterations (g,h,i).

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32 Chapter 2 Linear structures

The diffusion is made coherence enhancing by basing the diffusion tensor D on the structuretensor according to

D = d1v1vT1 + d2v2v

T2 , (2.34)

with v1,v2 the eigenvectors of the structure tensor and

d1 = α (2.35){

d2 = α λ1 = λ2

d2 = (1 − α)e−C

(λ1−λ2)2m λ1 > λ2

(2.36)

with λ1, λ2 the eigenvalues of the structure tensor and α, C, m parameters of the filter.In figure 2.12g,h,i we have shown the results of this filter (α = 0.01, C = 0.001, m = 1)after 5,10 and 20 iterations. These result are less ‘edge preserving’ than the generalizedKuwahara results.

2.4 Application: Automatic fault detection

In this section we will apply the image processing techniques described in this chapter tothe automatic detection of faults. The first step in each automation process is to studythe way things are done manually. In the case of fault detection this means that we shouldsummarize the methods used by the interpreter. The manual detection of faults consist oftwo steps. First the fault location is found using visual cues for reflector discontinuities. Ifthere is enough support, the fault is inspected more closely to estimate the fault parameters.This is done by matching seismic sequences from both sides of the fault. We will limitourself to the automatic localization of faults.

(a) (b)

Figure 2.13: Two schematic cross-section of faults showing a different vertical displacement ofa seismic sequence. A normal fault (a), and a reverse fault (b). The faults aredepicted by thick solid lines. The solid arrows indicate the direction of the dis-placement along the fault, and the dashed arrows show the forces in the subsurfacerocks.

Faults are fractures in the subsurface rock caused by tectonic forces. These forces oftenlead to a vertical displacement (throw) between the rock formations on both sides of

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2.4 Application: Automatic fault detection 33

the fracture. In seismic images, faults are recognized by abrupt vertical displacement ofseismic sequences along some plane, i.e. the fault plane. Two schematic examples of theappearance of faults in seismic images are depicted in figure 2.13. In practice, it is rarethat the reflectivity is identical on either side of the fault. Therefore there can be reflectordiscontinuities visible in the data, even if there is no measurable throw.

The most important observation from an image processing point of view, is that faultscause discontinuities in the reflectors. Flat continuous reflectors can be well described asplane-like linear structures and the value of Cplane from eq.(2.17) will be close to 1. Near afault, there are also intensity (seismic response) changes perpendicular to the fault plane,causing a decrease in the value of Cplane. We have computed Cplane over a part (1283 voxels)of a 3D seismic image, that contains many faults, and the result is shown in figure 2.14b.From this result we see that the faults are only partly highlighted by Cplane. The faultis not highlighted at locations where there are no clear reflectors present nor at locationswhere a the reflector from one side of the fault is continued by another reflector at theother side.

In three dimensional data it is possible distinguish between random structures and faultsusing the GST. Since a fault plane is two dimensional, there is still one orientation withonly small intensity changes. Namely perpendicular to both the reflector and the faultplane. The value of Cline will therefore be close to 1 near faults. We have summarized theobservations in the table below:

Random/isotropic: λ1 ≈ λ2 ≈ λ3 Cplane ≈ 0 Cline ≈ 0Fault: λ1 ≈ λ2 � λ3 Cplane ≈ 0 Cline ≈ 1

We have combined Cplane and Cline to create the fault confidence Cfault, which given by

Cfault = (1 − Cplane)Cline =2λ2(λ2 − λ3)

(λ1 + λ2)(λ2 + λ3). (2.37)

This new fault confidence measure takes values between 0 and 1, and should give fewerfalse positives on random structures.

A very distinguishing feature of faults is that they have a large vertical extent. Until nowwe have computed Cplane using isotropic windows. Increasing the isotropic window sizeto capture the vertical extent, will not yield a better signal to noise ratio1. Since a faultis a 2D signal inside a 3D window, the amount of noise inside the window grows morerapidly than the amount of signal. By elongating the tensor windows vertically along thefaults, we increase the SNR and we exploit the distinguishing vertical extent. Because weonly know the orientation of the reflector planes, we steer the elongated tensor windowsperpendicular to these planes. This is in many cases a reasonable approximation of theorientation of the fault. We computed Cfault on the 3D seismic image and the results areshown in figure 2.14c. This new method gives a more continuous fault highlighting withless false positives.

1 Here we define the faults as the signal and everything else as the noise.

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34 Chapter 2 Linear structures

(a) xt-slice (b) Cplane

(c) Cfault (d) fault detection

Figure 2.14: Detecting faults by measuring reflector discontinuity in a 3D seismic image. Axt-slice of this image is shown in (a). The result of the GST confidence measureCplane, computed using isotropic windows, is shown in (b) and the measure Cfault

computed using elongated tensor windows is shown in (c). The fault detectionusing based on (c) is shown in (d).

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2.4 Application: Automatic fault detection 35

(a) xy-slice

(b) yt-slice

Figure 2.15: Results of the fault detection algorithm on a 3D seismic image. The fault detectionsare indicated by black pixels. (a) A time slice (xy), (b) a yt-slice.

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36 Chapter 2 Linear structures

Figure 2.16: Result of the fault detection algorithm on a 3D seismic image. The images showa xt-slice of this image, with the fault detections indicated by black pixels.

The final task of our fault detection algorithm is to segment the seismic image into fault andnon-fault. This segmentation is based on Cfault and consists of two steps. First we applya non-maximum suppression. The candidate fault points are limited to the local maximaof Cfault, measured in one-dimensional windows steered perpendicular to the fault-planes.This perpendicular orientation is the maximum gradient orientation in the Cfault image.Next, we take the Cfault value of these local maxima into account using two thresholdvalues. The reduced Cfault image is segmented using a low threshold value Tl. The finalresult is obtained by accepting only those connected segments whose maximum Cfault valueis above a high threshold value Th.

The results of our automatic fault detection algorithm are show in figure 2.14d, 2.15 and2.16. For all the results we have used the threshold values Tl = 0.1 and Th = 0.4. Thesegmentation is visualized by making the fault points black in the seismic image. To allowa manual verification of the result, we have depicted the original images as well.

Now that the fault locations have been found, the estimation of fault parameters could beperformed as a subsequent processing step. One could for example perform a matchingpursuit between the seismic sequences at both sides of the fault to estimate the throw.This information could then be used to determine the extent of the fault, e.g. by splittingor merging of detected fault planes.

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2.5 Application: Structure enhancement 37

2.5 Application: Structure enhancement

A display of a subsurface layer with the same viewpoint as one has when watching theearth’s surface from an airplane, is called a map view. Map views have been used by seismicinterpreters for many years, and have become an essential tool for the inspection of lateralchanges in the geological structures. Horizons, the earth’s surface in previous geologicaltimes, form the natural bases for map views. The manual construction of horizons isa very time consuming task and is therefore automated by horizon-tracking algorithms.Ideally, a horizon-tracker is initiated by picking a single point on the target reflector, andit automatically finds all points in the image that belong to that same reflector. In todayspractice, the interpreter has to provide the algorithm with many start points to get anaccurate result.

Small deviations from the layered structure are often the cause of tracking errors. Thesedeviations can be the result of noise, pre-processing artifacts or subtle geological featuresthat are not important for the structural analysis. Enhancing the structural informationby suppressing these small deviations could greatly improve the performance of horizon-trackers. Reflectors can be locally described as planar linear structures using the structuretensor. Choosing a large tensor scale σT , makes this description insensitive to small devi-ations. Reflector surfaces are deformed by many geological processes, e.g tectonic forces,intrusive salt, or erosion. The tensor scale should therefore be chosen small enough, notto notice the deviations from the linear model due to curvature in the global structure,think of synclines and anticlines. In practice, the smallest scale should be coupled to thelowest frequency of seismic wavelet. A rough estimation can be made assuming a low signalfrequency of 10 Hz, corresponding to a time window of 100 ms to contain one cycle. If thethe wave velocity is 2000 m/s, then the window has a length 200 m. A time sampling of 4ms gives a window of 25 pixels corresponding to σT = 4.

Having estimated the local orientation of the structure using the GST, we can remove thesmall deviations by orientation adaptive filtering. A steered disc shaped filter (σ1 = σ2 �σ3) enhances the planar structure. We applied this intrinsically 2D orientation adaptivefilter to a 3D seismic image and the result is shown in figure 2.17b. The scale of the filterwas σ1 = σ2 = 2 and it was steered parallel to the reflectors using the first eigenvector ofthe GST (σg = 1, σT = 6). The adaptive filter has clearly reduced the deviations, and thereflectors are now much easier to follow. The filter has also smoothed the reflectors acrossthe faults, possible merging different reflectors. This should avoided and the filter shouldbe made edge preserving, or more accurately fault preserving.

Faults can be preserved using the generalized Kuwahara filter combined with the faultconfidence Cfault from the previous section. We applied a one-dimensional version of thegeneralized Kuwahara filter steered perpendicular to the fault-plane using the second eigen-vector of the GST. We used the disc shaped filter as the parameter estimate and Cfault asthe confidence value. The result is shown in figure 2.17c. We removed the false contourby controlling the selection process using eq.2.21. The final result is shown in figure 2.17d

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38 Chapter 2 Linear structures

and figure 2.18 for orthogonal slices through the data. These slices are same as the onesshown in the fault detection application.

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2.5 Application: Structure enhancement 39

(a) xt-slice (b)

(c) (d)

Figure 2.17: The result of the adaptive filter controlled by the improved orientation estimateapplied to a seismic image containing a lot of faults (a) is shown in (b). The faultpreserving version of (b) is shown in (c). (d) shows a spatially variant mix of (b)and (c) based on the GST confidence.

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40 Chapter 2 Linear structures

(a) xy-slice

(b) yt-slice

Figure 2.18: Results of the structure enhancement on a 3D seismic image. (a) A time slice (xy),(b) a yt-slice.

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3. Curvilinear structures

The GST model presented in chapter 2, describes images locally as a linear structure andyields estimates for orientation and confidence. The previous chapter shows that the GSThas a broad range of applications. We have shown that the orientation estimate is stableand suitable for controlling subsequent processing steps. The confidence estimate decreasesat image locations where the neighborhood deviates from the linear model. An exampleof such a neighborhood is the area around a border between two oriented textures. Theconfidence estimate can therefore be used to these highlight borders. Bending a straightstructure also causes a decrease of the confidence value, because the local structure is nolonger shift invariant. We have limited the influence of curvature in the previous chapter,by limiting the size of the analysis window, i.e. the tensor scale σT . However, the noiselevel in the image determines the minimum scale needed to obtain a confident descriptionof a neighborhood. In the case of a curved structure with a high curvature or a high noiselevel, the linear model does not suffice.

Verbeek et al. presented in [VvVvdW98, WVVG01] a curvature corrected confidenceestimate for 2D curvilinear structures by extending the structure tensor. A coordinatetransform is applied to locally transform the image in such a way that the rotationalinvariance of the local structure, becomes a translational one. Next, the structure tensoris applied in the new coordinates and transformed back to the original coordinates. Thisyields expressions for the eigenvalues of the tensor with the curvature κ as a free parameter.Maximization of the confidence measure yields the transformation that ‘closest resembles’the local structure. From the parameter of the optimal transformation κopt, an estimateof local curvature is obtained. We will extend this method to the curvature correctedconfidence estimation of 3D curvilinear structures.

In three dimensions there are two types of linear structures, namely plane-like and line-like. These structures are generalizations of an individual plane and an individual line.The curvilinear counterparts of planes and lines are surfaces and curves. Evidently, thereare also two types of curvilinear structures in 3D. Plane-like curvilinear structures arethe generalization of surfaces, and line-like curvilinear structures are the generalization ofindividual curves. A plane-like linear structure was defined in the previous chapter as shiftinvariance along a plane. A stack of straight isophote surfaces therefore has a plane-likelinear structure. A stack of curved isophote surfaces, on the other hand, has a plane-likecurvilinear structure. To be able to describe plane-like curvilinear structures using theGST, the gradient of the isophote surfaces should be continuous, since the GST analysis is

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42 Chapter 3 Curvilinear structures

based on differential geometry. In this chapter we will extend the local straight model ofthe GST to a local quadratic surface model. The curvature corrected confidence estimatefor 2D curvilinear structures will be derived as a special case of quadratic surfaces. Thecurvature corrected description of line-like curvilinear structures differs significantly andwill therefore be presented separately in the next chapter.

3.1 The GST for 3D plane-like curvilinear structures

3.1.1 The quadratic surface approximation

In the previous chapter we modeled a local neighborhood of an arbitrary surface S(x) = 0as a plane. This modeling can be interpreted as a first order polynomial approximation ofthis surface around a point P on this surface

S(x) ≈ b · x + c = 0, (3.1)

with x the local neighborhood coordinate with P as its origin, and b the unit normalvector (‖b‖ = 1) of the plane. Apart from translation, a plane can be described by twoangles. In this chapter we will extend this description with two principal curvatures. Thiscan be written as the addition of a quadratic term, making it a second order polynomialapproximation

S(x) ≈ xt ·A · x + b · x + c = 0, (3.2)

with A a symmetric matrix with two non-zero eigenvalues [FIS89]. In essence we limitourselves to the quadratic surfaces that, after rotation and translation, can be written as

S(x) ≈ λ1s2

1 + λ2s2

2 − s3 = 0, (3.3)

with s1, s2, s3 coordinates in the frame spanned by the eigenvectors of A.

Now, we want to determine the relation between the two eigenvalues of A and the twoprincipal curvatures (κ1, κ2) of the surface S at point P (s1 = s2 = s3 = 0). The principalcurvature κi (i ∈ 1, 2) can be found by computing the curvature of S(si) = λis

2i using the

definition [BSMM00]

κ =f ′′

[1 + (f ′)2]3/2, f ′ =

df(x)

dx(3.4)

This yields κi = 2λi, and we can write

S(x) ≈ 1

2κ1s

2

1 +1

2κ2s

2

2 − s3 = 0. (3.5)

Thus, the quadratic surface is either an elliptic paraboloid in case the signs of κ1, κ2 areequal, or a hyperbolic paraboloid (saddle), if κ1, κ2 have opposite sign.

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3.1 The GST for 3D plane-like curvilinear structures 43

We define a quadratic surface structure as shift invariant along a quadratic surface andnot shift invariant along the normal on each point of this surface. A quadratic surface canbe described by two curvatures (κ1, κ2) and a local frame. This frame is oriented alongthe normal vector and the two vectors that correspond to principal curvatures, i.e. theframe for which A is diagonal. Finding the optimal quadratic surface model thereforerequires the ‘simultaneous optimization of five parameters, namely κ1, κ2, and three anglesto determine the local frame.

A significant speedup of the optimization problem can be obtained by utilizing the localframe Q given by the eigenstructure of the GST.

[T]uvw = QT ·

f 2x fxfy fxfz

fxfy f 2y fyfz

fxfz fyfz f 2z

· Q =

f 2u 0 0

0 f 2v 0

0 0 f 2w

(3.6)

The definition of Q is given in appendix B of this chapter. In eq.(3.6), we labeled thegauge-coordinates of the GST: ‘uvw’. The coordinate u corresponds to dominant gradientorientation (the normal) and v, w to the directions of respectively κ1, κ2, with κ1 > κ2.This uvw-frame is schematically depicted in figure 3.1a. Without curvature, f 2

v = λ2 andf 2

w = λ3 are determined by noise. Increasing curvature (κ1, κ2) causes an increase of theeigenvalues (resp. λ2, λ3). Utilizing the gauge-coordinates of the GST, the optimization ofthe model reduces to optimization of two independent parameters κ1 and κ2.

u

wv

(a) uvw

u’

v’ w’

(b) u′v′w′

Figure 3.1: Deforming quadratic surfaces to planes using a coordinate transform.

3.1.2 The quadratic GST for 3D surfaces

The first step toward a curvature corrected GST, is to define the coordinate transformuvw → u′v′w′ that deforms a curvilinear surface into a plane, see fig 3.1. Using thequadratic surface approximation, the transform and its inverse are given by

u′ = u − 1

2κ1v

2 − 1

2κ2w

2

v′ = vw′ = w

u = u′ + 1

2κ1v

′2 + 1

2κ2w

′2

v = v′

w = w′

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44 Chapter 3 Curvilinear structures

This approximation is valid for small values of u, v or small values of κ1, κ2. The testresults will show the practical limitations of this approximation.

The inverse-transformation is used to express the derivatives in the u′v′w′-coordinates asa function of the derivatives in the uvw-coordinates.

fu′ = uu′fu + vu′fv + wu′fw = fu

fv′ = uv′fu + vv′fv + wv′fw = κ1vfu + fv

fw′ = uw′fu + vw′fv + ww′fw = κ2wfu + fw

(3.7)

We now consider curved surfaces in the uvw-space with as shown in figure 3.1a. By applyingthe traditional GST method to the u′v′w′-space for arbitrary κ1, κ2, we get the gradientstructure tensor for quadratic surfaces

TQS =

f 2u′ fu′fv′ fu′fw′

fu′fv′ f 2v′ fv′fw′

fu′fw′ fv′fw′ f 2w′

(3.8)

Using eq.(3.7) we can express TQS in the uvw-coordinates.

f 2u′ = f 2

u

f 2v′ = f 2

v + κ2

1a + 2κ1b

f 2w′ = f 2

w + κ2

2c + 2κ2d

fu′fw′ = fv′fw′ = fu′fv′ = 0

(3.9)

with the abbreviations

a ≡ v2f 2u , b ≡ vfufv

c ≡ w2f 2u , d ≡ wfufw.

(3.10)

The off-diagonal elements of TQS are all equal to zero, due to the symmetries in the model.The derivation and the details of the symmetry considerations are given in appendix A.Since the matrix of [TQS]uvw is diagonal, the eigenvalues of TQS are equal to the diagonalelements of this matrix

λ1 = f 2u , λ2(κ1) = f 2

v + κ2

1a + 2κ1b , λ3(κ2) = f 2w + κ2

2c + 2κ2d. (3.11)

For a plane-like linear structure the eigenvalues relate as λ1 � λ2 ≈ λ3. The optimalresemblance to the quadratic surface, can therefore be found by minimizing λ2 and λ3,with respect to κ1, κ2. The curvatures κ1,min, κ2,min that minimize λ2,λ3 yield an estimateof local curvature and the are given by

κ1 = κ1,min =−b

a, κ2 = κ2,min =

−d

c. (3.12)

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3.1 The GST for 3D plane-like curvilinear structures 45

Substitution of these curvatures in eq.(3.11), yields the following expressions

λ1 = f 2u , λ2 = f 2

v − b2

a, λ3 = f 2

w − d2

c. (3.13)

These curvature corrected eigenvalues can be combined to create a curvature correctedconfidence measure. Since λ2, λ3 of the GST are determined by the curvature and thenoise level, it is not guaranteed that λ2 > λ3, after removing the influence of curvature.This should be checked explicitly, to avoid an overestimation of the confidence value

C =λ1 − λ2

λ1 + λ2

. (3.14)

Note that for the limit κ1, κ2 → 0, the equations for λ1, λ2 reduce to those of the traditionalGST.

The computation of the structure tensor for quadratic surfaces TQS, consist of two steps.First, the GST is computed to obtain the gauge-coordinates uvw, and next the extra terms(a, b, c, d) of eq.(3.10), need to be computed. The direct computation of (a, b, c, d) resultsin a spatially variant operation, since the gauge-coordinates change over the image. Forlarge filter sizes, this can lead to a high computational demand. Appendix B shows howthese terms can be computed efficiently using linear combinations of convolutions.

3.1.3 Experimental tests and results

The structure tensor for quadratic surfaces TQS, described above models a local neigh-borhood as a linear structure after a quadratic transformation. The parameters of theoptimal transformation κ1, κ2 yield an estimate of the local curvature. The resemblancebetween the local model and the image data is given by the confidence estimate C. Thevalue of the confidence estimate can be decreased by two factors; noise and a deviation ofthe local structure from the supposed model. In this section we will experimentally testthe structure tensor for quadratic surfaces TQS, by means of several measurements on testimages.

Quadratic structure

We will start with a test of the correctness of the theory and its implementation using the‘ideal’ test image It1 given by

It1(x, y, z; κ1, κ2) = I ′

t1(u, v, w; κ1, κ2) = u − 1

2κ1v

2 − 1

2κ2w

2, (3.15)

where the transformation of the xyz to the uvw-coordinates is given by an arbitrary ro-tation. The neighborhood around the origin (x, y, z) = (0, 0, 0) of the test image has a

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46 Chapter 3 Curvilinear structures

quadratic structure. The tensor TQS should therefore give an exact description of theimage structure at the origin. We will experimentally verify this proposition in two steps.

First we verify that the first eigenvector of the structure tensor is normal to the shiftinvariant surfaces and that the second eigenvector is aligned with the orientation of maximalcurvature in the tangent plane. At the origin of the test image It1 this means that the firsteigenvector should be parallel to the u-axis. The second eigenvector should be parallel tothe v-axis if κ1 > κ2 and parallel to the w-axis if κ2 > κ1. We found that the eigenvectorsof the GST are exactly parallel to the uvw-axes for all rotations and curvatures1. Thereis however a value of the curvature κi > κmax, where first eigenvector of the GST swapswith the second eigenvector. The value of κmax depends on the tensor scale σT , and weexperimentally verified that

κmax(σT ) =1

σT. (3.16)

For example, we found the maximum curvature κmax = 0.2 for σT = 5 and κmax = 0.5 forσT = 2.

Next we applied TQS to the test image It1 to verify the exactness of the description at theorigin. We found for the curvature and confidence estimates that

C = 1 , κi = κi , i ∈ {1, 2}, (3.17)

within the precision of the measurement; 10−7 of 32-bit floating point numbers. Thereforewe have shown that theory and the implementation of TQS are correct.

Spherical structure

To test the robustness with respect to noise and with respect to deviations of the structurefrom the local model, we use a second test image of noisy concentric spherical surfacesdefined by

It2(x, y, z; p, σn) = It2(r; p, σn) = cos

(

pr

)

+ N(0, σ2

n) , r =√

x2 + y2 + z2, (3.18)

where σn is the standard deviation of the normally distributed noise N . First we appliedTQS with (σg = 1, σT = 5) to the test image It2(r; 8, 0). This noise-free version allowsus to determine the estimation errors due to the deviation of the spherical structure fromthe quadratic model. We measured the resulting estimates for curvature and confidenceaccording to eqs.(3.12) and (3.14) as a function of r, and the results are shown in figure 3.3aand b. In figure 3.3a we compared the confidence estimate (QGST) with the confidenceestimate of the structure tensor for linear structures (GST). We have also plotted in thisfigure the improvement in the confidence estimate

∆C = (C)QGST − (C)GST , (3.19)1 All curvatures for which the corresponding surfaces can be properly sampled.

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3.1 The GST for 3D plane-like curvilinear structures 47

due to the curvature correction. For the evaluation of the curvature estimate we haveplotted the ‘true κ’ in figure 3.3a. This true curvature is the curvature of the surface of asphere along some arbitrary tangent, and is given by

κ =1

r, (3.20)

with r the radius. The estimation error is therefore the difference between κ and theestimated value. When we analyze the estimation error in figure 3.3a, we can distinguishtwo parts. For the first part (r > 15), the error increases slowly for decreasing r from10−5 up to 10−3 when r approaches 15. In the second part (r < 15), the estimation errorincreases rapidly for decreasing r and becomes very larger. The confidence estimate showsapproximately the same behavior; close to 1 for r > 15, and quickly decreasing to 0 forr < 15. We have checked this behavior for several values of σT , and we found that thisrapid decrease in confidence occurs for r < 3σT .

(a) r = 0 (b) r = 15 (c) r = 31

Figure 3.2: Three Gaussian windowed (σ=5, 313 pixels) neighborhoods of the noise-free testimage around points at r=0,15,31. The corresponding confidence estimates usingthe quadratic model are C=0.00, 0.96, 1.00.

To give an impression of the underlying data, we selected three neighborhoods of 313 pixelsfrom the noise-free test image It2(r; 8, 0) around points at r=0,15,31. The neighborhoodsare multiplied with a Gaussian window (σ = 5) to show the input-data for the local analysiswith the quadratic structure tensor TQS. A two-dimensional cross-section of each of theseneighborhoods, is shown in figure 3.2. At r=0 the neighborhood consists of concentricspheres, that constitute an isotropic structure and the confidence value is C=0. At r=15the center of the spheres is at the edge of the neighborhood and the structure consists ofhalf spheres. The quadratic model is able to describe this neighborhood with a confidenceC=0.96. The value of the confidence estimate of the third neighborhood is C=1.00.

The deviations of the local structures in It2 from the quadratic model, consist of two parts.The exact description of a spherical surface requires a higher order polynomial. Second,the quadratic model assumes a constant curvature along the u-coordinate. The u-axisis spanned by the first eigenvector of the GST. If we describe the spherical test imagewith spherical coordinates It2(r, φ, θ), than the normal vector at each point in the imageis parallel to the r-axis. The u-axis is parallel to the r-axis, since the first eigenvector ofthe GST is parallel to the normal vector Using eq.(3.20) we can write the curvature as a

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48 Chapter 3 Curvilinear structures

0 20 40 60 80 100Radius

0

0.2

0.4

0.6

0.8

1

Con

fiden

ce

QGSTGSTQGST−GST

(a) Confidence, noise-free

0 20 40 60 80 100Radius

0

0.02

0.04

0.06

0.08

0.1

0.12

Cur

vatu

re

True κκ1

(b) Curvature, noise-free

15 20 25 30 35 40 45 50Radius

0.5

0.6

0.7

0.8

0.9

1

Con

fiden

ce

QGSTGST

10 dB

3 dB

20 dB

(c) Confidence, noise dependence

Figure 3.3: Experimental results of estimation of curvature and confidence (σT = 5) on a noise-free spherical structure using the quadratic model (a,b). Experimental results ofthe test of robustness with respect to noise (3,10,20 dB) of the confidence estimate(c). The measurements in (c) are averaged over 20 different noise realizations, andthe error-bars denote the corresponding standard deviation.

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3.1 The GST for 3D plane-like curvilinear structures 49

15 20 25 30 35 40 45 50Radius

0.01

0.02

0.03

0.04

0.05

0.06

0.07

κ 1

True κ20 dB10 dB3dB

(a) Curvature κ1

15 20 25 30 35 40 45 50Radius

0.01

0.02

0.03

0.04

0.05

0.06

0.07

κ 2

True κ20 dB10 dB3dB

(b) Curvature κ2

Figure 3.4: Experimental results of the test of the robustness with respect to noise of the curva-ture estimates κ1 (a) and κ2 (b). The measurements are performed at scale (σT = 5)for 3 noise levels, 3,10,20 dB. The measurements are averaged over 20 different noiserealizations and the error-bars denote the corresponding standard deviation.

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50 Chapter 3 Curvilinear structures

function of u

κ(|u|) =1

|u|. (3.21)

The curvature is clearly not constant along the u-coordinate. Since the GST is a quadraticform, the sign of the eigenvectors is not defined by the image structure, and thereforewe have written the curvature as a function of the absolute value of u. For use of theeigenvectors in subsequent processing steps, their signs are chosen in a consistent manner.

To explain why the curvature estimate approaches zero when the analysis window ap-proaches the center of the test image, we consider the neighborhood (r = 0) in figure 3.2a.One half of this neighborhood (u > 0) contributes to a positive curvature and the otherhalf (u < 0) to a negative curvature, resulting in a curvature estimate κ1,2 = 0. For radii rsmaller than the effective radius of the analysis window, approximately 3σ for a Gaussianwindow, the structure of spherical surfaces can no longer be straightened by a coordinatetransform of the form

u′ = u − F (v, w) , v′ = v , w′ = w, (3.22)

with F (v, w) and arbitrary function. Even the optimally transformed structure becomesisotropic as r → 0.

Next, we tested the robustness with respect to noise, by applying the tensor TQS at aconstant scale (σg = 1, σT = 5) to the test image It2(r; 8, σn) at three different noiselevels σ2

n = {0.04, 0.4, 2}, corresponding to the SNR values 20,10,3 dB. Again we measuredthe curvature and confidence estimate as a function of the radius (15 < r < 50). Themeasurements are averaged over 20 different noise realizations, and the results are shownin figure 3.3b and 3.4. The error-bars in these figures denote the corresponding standarddeviation.

From figure 3.3b it is clear that the confidence value decreases with an increasing noise level,for both the quadratic (QGST) and the linear (GST) model. If we consider a neighborhoodwhere f = p + n, with p the pattern and n the noise, then we get the following noisedependency of the GST confidence

(C(n))GST =f 2

u − f 2v

f 2u + f 2

v

=p2

u − p2v

p2u + p2

v + 2n′2, (3.23)

with n2u = n2

v = n′2 and punu = pvnv = 0. It is clear that as the standard deviation of thenoise becomes larger the confidence measure becomes smaller. The noise dependency ofconfidence estimate of the QGST can be found using eq.(3.13).

(C(n))QGST =p2

u − p2v + K(n)

p2u + p2

v + 2n′2 − K(n)(3.24)

with

K(n) =vfufv

2

v2f 2v

=v(pupv + punv + pvnu + nunv)

2

v2(p2v + 2pvnv + n2

v)=

vpupv2

v2p2v + v2 n2

v

. (3.25)

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3.2 The GST for 2D curvilinear structures 51

Equation (3.25) shows that K(n) decreases for an increasing noise level. Therefore it is clearfrom eq.(3.24) that (C(n))QGST decreases if n2 increases. Furthermore the improvementof the confidence estimate due to the curvature correction ∆C, decreases as well. This isconsistent with the results shown in figure 3.3b.

The experimental results in figure 3.4 show that the magnitude of the curvature estimatebecomes smaller with an increasing noise level. This under-estimation of the curvature canbe explained in a similar fashion as the decrease in the confidence estimate due to noise.If we substitute f = p + n in eq.(3.12), we obtain

κ1(n) =vfufv

v2f 2v

=vpupv

v2p2v + v2 n2

v

, κ2(n) =wfufv

w2f 2v

=wpupv

w2p2v + w2 n2

v

, (3.26)

and we see that both κ1(n) and κ2(n) decrease if n2 increases.

Discussion

Summarizing the results, we can say that the estimators for curvature and confidencegive error-free results on the ‘ideal’ test image, and we can therefore conclude that thetheory and its implementation are correct. By applying the quadratic structure tensor toa second test image that has a structure of spherical surfaces, we showed that deviationsfrom the quadratic model only result in significant errors, if the radius of curvature issmaller than the effective radius of the analysis window. We experimentally determinedthe noise dependency of the estimators, and found that both the confidence estimate andthe improvement in the confidence estimate due to the curvature correction, decrease foran increasing noise level. For signal-to-noise ratios smaller than 10 dB, the curvatureestimates becomes significantly biased, i.e. they give an under-estimation.

3.2 The GST for 2D curvilinear structures

The two-dimensional curvature corrected GST can be derived from the structure tensorfor quadratic surfaces TQS, in a straight forward manner by considering a 2D sub-sectionof the uvw-space. Convenient sub-sections for the derivation of the 2D estimators arethe uv-plane or the uw-plane. If we consider the uv-plane, then the transformation thatstraightens the curved structure in this plane is the coordinate transform of quadraticsurface model with κ2 = 0, w = 0. The inverse transformation can be derived similarly,since κ1 and κ2 of the quadratic surface model are independent.

u′ = u − 1

2κ1v

2

v′ = vu = u′ + 1

2κ1v

′2

v = v′

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52 Chapter 3 Curvilinear structures

Applying the traditional GST to the u′v′-space, yields the GST for 2D curvilinear structures

Tcurv2D =

[

f 2u′ fu′fv′

fu′fv′ f 2v′

]

. (3.27)

Expressions for the tensor element in the uv-space are found using the inverse transforma-tion and they are given in eq.(3.9). The optimal resemblance between the model and thelocal image structure is found by maximizing the confidence measure

C =λ1 − λ2

λ1 + λ2

, (3.28)

and thus by minimizing λ2(κ1) with respect to κ1. This minimization yields the followingresults

λ1 = f 2u , λ2 = f 2

v − b2

a, κ1 = κ1,min =

−b

a(3.29)

with the abbreviationsa ≡ v2f 2

u , b ≡ vfufv (3.30)

Applying the GST to the u′w′-plane gives the same results, replacing κ1 ↔ κ2 and λ2 ↔λ3. The direct derivation of the curvature corrected structure tensor for 2D curvilinearstructure including test results and an application to finger-print recognition is presentedin [VvVvdW98]. The implementation shown in appendix B is also applicable to the twodimensional terms (a, b).

In [WVVG01], Weijer et al. derive two-dimensional curvature corrected confidence mea-sures by applying normalized curvilinear models to the gradient vector field. This allowsfor the quantitative comparison of different models, i.e. coordinate transformations. Theycompared the quadratic model (second order polynomial) with the circular model, andshow that the latter gives better results on circular oriented textures for small radii ofcurvature. They also present an application to interference patterns of a vibrating plateand to images of growth rings of trees.

3.3 Curvature adaptive filtering

In the previous chapter, we showed that noise in oriented texture domains can be reducedby anisotropic low-pass filtering along the shift invariant orientations, i.e. by orientationadaptive filtering (OAF). The motivation for using adaptive filter windows is that increasingthe window size isotropically does not yield a higher SNR, but increasing the size of anadaptive window that matches the local structure does. We have already encountereda limitation of elongated OAF windows. They are not able to match a curved orientedtexture, causing an increasing filter error with increasing curvature. Using the parameterestimates of the curvature corrected structure tensor, we can make the filter windows both

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3.3 Curvature adaptive filtering 53

curvature and orientation adaptive (COAF). In this section we present a two-dimensionalcurvature adaptive filter, and its application to noise reduction in curved oriented textures.Although we present the filtering method in 2D, we have applied it in 3D as well.

The COA filter windows can be created in the same way as we created the OA filter windowsin the previous chapter. The appropriate scale σ is selected by applying an isotropic filter,and the filter is elongated by sampling the isotropic responses at regular intervals alonga curved line, see figure 3.5. This can be implemented by locally transforming the datausing the quadratic coordinate transform of Tcurv2D from the xy to the u′v′-coordinates,and sampling of the filter along the u′-axis. The weighted average of the samples yieldsthe adaptive filter result. Below we will use a Gaussian weighting with a width σ1.

(a) (b)

Figure 3.5: (a) Creating a curved elongated filter by combining isotropic filters. (b) Adaptingthe curved elongated filter to match the local structure.

To demonstrate the COA filter, we will use it to reduce the noise in a test image It of acurved oriented texture.

It(x, y; p, σn) = It(r) = cos

(

2πr

p

)

+ N(0, σ2

n) , r =√

x2 + y2, (3.31)

where N is normally distributed noise. A realization of It(x, y; p, σn) of 128*128 pixels with(p = 8, σ2

n = 1/2) is shown in figure 3.6a. The first step in reducing the noise in curvedoriented textures is to apply the tensor Tcurv2D. The resulting parameter estimates onIt at scale (σg = 1, σT = 5), are shown in figures 3.6d-f. Since the tensor is based ona quadratic gradient model, the model error becomes large on circular structures with aradius R ≤ 3σT . This is reflected by the lower confidence values in the center of figure3.6f. The sign of the curvature estimate in figure 3.6e is defined relative to the gauge-coordinates uv of the structure tensor, and is therefore only meaningful in combinationwith the orientation estimate. Note that the sign of the curvature estimate changes at thesame locations as the ‘jumps’ in the orientation estimate occur. The improvement in theconfidence estimate ∆C given by

∆C = (C)curv2D − (C)GST , (3.32)

is shown in figure 3.6g. From this result it is clear that the improvement has the highestvalues for small radii. The maximum improvement in confidence for the noise-free case,occurs at radius R = 8.

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54 Chapter 3 Curvilinear structures

(a) Test image (b) COAF (c) OAF

(d) Orientation (e) Curvature (f) Confidence

(g) ∆C

Figure 3.6: Using adaptive filters (b),(c) to improve the SNR of curved oriented textures (a).The parameter estimates derived from Tcurv2D are shown in (d)-(f), and the im-provement in the confidence estimate by curvature correction ∆C is shown in (g).

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3.4 Appendix A: Symmetry 55

Using the curvature and the orientation estimate of the tensor Tcurv2D, we can control aCOA filter. The result of a Gaussian COA filter with σ1 = 5 is shown in figure 3.6b. Forvisual comparison, we have shown the result of a Gaussian OA filter (σ1 = 5) in figure 3.6c.A quantitative comparison of the filter results can be made by measuring the signal-to-noiseratios of the filtered test images. The noise in the test image is white and additive, andtherefore we estimate the standard deviation of the noise σn of a filtered image filter(It)by

σn = STDDEV[filter(It(x, y; p, σn)) − It(x, y; p, 0)]. (3.33)

Since the intensities of the noise-free image are bounded between -1 and 1, we use thefollowing definition for the signal-to-noise ratio

SNR = 20 log10

(

max(I) − min(I)

σn

)

dB, (3.34)

with I the noise-free image. The resulting global SNR’s of the test image after reducingthe noise using different filter types are given in table 3.1.

Filter σn SNR (dB)Ideal 0.147 22.7

COAF 0.162 21.8OAF 0.238 18.5

Isotropic 0.559 11.1None 0.710 9.0

Table 3.1: A quantitative comparison of adaptive filters for noise reduction in a circular orientedtexture.

All the filters of table 5.1 use a Gaussian window. The SNR of the test image withoutfiltering (None), is 9 dB. The isotropic filter is applied at scale σ =

√5. This scale is chosen

such that the area of the isotropic window is equal to the area of the windows of the adaptivefilters. we applied the ‘ideal’ filter to get an estimation of the maximal attainable SNRusing a local filter. The shape ideal filter window matches the shape local structure, whichis in this case circular. As the ideal filter we applied a circular shaped Gaussian filter atthe same scale as the adaptive filters, controlled by the true local parameter values of thenoise-free test image. The SNR’s of COAF and OAF are computed from the results shownin figure 3.6. The visible improvement of the COAF over the OAF is supported by thequantitative results in table 5.1.

3.4 Appendix A: Symmetry

The averaging of a tensor component Tij in 3D, can be written as a triple integral of thecomponent times a symmetric window function w

Tij(u, v, w) =

∫ ∫ ∫

w(u, v, w)Tij(u, v, w)dwdvdu. (3.35)

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56 Chapter 3 Curvilinear structures

The integration of an anti-symmetric function fasym(u, v, w) ≡ −fasym(u,−v, w), equalszero

∫ ∫ ∫

fasym(u, v, w)dwdvdu = 0. (3.36)

Writing an anti-symmetric function as

fasym(x) = sign(x)fsym(x), (3.37)

it is easy to verify that the multiplication of (anti-)symmetric functions leads to the fol-lowing rules

fasymfsym = f ′

asym , fsymfsym = f ′

sym , fasymfasym = f ′

sym. (3.38)

The gradients in the quadratic surface model have the following symmetries with respectto the uvw-coordinates

fu(u, v, w) = fu(u,−v, w) , fu(u, v, w) = fu(u, v,−w)

fv(u, v, w) = −fv(u,−v, w) , fv(u, v, w) = fv(u, v,−w)

fw(u, v, w) = fw(u,−v, w) , fw(u, v, w) = −fw(u, v,−w)

(3.39)

The combination of eqs.(3.39), (3.36), and (3.38) yields the following results

fu′fv′ = κ1vf 2ufv + fufv = 0

fu′fw′ = κ2wf 2ufw + fufw = 0

fv′fw′ = κ1κ2vwf 2ufvfw + κ1vfufvfw + κ2wfufvfw + fvfw = 0

(3.40)

3.5 Appendix B: Implementation

The elements of the three-dimensional quadratic structure tensor

abfcfd , afcfd , a, b, c, d ∈ {u, v, w} (3.41)

can be implemented using convolutions, by transforming them from uvw-coordinates backto xyz-coordinates before averaging. The terms in eq.(3.41) consist of two components,a (quadratic) coordinate and a structure tensor component. For an arbitrary coordinatesystem (x1, x2, · · · , xn), we can consider the quadratic coordinates xixj as elements of atensor of order 2. The definition of tensors and their elementary operations are given in[BSMM00]. The elements of the second order tensor X, used to describe the quadraticcoordinates are given by

Xij = xixj. (3.42)

Equation (3.42) is the definition of the dyadic product, a special case of the tensor productthat maps two tensors of order 1 to a tensor or order 2. Therefore we can define thequadratic coordinate X using the tensor product by

X = xx. (3.43)

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3.5 Appendix B: Implementation 57

Since we will only use the tensors under a coordinate transformation, there is a practicalidentity between a vector and a tensor of order 1, and between a matrix and a tensor oforder 2, see [Gol80]. The coordinate tensors x,X and the gradient tensor T can thereforebe transformed to a different coordinate system [FIS89] using

v = QT · x (3.44)

V = QT · X · Q (3.45)

[T]uvw = QT · T · Q, (3.46)

where the dot denotes the standard matrix product and with the definitions

T ≡ [T]xyz , X = xx , V = vv , x =

xyz

,v =

uvw

. (3.47)

The orthonormal transformation matrix Q is given by

Q = [euevew] , eu =

xu

yu

zu

, ev =

xv

yv

zv

, ew =

xw

yw

zw

, (3.48)

with (eu, ev, ew) the eigenvectors of the traditional GST.

Now we are ready to transform the quadratic structure tensor elements to xyz-coordinates.We start with the terms that contain quadratic coordinates

abfcfd = (ea · X · eb)(ec · T · ed) (3.49)

= ea · (X(ec · T · ed)) · eb (3.50)

= ea · (X(T ◦ eced)) · eb (3.51)

= ea · (XT ◦ eced) · eb , a, b, c, d ∈ {u, v, w} (3.52)

This results in a linear combination of the 36 terms

ijfkfl , i, j, k, l ∈ {x, y, z}. (3.53)

In eq.(3.51) we introduced the operator (◦) that performs the sum over two indices afteran element-wise multiplication, thereby reducing the order n of the tensor by two (n− 2).It maps the two second order tensors T, eced to a scalar. If Tij are the elements of T

and Eij are the elements of eced then the scalar s = T ◦ eced is given by the sum of theelement-wise multiplication

s =∑

i,j

TijEij (3.54)

For clarity we give a two-dimensional example:(

a11 a12

a21 a22

)

◦(

b11 b12

b21 b22

)

= a11b11 + a12b12 + a21b21 + a22b22. (3.55)

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58 Chapter 3 Curvilinear structures

In eq.(3.52) we create the fourth order tensor XT, that contains all the elements of eq.(3.53). A fourth order tensor in 3D has 34 = 81 elements, but since X,T are bothsymmetrical only 36 are unique. If we define the elements of this tensor Mijkl = TijXkl,then the elements Lkl of the second order tensor XT ◦ eced are given by

Lkl =∑

i,j

MijklEij (3.56)

The transformation of the linear coordinate terms can be performed similarly

afcfd = (x · ea)(ec · T · ed)

= ea · (x(ec · T · ed))

= ea · (x(T ◦ eced))

= ea · (xT ◦ eced) , a, c, d ∈ {u, v, w},

(3.57)

and results in the linear combination of the 18 unique terms

ifkfl , i, k, l ∈ {x, y, z}, (3.58)

of the third order tensor xT. Writing the transformation of the quadratic structure tensorelements in tensor form, has a number of advantages over the straight forward expansion.It is a compact representation. The coefficients for the linear combination of the termsin eq.(3.53) and (3.58) do not have to be computed individually, but can be computedby applying the tensor product and the matrix product to the eigenvectors of the GST.The most important advantage, however, is that eq.(3.52) and (3.57) hold for the two-dimensional case as well. The only difference is that a, b, c, d ∈ {u, v}.The averaging of the terms in eqs.(3.53) and (3.58), is linear and shift invariant, since thexyz-coordinates are the same for each point in the image. The computation of these termscan now be written as the following convolutions

Tkl ⊗ K(x) = Tkl ⊗ (ijG(x; σT )) , i, j, k, l ∈ {x, y, z}, (3.59)

where G is a Gaussian at scale σT . Since the convolution kernel K is xyz-separable,the convolution can be perform efficiently in the spatial domain using FIR-filters. Theconvolution can also be performed in the Fourier domain using a fast Fourier-transform,for example the FFTW described in [FJ97] or see http://www.fftw.org. For a comparisonof the computational speed of convolutions performed with FIR-filters and convolutionsusing a FFT-algorithm see [YGV98].

We finish with a rough estimation of the difference in computation demand in 3D, betweenthe implementation using spatially variant operations and the method described above. Forthe computation of the four extra terms in eq.(3.10) of the structure tensor for quadraticsurfaces, four spatially variant operations are needed, or 36+18=52 spatially invariantconvolutions. We limit our estimation to the FIR implementation of the convolutions.

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3.5 Appendix B: Implementation 59

Furthermore, we assume that the computational demands of the coordinate transforma-tion is equal in both methods. A xyz-separable convolution kernel with a width of N pixels,requires approximately 3N multiplications and additions. Where as the non-separable ver-sion requires approximately N 3 of these operations. If we combine these complexities withthe number of convolutions needed (4N 3 = 156N), then the filter width that gives an equalnumber of operations for both methods is N ≈ 6. Thus for a filter width N > 6, approx.σ > 1 for a Gaussian window, the spatially variant implementation uses approximately4N3 − 156N more operations than the method presented in this appendix.

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4. Line-like curvilinear structures

The Gradient Structure Tensor (GST) models an image locally as a linear structure. Inthree dimensions there are two types of linear structures: plane-like and line-like. Curvaturein the local structure causes a decrease of the confidence value of the GST. In the previouschapter we have shown that a curvature corrected confidence estimate can be obtained,by extending the local linear model to a local quadratic model. The structure tensor forquadratic surfaces was presented and tested. In this chapter we will extend the structuretensor to a curvature corrected descriptor of line-like curvilinear structures.

In chapter 2, we defined a line-like linear structure as shift invariant along a single orienta-tion. The natural generalization to line-like curvilinear structures is shift invariance alonga space curve. The isophotes or level-sets of the local image form parallel space curves.Examples of images that contain these structures are 3D medical images of blood-vessels,such as CT or MRI. Another example is time series of 2D images. Following an objectin the 2D image through time yields a space curve, the curvature of which is related tothe acceleration of the object. Channels in seismic images can also manifest themselves asspace curves.

For the local analysis of these structures, we will locally approximate a space curve with aquadratic curve. This curve will be straightened with a coordinate transform, and the GSTwill be applied to this straightened structure. Transforming the elements of the structuretensor back to original coordinates and maximizing the confidence value, yields an estimatefor the curvature of the space curve and a curvature corrected confidence estimate. We willextensively test the new estimators by applying them to images of various space curves.We will conclude this chapter by applying the curvature corrected confidence estimate tothe detection of channels. This chapter is based on the previously published article fromthe author [BVV01].

4.1 The GST for 3D line-like curvilinear structures

4.1.1 The quadratic curve approximation

In chapter 2, we implicitly modeled a local neighborhood of an arbitrary space curve c, bya straight line. This modeling can be interpreted as a first order polynomial approximationof this curve around point p on this curve. The approximation of curve c can be expressed

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62 Chapter 4 Line-like curvilinear structures

in parametric form byc(t) ≈ bt + p, (4.1)

with b the tangent vector and t the parameter of c. In this chapter we will extend thisapproximation with a quadratic term, making it a second order polynomial approximation.This is equivalent to the local modeling of c with a quadratic curve, and can be written inparametric form as

c(t) ≈ at2 + bt + p, (4.2)

where a contains the quadratic coefficients. There exists a coordinate system for eachquadratic curve such that only one element of a is non-zero. Writing the curve in thissystem gives

c(t) ≈ e1a1t2 + e2t, (4.3)

with ei the ith basis vector of the coordinate system. This local coordinate system in pointp corresponds to the Frenet-frame [Spi79] of the quadratic curve, where e2 is the tangentand e1 the normal. Using the arc length parameterized definitions we obtain

ds

dt=

dc

dt

∥, e1 = c′(s) ≡ dc(s)

ds=

dc(t)

dt

dt

ds(4.4)

e2 =c′′(s)

‖c′′(s)‖ , e3 = e1 × e2 (4.5)

The vector e3 is used for the 3D analysis and is called the bi-normal. This direction isneeded for the computation of torsion τ , and is therefore not required for the quadraticcurve approximation.

κ ≡ ‖c′′(s)‖ =∥

de1

ds

∥=

de1

dt

dt

ds

∥,

de3

ds= −τe2 (4.6)

The relation between a1 and the curvature κ in point p on the curve is found using thedefinition of curvature for parameterized curves.

k2 =[(2a1t)

2 + 1](2a1)2 − [(2a1)

2t]2

[(2a1t)2 + 1]3t=0−−→ k = 2a1 (4.7)

Using this relation we can write

c(t) ≈ e1

1

2κt2 + e2t. (4.8)

A quadratic curve can be determined by one curvature κ and the Frenet-frame. The opti-mization of the quadratic curve model therefore consists of the simultaneous optimizationof four parameters.

If the structure tensor is applied to a noise-free image of a straight line, then the eigenvaluesof the tensor for a point on the line relate as λ1 = λ2 > λ3 = 0, and the third eigenvectoris aligned along the tangent. Due to the degeneracy, there is still one degree of freedom in

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4.1 The GST for 3D line-like curvilinear structures 63

the first two eigenvectors. Bending the line increases the third eigenvalue. Assuming thatthe total gradient energy is constant we have

λ1 > λ′

2 > λ′

3 > 0. (4.9)

The second eigenvector is now aligned along the normal, and the first eigenvector alongthe bi-normal. However, if the cross-section of the line is not circular symmetric, then wehave λ1 > λ2, even without bending. In this case it is not guaranteed that the secondeigenvector is parallel with the normal, and therefore we can only use the third eigenvectoras an estimate of the tangent. Still, if we use the third eigenvalue of the GST as the tangent,we reduce the optimization of the quadratic curve model to the simultaneous optimizationof two parameters. Namely, the curvature and an angle to determine the normal.

In the next section we will use an alternative description of the quadratic curve. We usethe gauge-coordinates of the GST ‘uvw’ defined by the local frame of the GST Q,

[T]uvw = QT ·

f 2x fxfy fxfz

fxfy f 2y fyfz

fxfz fyfz f 2z

· Q =

f 2u 0 0

0 f 2v 0

0 0 f 2w

, (4.10)

where the w-axis is parallel to the tangent. The definition of Q is given in appendix B ofthe previous chapter. A schematic example of a quadratic curve and its local uvw-frameis shown in figure 4.1b. Instead of estimating the curvature κ and the angle between thenormal and the v-axis, we will estimate the curvatures κ1 and κ2. The curvature κ1 isdefined as the curvature of the projection of the quadratic curve c on the uw-plane, andκ2 as the curvature of the projection of c on the vw-plane, see 4.1a. The relation betweenthe projected curvatures and the total curvature is given by

κ =√

κ21 + κ2

2. (4.11)

4.1.2 The quadratic GST for space curves

The first step towards a curvature corrected GST, is to define the coordinate transformuvw → u′v′w′ that deforms an arbitrary curve into a straight line, see fig 4.1b. Using thequadratic curve approximation and the κ1, κ2 description, the coordinate transform andits inverse are given by

u′ = u − 1

2κ1w

2

v′ = v − 1

2κ2w

2

w′ = w

u = u′ + 1

2κ1w

′2

v = v′ + 1

2κ2w

′2

w = w′

(4.12)

This approximation is valid for small values of κ1 and κ2. The test results will showthe practical limitations of this approximation. The inverse-transformation is used to

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64 Chapter 4 Line-like curvilinear structures

u

12

w

c

v

κκ

(a) projections

w’w

v

u u’

v’

(b) transformation

Figure 4.1: A schematic representation of the projections of the curve c on the uw and thevw-plane and the corresponding curvatures κ1, κ2 (a). Deforming a quadratic curveto a straight line using a coordinate transform (b).

express the derivatives in the u′v′w′-coordinates as a function of the derivatives in theuvw-coordinates.

fu′ = uu′fu + vu′fv + wu′fw = fu

fv′ = uv′fu + vv′fv + wv′fw = fv

fw′ = uw′fu + vw′fv + ww′fw

= κ1wfu + κ2wfv + fw

(4.13)

We will now consider a curved line through the origin of the uvw-space with its tangent inthe origin parallel to the w-axis, see figure 4.1b. By applying the traditional GST to theu′v′w′-space for arbitrary κ1 and κ2, we obtain the gradient structure tensor for quadraticcurves

TQC =

f 2u′ fu′fv′ fu′fw′

fu′fv′ f 2v′ fv′fw′

fu′fw′ fv′fw′ f 2w′

(4.14)

Using eq.(4.13) we can express the elements of TQC in the uvw-coordinates.

f 2u′ = f 2

u

f 2v′ = f 2

v

f 2w′ = f 2

w + κ2

1a + 2κ1κ2b + κ2

2c + 2κ1d + 2κ2e

fu′fw′ = fv′fw′ = fu′fv′ = 0

(4.15)

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4.1 The GST for 3D line-like curvilinear structures 65

with the abbreviations

a ≡ w2f 2u , b ≡ w2fufv , c ≡ w2f 2

v

d ≡ wfufw , e ≡ wfvfw.(4.16)

Note that the terms a, b, c, d differ from those presented in the previous chapter.

The off-diagonal elements of TQC are all equal to zero, due to the symmetry f(u, v, w) =f(u, v,−w) in the model. The derivation and the details of the symmetry considerationsare given in appendix A. Since the matrix of TQC is diagonal, the eigenvalues are equal tothe diagonal elements.

λ1 = f 2u , λ2 = f 2

v , λ3(κ1, κ2) = f 2w + κ2

1a + 2κ1κ2b + κ2

2c + 2κ1d + 2κ2e (4.17)

For a line-like linear structure the eigenvalues relate as λ1 ≈ λ2 � λ3. The optimalresemblance to a quadratic curve structure can therefore be found by minimizing λ3 withrespect to κ1 and κ2. In contrary to the quadratic surface model, κ1 and κ2 are coupledand the minimization should be performed two-dimensional. The curvatures κ1,min andκ2,min that minimize λ3 yield an estimate of local curvature and the are given by

κ1 = κ1,min =be − cd

ac − b2, κ2 = κ2,min =

bd − ae

ac − b2. (4.18)

Substituting these curvatures in eq.(4.17), we get the following expressions

λ1 = f 2u , λ2 = f 2

v , λ3 = f 2w − K. (4.19)

with the abbreviationK ≡ aκ2

1 + bκ1κ2 + cκ2

2 (4.20)

The curvature corrected third eigenvalue can be used to create the curvature correctedline-confidence measure

Cline =λ2 − λ3

λ2 + λ3

. (4.21)

Note that for the limit κ1, κ2 → 0, the equation for λ3 reduces to the third eigenvalue ofthe traditional GST. An estimate of the total curvature κ can be constructed from theestimates of the projected curvatures using eq.(4.11)

κ =√

κ21 + κ2

2. (4.22)

The computation of the structure tensor for quadratic curves TQC , consist of two parts.First, the GST is computed to obtain the gauge-coordinates uvw, and next the extra terms(a, b, c, d, e) of eq. (4.16) need to be computed. The direct computation of (a, b, c, d, e)results in a spatially variant convolution, since the gauge-coordinates change over theimage. For large filter sizes, this can lead to a high computational demand. The extra

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66 Chapter 4 Line-like curvilinear structures

terms have the same form as the correction terms of the structure tensor for quadraticsurfaces, namely

abfcfd , afcfd , a, b, c, d ∈ {u, v, w}. (4.23)

The terms (a, b, c, d, e) can therefore be computed as linear combinations of convolutions,as shown in Appendix B of chapter 3.

4.2 Experimental tests and results

The structure tensor for three-dimensional space curves TQC described above, locally mod-els an arbitrary space curve as a line-like linear structure after a quadratic coordinatetransformation. The resemblance between the model and the image data is given by theconfidence value Cline. The parameters of the model that maximize the confidence yieldan estimate of the curvature κ. The confidence of the model fit can be decreased by twofactors: noise and a deviation of the image structure from the model. In this section, wewill analyze the estimates for curvature and confidence of the tensor TQC, by performingseveral measurements on test images.

For these measurements we use images of several types of space curves. The shape ofan arbitrary space curve in 3D is determined by the curvature and the torsion along thecurve. Since our quadratic model only corrects for curvature we will first examine imagesof circles with various radii. Next we will measure the influence of torsion by applying thetensor TQC to images of a helix. The quadratic curve model assumes a curve structurethat is symmetric along the w-axis. As a final test we will examine the influence of localasymmetry using images of an ellipse.

Creating an image of a space curve means sampling the Euclidean space in the neighbor-hood of the curve. To be able to sample the curve it should have a finite thickness1. Oneshould make sure that the spectrum of space curve is band-limited, to be able to sam-ple it according to the sample theorem of Shannon [Sha49]. An example of a practicallyband-limited image of a curve, is a curve with a Gaussian intensity cross-section

Icross(u, v) = e−r2/2σ2c , r2 = u2 + v2, (4.24)

where r can be obtained by computing for each point in the image I(x, y, z), the distanced = r to the space curve c. A straightforward way to compute the distance to the spacecurve c is to find a parameterization for the curve c(t), and sample it for equal intervalsti. The distance d for each point in the image can now be computed by

d = mini‖c(ti) − x‖ , x = (x, y, z) (4.25)

As a 2D example, a 128*128 pixel image of a circle with a radius of 32 pixels is shown infigure 4.2. The computed distance to the circle, shown in figure 4.2a, is used as the input

1 Strictly speaking it is no longer a space curve since a mathematical curve is infinitely thin.

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4.2 Experimental tests and results 67

(r) to eq. (4.24) with σc = 2 to create to image shown in figure 4.2b. Although we describethe curves using a parameterization, the estimation of curvature and confidence does notmake use of this parameterization in any way. Furthermore, an arbitrary translation androtation of the curves does not affect the estimates for curvature and confidence.

(a) (b)

Figure 4.2: The creation of a band-limited image of a space curve. For each point in the image,the distance to the curve is computed (a), to create a space curve with a Gaussianprofile (b).

We chose the width of the intensity cross-section σc = 2 voxel for all images. For both thegradient regularization and the local averaging of the structure tensor a Gaussian windowis used. We computed the gradient at scale σg = 1 and the tensor smoothing at scaleσT = 4 for all measurements. The scale σT is chosen such that an optimal SNR over allmeasurements is achieved. We will elaborate on the scale selection of the tensor analysiswindow, in the discussion section below.

To test the robustness of the estimators with respect to noise, uncorrelated normally dis-tributed noise N(0, σ2

n) is added to the test images. The noise dependency is measuredby repeating the measurements for three different noise levels σ2

n = {0.01, 0.1, 0.5}, corre-sponding to the signal-to-noise ratios 20,10,3 dB. Since the intensity of the test images arebounded, 0 ≤ I(x, y, z) ≤ 1, we use the following definition for the signal-to-noise ratio

SNR = 20 log10

(

max(I) − min(I)

σn

)

dB = −20 log10(σn)dB. (4.26)

All measurements are averaged over 40 different realizations of the noise, and the corre-sponding standard deviations are indicated by error-bars. To show the improvement inthe confidence estimation, we will also compute the confidence estimate of the traditionalGST.

4.2.1 Circle image

The first type of test images Icircle(x, y, z; r, σc, σn) are images of circle with radius r,created using equations (4.25) and (4.24), with normally distributed noise (σn) added.The parameterization used for the circle in 3D is given by

ccircle(t; r) = (r cos t, r sin t, 0). (4.27)

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68 Chapter 4 Line-like curvilinear structures

The shape of a circle is fully determined by its constant curvature. The curvature of thecircle can be derived from the parameterization using equation (4.6), and is given by thesimple relation

κ =1

r, (4.28)

where r is the radius of the circle. We will measure both the curvature and the confidenceestimate as a function of r. To give a visual impression of the circle images, figure 4.3shows for each noise level the xy-slice (z = 0) of the test image Icircle(x, y, z; 20, 2, σn).

(a) ∞ dB (b) 20 dB (c) 10 dB (d) 3 dB

Figure 4.3: Normally distributed noise is added at three different levels for the measurement ofthe noise dependency.

The test image Icircle could also be interpreted as an image of a solid torus, with a Gaussianshaped intensity profile. A surface rendering of this torus is shown in figure 4.4.

R2σ

Figure 4.4: A surface rendering of one half of a solid torus with radius R and Gaussian intensitycross-section of width σ.

The results of the curvature and confidence estimation on the circle images are depicted infigure 4.5. From the results in 4.5b we see that the curvature correction of the confidenceestimate yields a significant improvement. The estimators show approximately the samenoise dependency as the estimators of the quadratic surface model. The confidence estimatedecreases for an increasing noise level, and the curvature estimate becomes biased for lowsignal-to-noise ratios, i.e. the curvature is under-estimated. The increase of the bias in thecurvature estimation for decreasing SNR, is explained in appendix B.

There is another bias visible in the curvature estimation in figure 4.5a. For small radii thecurvature is over-estimated. This is due to the model error we make by locally modelingthe circle with a quadratic curve. Since we keep the size of the analysis window constant,the quadratic curve must describe a larger part of the circles with smaller radii. The

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4.2 Experimental tests and results 69

5 10 15 20 25 30 35 40radius

0

0.05

0.1

0.15

0.2

κ

True κ20 dB10 dB3 dB

5 10 15 20 25 30 35 40radius

0

0.2

0.4

0.6

0.8

1

Clin

e

GSTPGST

20 dB

10 dB

3 dB

Figure 4.5: Curvature κ and confidence Cline estimation as a function of the radius on the circle-images (σT = 4). The measurements are performed at 3,10,20 dB, and averagedover 40 different noise realizations. The error-bars denote the corresponding stan-dard deviation. The small horizontal shift between the different labels is artificiallyintroduced to improve their visibility.

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70 Chapter 4 Line-like curvilinear structures

fourth order term of the polynomial approximation of a circle becomes more importantand causes the over-estimation. Note that (Cline)GST ≤ (Cline)QGST for all radii. This canbe explained by the fact that adding an extra parameter to the model, always results in abetter fit.

4.2.2 Helix image

To test the influence of torsion on the estimators, we will apply the structure tensor forquadratic curves to images of a helix Ihelix(x, y, z; r, h, σc, σn). The parameterization weused for the creation of the helix images is given by

chelix(t) = (r cos t, r sin t, ht). (4.29)

the parameter h is called the pitch and its geometrical meaning is indicated in figure 4.6.The curvature κ and the torsion τ of the helix are given by

κ =r

r2 + h2, τ =

h

r2 + h2. (4.30)

The curvature of the helix approaches the curvature of a circle for h → 0. for an increasingpitch h, the curvature decreases and the torsion increases. We will measure both thecurvature and the confidence estimate as a function of the radius r for a fixed pitch h = 10.The curvature κ(r) as function of the radius is maximal for r = h.

z

yx

h

r

Figure 4.6: A Graph of a helix with its two constant parameters, radius r and pitch h, indicated.

The results of the curvature and confidence estimation on the helix-images are depictedin figure 4.8. In figure figure 4.8a we can see that curvature estimate at 20 dB is notsignificantly biased. The curvature values in the helix-images are κ ≤ 0.05. The curvatureestimates on the circle-images for this curvature range are unbiased too. We have checkedthe curvature estimate on noise-free helix-images for a broad range of torsion values andwe obtained an unbiased result in all cases. The improvement in the confidence estimate iscomparable to the improvements measured on the circle-images, as is the noise dependencyof the estimators.

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4.3 Discussion 71

4.2.3 Ellipse image

For the derivation of the estimators from the structure tensor for quadratic curves, it was as-sumed that the curve is locally symmetric along the w-axis: f(u, v, w) = f(u, v,−w). Thisassumption was valid for the circle and the helix images. To test the influence of local asym-metry on the estimators, we will apply them to an image of a ellipse Iellipse(x, y, z; a, b, σc, σn).

a

b

Figure 4.7: A Graph of an ellipse with its radii a and b indicated.

The parameterization used for the three-dimensional ellipse-curve is given by

cellipse(t) = (a cos t, b sin t, 0), (4.31)

where a, b are the radii of the ellipse, see figure 4.7. The ellipse-curve is only symmetricalat the extrema of the ellipse, i.e. for t ∈ {0, 1

2π, π, 3

2π}. The non-constant curvature κ

along the ellipse-curve is given by

κ(t) =ab

(a2 sin2 t + b2 cos2 t)3/2. (4.32)

The curvature κ(t) of the ellipse approaches the curvature of a circle for |a−b| → 0. We willmeasure both the curvature and the confidence estimate as a function of the parametert for fixed radii a = 60, b = 30. Due to the symmetry of the ellipse, we can limit themeasurements to one quadrant of the ellipse 0 < t < 1

2π.

If we examine the the results in figure 4.9, we see that at 20 dB the curvature estimateis unbiased for the extrema of the ellipse (t = 0, 1

2π). For some of the estimated values in

between, we see that the local asymmetry causes a small over-estimation of the curvature.This bias is partly due to a bias in the tangent estimate. For a neighborhood around anasymmetric curve the third eigenvector of the GST is not parallel to the tangent. Thepresence of noise causes an under-estimation of the curvature. The noise dependency ofthe estimators is comparable to the noise dependency measured on the circle-images, andso is the improvement of the confidence estimate.

4.3 Discussion

For the description of line-like curvilinear structures we have presented a curvature cor-rected gradient structure tensor TQC . A line-like curvilinear structure was defined as shiftinvariant along a space curve. The curvature correction is based on a local approximation

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72 Chapter 4 Line-like curvilinear structures

5 10 15 20 25 30 35 40radius

0

0.01

0.02

0.03

0.04

0.05

0.06

κ

True κ20 dB10 dB3 dB

5 10 15 20 25 30 35 40radius

0

0.2

0.4

0.6

0.8

1

Clin

e

GSTPGST

20 dB

10 dB

3 dB

Figure 4.8: Curvature κ and confidence Cline estimation on the helix-image as a function of theradius r, with a constant pitch h = 10 (σT = 4). The measurements are performedat 3,10,20 dB, and averaged over 40 different noise realizations. The error-barsdenote the corresponding standard deviation. The small horizontal shift betweenthe different labels is artificially introduced to improve their visibility.

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4.3 Discussion 73

−0.1 0.3 0.7 1.1 1.5t

0

0.02

0.04

0.06

0.08

κ

True κ20 dB10 dB3 dB

−0.1 0.3 0.7 1.1 1.5t

0

0.2

0.4

0.6

0.8

1

Clin

e

GSTPGST

20 dB

10 dB

3 dB

Figure 4.9: Curvature κ and confidence Cline estimation on the ellipse-image as a function ofthe parameter t, with constant radii a = 60, b = 30 (σT = 4). The measurementsare performed at 3,10,20 dB, and averaged over 40 different noise realizations. Theerror-bars denote the corresponding standard deviation. The small horizontal shiftbetween the different labels is artificially introduced to improve their visibility.

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74 Chapter 4 Line-like curvilinear structures

of a space curve with a quadratic curve. The complete representation of the shape of athree-dimensional space curve requires two parameters for each point on the curve, namelycurvature κ and torsion τ . Our model, on the other hand, only incorporates curvature.Furthermore, mirror symmetry with respect to the plane normal to the tangent vector ofthe curve is assumed.

To test the robustness of the estimators for curvature and confidence derived from TQC

with respect to deviations of the image structure from the local model, we applied them toimages of a circle, a helix, and an ellipse. Furthermore, we added noise to the space curveimages to measure the robustness with respect to noise. For all measurements we keptthe thickness of the space curves and the scale of the tensor window constant. Increasingthe tensor scale σT suppresses the influence of noise, i.e. the standard deviation of themeasurements decreases. However, a larger analysis window requires the modeling of alarger part of the space curve. The model errors become more apparent and the curvatureestimate becomes biased. Another draw-back of increasing σT is a decrease in signal-to-noise ratio. Consider an individual line in an isotropic three-dimensional window withradius R, with noise. Increasing the window size causes the total amount of signal energy toincrease linearly with the radius of the window. The total noise energy, however, increaseswith R3/2, assuming uncorrelated normally distributed noise.

The estimators show approximately the same noise dependency as the estimators of thequadratic surface model. The confidence estimate and its improvement with respect to thetraditional GST confidence, decrease for an increasing noise level. The curvature estimatebecomes biased for low signal-to-noise ratios, i.e. it gives an under-estimation. The mea-surements on the helix-images show that a constant torsion has no measurable effect onthe estimators. The measurements on the ellipse-images show that local asymmetry doesinfluence the estimation of curvature, i.e. it causes an over-estimation of the curvature.

4.4 Application: Channel detection

The occurrence of sedimentary structures, such as channels, is an important cue for thegeological model of a subsurface region. The manual localization of channels in often vastamounts of 3D seismic data, is very time consuming. One way of automating this processis to find a suitable mathematical structure model for channels. The resemblance betweenthe model and the structure that is being analyzed should provide a measure of how muchthis structure resembles a channel. A characteristic feature of channels is their meanderingshape. Therefore, a parametric model based on shape parameters such as curvature doessuggest itself. In this section we will study the applicability of the structure tensor to thedetection of channels.

The image data we are going to analyze is a region (256*128*64 voxels) of a 3D seismicimage I(x, y, t) around a channel. The intensity of the image corresponds to the amplitude

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4.4 Application: Channel detection 75

of the reflected seismic wavelet. The xy-axes are spatial and the t-axis indicates thetravel time of the seismic wavelet. The t-axis can be inverted to depth (spatial) if thevelocity of the acoustic wave is known at each depth. To give an impression of howthis channel manifests itself in this data, we have visualized it by 2-D cross-sections infigure 4.10. The (time) xy-slice shows the horizontal extend of the channel, that has thecharacteristic meandering shape. The yt-slice shows a cross-section of the channel. Figure4.10d depicts the t-axis along the centre of the channel showing the depth (time) variationof the channel. As a preprocessing step, we create an attribute volume by computing thestandard deviation within a window of (5*5*9) voxels, for each point in the seismic image.A cross-section of this volume, corresponding to figure 4.10a, is depicted in figure 4.10c.The advantage of this attribute volume is that the channel manifests itself as a space curve.

The meandering nature of channels suggests that the curvature corrected model gives amore accurate description of a channel than the traditional straight model. Thereforewe expect that the curvature corrected confidence estimate has a significantly higher valuethan the confidence estimate of the traditional GST. To test this hypothesis we have appliedboth the GST and structure tensor for quadratic curves (QGST) to the seismic image. Inboth cases we used a gradient smoothing σg = 1 and a tensor smoothing σT = 3.5. Thechoice of the tensor scale is a trade-off as pointed out in the discussion in section 4.3. Forthe analysis of the space curve images we used a tensor scale that was twice the width ofthe curves (σT = 2σc). Here we use the same rule, and we roughly estimated the widthof the channel. The results of the QGST confidence estimate and the improvement in theconfidence estimate are shown in figure 4.10e and f. Due to the contract-stretch it is notpossible to determine the relative improvement from these image. The purpose of (f) is toshow where the improvement is achieved. The relative improvement in the bright spots in(f) is approximately 50%.

For the evaluation of the result, we defined three regions in the seismic image. The firstregion contains the channel and is indicated in figure 4.11a with a thick white contour.The second region contains a straight sedimentary structure and is depicted by a blackcontour in figure 4.11a. The last region consist of points from an area without sedimentarystructure, and represents the noise in the data. This area is indicated by a white rectanglein figure 4.11b. For all three regions we computed the cumulative frequency distribution ofthe confidence values in that region of both the GST and the QGST confidence estimate.These cumulative distributions are displayed in figure 4.11c.

We will use the cumulative distributions to verify our hypothesis that the curvature cor-rection yield a significant better description of channels. For notational convenience wedenote the improvement in the confidence value as

∆C = (Cline)QGST − (Cline)GST . (4.33)

Inspection of the confidence distributions of the channel region in figure 4.11c, shows thatthe improvement in this region ∆C is larger than zero. However, this does not necessarilymean that the improvement is significant. As we have already seen in the test results

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76 Chapter 4 Line-like curvilinear structures

A

A’

(a) xy-slice (t = 33)

A A’

(b) yt-slice (x = 133)

(c) attribute volume (t = 33) (d) t along channel

(e) (Cline)QGST (t = 33) (f) ∆C (t = 33)

Figure 4.10: Seismic data volume used to demonstrate channel detection (a,b,d), and a sliceof the attribute volume (c). The resulting confidence estimate and the confidenceimprovement are shown in (e) and (f).

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4.5 Appendix A: Symmetry 77

that the QGST confidence estimate always yields a higher value than the GST confidenceestimate. Since ∆C > 0 in the noise region, the significance of ∆C along the channelshould be proven by correcting it for the improvement due to a better description of thenoise ∆Cn.

For the following reasoning we will assume that the noise has the same distribution at eachposition in the volume. The improvement ∆C ≈ 0.03 in the noise region (SNR = −∞)gives an estimation of the improvement due to noise ∆Cn. However it should be taken intoaccount that the value of ∆Cn is not constant for all Cline values. Imagine for example anoise-free neighborhood around a straight line (SNR = ∞). In that case the line confidenceCline = 1 and improvement ∆C = 0 and thus ∆Cn = 0. To get an impression of how ∆Cn

changes for higher values of Cline we have included the ”straight structure” region. Inthis region the curvature correction improves the fit by giving a better description of thenoise. Therefore we have ∆Cn ≈ 0.01 at Cline = 0.55. We can now conclude that thenoise contribution ∆Cn to the improvement ∆C ≈ 0.05 in the channel region is smallerthan 0.01. The curvature correction therefore yields a significant better description of themeandering structure of the channel.

Summarizing the results we conclude that the detection of channels, based on the estimatorsof structure tensor alone is not feasible. This is most likely due to the local nature of thetensor analysis. Increasing the size of the analysis window isotropically does not result intoa better description, for the same reasons we mentioned in the discussion of the analysisof space curve images. For a more accurate analysis of channels, windows that adaptto the shape of the channel are needed. We have shown that the curvature correctedstructure tensor for space curves TQC , gives a better description of channels than theGST. Therefore TQC should be used to control a subsequent processing step with adaptivewindows. Windows that are able to match the shape of a channel will be presented in thenext chapter.

4.5 Appendix A: Symmetry

The symmetry in the quadratic curve model f(u, v, w) = f(u, v,−w) with respect to theuvw-coordinates, gives rise to the following symmetries in the gradients

fu(u, v, w) = fu(u, v,−w)

fv(u, v, w) = fv(u, v,−w)

fw(u, v, w) = −fw(u, v,−w)

(4.34)

We will use the fact that the integration of an anti-symmetric function equals zero, to showthat the off-diagonal elements of the quadratic structure tensor for curves, equal zero. Thecombination of eqs.(4.34) and the equations (3.36), (3.38) for (anti-)symmetric functions

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78 Chapter 4 Line-like curvilinear structures

(a) xy-slice (t = 33) (b) xy-slice (t = 47)

0 0.2 0.4 0.6 0.8 1Cline

0

0.2

0.4

0.6

0.8

1

GSTQGST

Noise ChannelStraightstructure

(c) cumulative distributions

Figure 4.11: The cumulative frequency distributions of the confidence values (c) in selectedregions of the seismic volume (a)+(b). The noise region is indicated by a whiterectangle in (b). The region with the straight structure is indicated with a blackcontour and the channel region with a thick white contour in (a).

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4.6 Appendix B: Bias 79

from appendix A of chapter 3, yields the following results

fu′fw′ = fufw + κ1wf 2u + κ2wfufv = 0

fv′fw′ = fvfw + κ1wfufv + κ2wf 2v = 0

fu′fv′ = fufv = 0

(4.35)

The last equation requires mirror symmetry with respect to an arbitrary plane perpendic-ular to the uv-plane.

4.6 Appendix B: Bias

From figure 4.5a it is clear that noise introduces a bias in the curvature estimation. Theexpectation value of the curvature estimator E[κ] is smaller than the true curvature. Thiscan be explained by examining κ1 and κ2.

κ1 =w2fufv wfvfw − w2f 2

v wfufw

w2f 2u w2f 2

v − (w2fufv)2(4.36)

κ2 =w2fufv wfufw − w2f 2

u wfvfw

w2f 2u w2f 2

v − (w2fufv)2(4.37)

κ =√

κ21 + κ2

2 (4.38)

The terms w2f 2i , i ∈ {u, v}, increase when noise is added, while the other terms do not

change. Since the increasing terms appear quadratic in the denominator and linear in thenominator, the curvature estimate κ becomes smaller for an increasing noise level. Thesign of κ1 and κ2 is lost in eq.(4.38).

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5. Structural analysis using a non-parametric

description

The local structural analysis of images is based on the idea that complex image structurescan be described at a small scale using a simple geometric model. We started the structuralanalysis of images by assuming local shift invariance, and we called an image structure withshift invariance in one or more orientations a linear structure. The structure tensor waspresented as a versatile tool for the analysis of these structures. Increasing the analysisscale leads to the description of a larger part of the complex structure. The curvature of thestructure becomes gradually more important, and therefore we introduced the curvaturecorrected structure tensor based on a quadratic model. The application of the confidencevalue of this model to the detection of channels showed that the local structure analysis isnot sufficient for this detection task.

An important feature of the channel structure is its spatial continuity. At the scale neededto capture this continuity, the quadratic model is not flexible enough to describe the struc-ture. Increasing the flexibility of a parametric model means increasing the number ofparameters, and thereby increasing the computational demand. Instead of introducing amore complex parametric model with more parameters, we will turn to a non-parametricmodel. The local parameter estimates obtained from the simple geometric models can beused for a piece-wise description of curvilinear image structures. The shape of the structurecan be found by the tracking of this piece-wise description. The shape of two-dimensionaland line-like curvilinear structures can be described using a single space curve. This shapecan therefore be found by the tracking of an individual curve. The shape of a plane-likecurvilinear structure is defined by a surface. The tracking of a surface is significantly morecomplex than the tracking of a curve, and falls outside the scope of this thesis. The non-parametric description of plane-like curvilinear structures is therefore not discussed in thisthesis.

This chapter begins with the description of the tracking method. The method will bedemonstrated and tested by applying it to the tracking of growth rings and channels.Next, the tracking method is used for the creation of a non-parametric adaptive filter. Theresults of this filter applied to the noise reduction in an oriented texture, will be comparedto the results of the adaptive filters presented in chapter 3. We will conclude this chapterwith the estimation of a non-parametric confidence measure, and its application to channeldetection.

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82 Chapter 5 Structural analysis using a non-parametric description

5.1 The tracking of line-like curvilinear structures

In chapter 4 we generalized line-like linear structures to line-like curvilinear structures. Animage neighborhood has a line-like curvilinear structure if the isophotes or the level-sets inthat neighborhood can be described as an individual space curve or a bundle of parallel ofspace curves. In this section we will present a method for the tracking of these structures.With the word ‘tracking’, we refer to the iterative process of following the spatial extentof a selected object. In the case of curvilinear structures, the objects are the isophotes.Since the isophotes form space curves, we start with the tracking of space curves.

Consider a space curve c(t) and two points t=P and t=Q on this curve, as depicted infigure 5.1a. We select point P as the start position. For the tracking of the curve we needto find the displacement vector

∆c = c(Q) − c(P ), (5.1)

to make the first step to the next point on the curve, Q. Since the parameterization of thecurves formed by isophotes is in general not known, we need to estimate the displacementvector ∆c using a local geometric model. From differential geometry [Spi79], we know thatthe shape of an arbitrary space curve in 3D is fully determined by the curvature and thetorsion along the curve. Therefore, if we have an estimation of the shape parameters atpoint c(P ), then we can use these parameters to estimate the displacement vector ∆c.

P

Q

nt

∆c

c

(a)

, 33 κφ )(, 11 κφ )(

1t 2t

3tt 4

t 5c

(b)

Figure 5.1: Tracking a space curve c by following the displacement vectors ∆c. The displace-ment vectors are computed using the locally estimated parameters orientation andcurvature (φi, κi).

We start by applying this idea to the tracking of a space curve c in two dimensions. If weassume that the tangent vector t and the normal vector n are known in each point, thenthe displacement vector can be approximated by

∆c ≈ n1

2κ(∆t)2 + t(∆t), (5.2)

with ∆t the step size. Suppose that we have a sampled version of the space curve c(ti),then we can use eq.(5.2) to estimate the displacement vectors between the sample points ti

and ti+1. Following these vectors ‘head-to-tail’ as shown in figure 5.1b, we track the curve.

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5.1 The tracking of line-like curvilinear structures 83

Translating this method to the tracking of a two-dimensional curvilinear image structureis not difficult. An estimate for the tangent and the normal vector for each point of thestructure, is given by the eigenvectors of the structure tensor {e1, e2}. The curvatureestimate κ of the curvature corrected structure tensor can be used as the curvature ineq.(5.2). The frame of the GST in 2D {e1, e2} can be fully determined by one anglebetween the first eigenvector and the x-axis of the image. Therefore we can track theimage structure by following the path that is piece-wise defined by the local orientationand curvature estimates (φi, κi), as depicted in 5.1b.

The tracking of a space curve in three dimensions requires more parameter estimates. Afull local description of the curve is given by the Frenet-frame and the shape parameterscurvature and torsion [Spi79]. We can approximate the curve using the quadratic modelpresented in chapter 4. The quadratic curve was parameterized using the local frame ofthe structure tensor {e1, e2, e3} and two curvatures κ1, κ2. The third eigenvector e3 is anestimate for the tangent, and the curvatures κ1 and κ2 are defined along resp. e1 ande2. In chapter 4, we have experimentally shown that the quadratic curve model is notsignificantly hampered by torsion. If we start at an arbitrary point ti on the space curvec then we can track the curve by following the displacement vectors

∆c(ti) ≈ e1

1

2κ1(∆ti)

2 + e2

1

2κ2(∆ti)

2 + e3(∆ti), (5.3)

where ∆ti is the step size that can be chosen freely. Increasing the step size however, canincrease the model error and thereby the tracking error.

So far we have assumed that the start point from which we start the tracking is knownor manually selected. If the starting position is not known and manual selection is notdesirable, the curve needs to be detected first. The automatic detection of a space curveconstitutes a more difficult problem than the tracking of the curve. Under favorable cir-cumstances the confidence estimate of the local models can be used for detection. We willinvestigate the detection capabilities of the confidence estimates later on in this chapter.

5.1.1 Application to the tracking of growth rings

To demonstrate the tracking method described above, we applied it to the tracking of thegrowth rings of a tree. A 2D cross-section of a CT image of a tree-trunk that shows thegrowth rings, is depicted in figure 5.2a. We started the analysis of this 2D cross-section byapplying the curvature corrected structure tensor at scale (σg = 1, σT = 3). The resultingestimates for orientation, curvature and confidence are depicted in figures 5.2e-g. Weselected three starting points for the tracking algorithm, and these points are indicated bywhite arrows in figure 5.2d. The tracking results based on the orientation and the curvatureestimate are shown in figure 5.2b as black curves super-imposed on the original image. Forcomparison we have shown the tracking result using only the orientation estimate in figure

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84 Chapter 5 Structural analysis using a non-parametric description

5.2c. The difference between the tracked paths of the two methods is shown in 5.2d. Astep size of one pixel is used in both cases, and we allowed sub-pixel positions.

Evaluation of the tracking of the two inner rings (1,2), shows that following the estimatedorientation and curvature according the eq. (5.3), results in a closed path that matches thecorresponding growth rings. If the curvature estimate is not used, then the tracked path isno longer closed but has a spiral shape. Evaluation of the tracking of the third outer ring(3), shows that the tensor analysis is based on the differential geometry of the intensityvalues, i.e. it is a gradient model. The parameter estimators are only sensitive to intensitychanges not to the intensity value itself. The dominant gradients in the neighborhood of thethird ring are those belonging to the change in the background, not those on the slopes ofthe local intensity minimum that constitutes the growth ring. The resulting path thereforematches the shape of the background transition. Again we see that the path based onorientation alone is not able to keep up with the curvature of the structure. Eventually itfollows a more outward laying growth ring uninfluenced by the background transition.

The purpose of this section was to show that the parameter estimates of the curvaturecorrected structure tensor can be used for the piece-wise definition of paths that followthe local structure. A more accurate tracking of the growth rings could be obtained byutilizing more features. To give a simple example: one could check the intensity valuealong the path. If the intensity value of a new point differs too much from some averagevalue along the path, then the path has probably taken a wrong turn.

5.1.2 Application to the tracking of sedimentary structures

In the previous chapter we showed that the confidence value of the quadratic structuretensor applied to a 3D seismic variance attribute volume, can detect channels. The curva-ture correction improved the detection of the strongly bent parts of the channel. However,this confidence based detector gives a lot of false positives, i.e. there are still a lot ofnon-channel structures that give rise to a high confidence value. The analysis of the chan-nels at a relatively small scale, approximately two times the width of the channel, is notsufficient to distinguish between channel and non-channel. Taking the spatial continuityof the channel into account could improve the channel detection. One way to incorporatethe continuity of channels into the detection, is to extend the analysis window along thestructure. To accomplish this, a technique for the tracking of channels is needed. In thissection we will test the ability of the tracking method described above, to track channelsand related sedimentary structures.

We will apply the tracking method to a horizontal 2D cross-section (512*256 pixels) of a 3Dseismic variance attribute image. The planar cross-section shown in figure 5.3a, is chosensuch that it approximates the depositional surface, and it thereby makes the sedimentarystructures visible. We selected three clearly visible channels and show them in figure 5.3b.The structure tensor analysis can be interpreted as imposing a local parametric model on

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5.1 The tracking of line-like curvilinear structures 85

(a) Wood (b) Tracking O+C (c) Tracking O

12

3

(d) Difference (e) Orientation (f) Curvature

(g) Confidence

Figure 5.2: Tracking the growth rings in a 2D cross-section of a CT image of a tree-trunk. Theresults are shown using only orientation (c) and using orientation and curvature (b).(d) shows the difference between the two methods and the starting points indicatedby white arrows.

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86 Chapter 5 Structural analysis using a non-parametric description

the gradients. To give an impression of the gradient information, we computed the gradientmagnitude at scale (σg = 2) and depict it in figure 5.3c.

Instead of manually selecting a starting point for the tracking algorithm for each structurewe want to track, we will use all ‘edge points’ as start positions. The edge points are allpoints in the image that, according to an edge detector, belong to an edge. The edge pointsare usually found by locating the points where the gradient magnitude has a maximum[Can86]. The traditional way to find this maximum is to locate the zero-crossings of thesecond derivative in the gradient direction. We applied a one-dimensional window (7*1pixel) steered along the gradient direction to each point in the gradient magnitude image.If the intensity value of the current pixel is equal to the maximum value inside this window,then it belongs to an edge. The result of this edge detection is shown in figure 5.3d.

Next, we apply the curvature corrected structure tensor at scale (σg = 2, σT = 4) to the 2Dattribute image shown in figure 5.3a, to estimate the orientation and curvature (φ, κ) ateach point in the image. The angle φ is the angle between the x-axis of the image and thesecond eigenvector of the GST, and gives an estimate of the tangent direction. Consideran arbitrary edge point as starting position P0 and the corresponding orientation estimateφ0, see figure 5.4. Since the structure continues in both the φ0 and −φ0 direction, weinitiate the tracking algorithm in both directions. The only stop criterion for the trackingis a maximum number of steps n. The result of this tracking experiment using a fixed stepsize of one pixel and n = 100, is shown in figure 5.3e. The tracked paths are visualized byincreasing the intensity value of the output image at all the points of all the paths witha constant value 1. Thus, if forty paths cross the point (x, y), then the intensity value atthat point I(x, y) = 40.

If we examine the tracking result near the channel on the left and the channel in themiddle, we see that the tracked paths are parallel to the channel banks. This means thatthe tracking method is able to follow these channels. The tracking of the bottom-leftchannel is more difficult and fails near the ‘hairpin turns’. At the locations where thechannel banks constitute salient edges in the attribute image, the tracking results in lineswith a relatively high intensity value. This intensity value is not related to the gradientmagnitude value, but indicates that all the paths started on that edge, follow the sameroute.

To enhance the edges where most of the tracked paths coincide, we added two extra stopcriteria based on heuristics to the tracking algorithm. If during the tracking the gradientmagnitude value of a new point differs more than 40% from the average value along thepath, the tracking is stopped. Assuming that an edge ends if the gradient magnitude valuechanges rapidly, this should prevent the path from continuing when it is no longer on theedge. If the angle between two subsequent displacement vectors is larger than 40 degrees,the tracking is stopped as well to eliminate incoherent paths. The result of this conditionaltracking algorithm is shown in figure 5.3f. The edge where the paths are consistentlydefined are clearly visible in this image. The higher the intensity value the more consistentthe tracking. Again we see that the tracking algorithm is able to follow the banks of the

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5.1 The tracking of line-like curvilinear structures 87

(a) Attribute image (b) Three channels

(c) Gradient magnitude (d) Edge detect

(e) Tracking (f) Conditional tracking

Figure 5.3: Tracking of the structures in a 2D cross-section (a) of a seismic attribute image.Three channels in this cross-section are manually highlighted in (b). The trackingresult using the edge points (d) as the starting positions is shown in (e). Theconsistently tracked edges are enhanced by applying extra stop criteria (f). In bothcases we use a maximum number of step n = 100, and a step-size of one pixel.

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88 Chapter 5 Structural analysis using a non-parametric description

0P, 00 κφ )(

P2P1 Pn

−1P−2P

−nP

Figure 5.4: Tracking a structure by following the displacement vectors. The structure is trackedfrom the starting point P0 in both directions φ0 and −φ0. The tracking is stoppedafter n steps.

channel on the left and the channel in the middle, but fails on the bottom-right channel.

5.2 Non-parametric adaptive filtering

In the previous chapters we have shown how the parameter estimates of the local structuremodels can be used to control adaptive filters. In chapter 3 we applied four differentfilter types to the noise reduction in curved oriented textures. These filter types werein order of increasing complexity: isotropic, orientation adaptive (OAF), curvature andorientation adaptive (COAF), and ideal. The ‘ideal’ filter is the filter that exactly matchesthe underlying structure of the data. To obtain the most accurate filter result, the adaptivefilter should approximate the ideal filter as much as possible. On the other hand, thecomputational burden of the filter and the required parameter estimates should remainpractical.

In figure 5.5a we have drawn a space curve c, and three isotropic analysis windows atthree different scales (1-3). As mentioned before, the goal of the adaptation of the filter,is to approximate the shape of the structure a well as possible. Figuratively speakingwe want to ‘deflate’ these isotropic windows so they wrap around the curve tightly. Tobe able to do so, we need a mathematical description of the curve inside the isotropicwindow. Describing the space curve at scale 1 using a linear model will probably resultin a high confidence value. At a larger scale (2) the curvature of the curve becomes moreimportant and a quadratic model is required to obtain a confident description. Making theparametric model flexible enough to describe the part of the space curve inside the thirdwindow, requires more parameters. The estimation of these extra parameters increasesthe computational burden, and the applicability of the higher order parameter estimatesis limited.

Instead of using a more flexible parametric description, we choose to use a non-parametricdescription. The parameter estimates of the quadratic local model at scale 2 are used todescribe the curve at larger scales (e.g. 3) as piece-wise quadratic. The actual shape of the

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5.2 Non-parametric adaptive filtering 89

12

3

c

(a) Window scales (b) Non-parametric window

Figure 5.5: Analyzing a space curve at three different scales (a). The largest scale requires anon-parametric window shape (b).

structure is found by tracking. The tracking is performed as described in section 5.1.2 andis schematically depicted in figure 5.4. If P0 is the starting point and t0 is the estimatedtangent at this point, then the tracking algorithm is initiated in both the t0 direction andin the −t0 direction. The tracking is stopped after a fixed number of steps n and noadditional stop criteria are used. The non-parametric adaptive filter is created in the sameway as the adaptive filters presented in chapter 2 and 3. First the appropriate filter scaleis selected by applying an isotropic filter to the image. Next, a non-parametric filter iscreated at each point in the image by sampling and a weighted averaging of the isotropicoutputs along a tracked path that is piece-wise defined by the parameter estimates of the(curvature corrected) structure tensor. The resulting non-parametric filter shape for thecurve c of figure 5.5a is shown as an example in figure 5.5b.

For the visual and quantitative comparison of the non-parametric filter with the otheradaptive filters, we will use the 2D test image It of a curved oriented texture given by

It(x, y; σn) = It(r) = cos

(

8r

)

+ N(0, σ2

n) , r =√

x2 + y2, (5.4)

where N is normally distributed noise. A realization of It(x, y; σn) with σ2n = 0.5, in a

128*128 pixel image, is shown in figure 5.6a. The orientation and curvature estimates usedto control the filters are computed by applying the tensor Tcurv2D at scale (σg = 1, σT = 5),and they are depicted in figure 5.6e and f. The result of the non-parametric adaptive filter(NPAF) applied to the test image, is shown in figure 5.6b. For the visual comparisonof the different adaptive filters we have shown the results of the COAF and the OAF infigure 5.6c and d. In all three filter results we have indicated one instance of the filterwith a white drawing. From these drawings one can see that in the high curvature areathe non-parametric filter is able to match the local structure, but the COAF is not. As aresult the NPAF yields visibly better result in this area.

We performed a quantitative comparison of the filters by computing the signal-to-noiseratios (SNR) of the filtered images. Since the noise-free version of the test image I =

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90 Chapter 5 Structural analysis using a non-parametric description

(a) Test image (b) NPAF (c) COAF

(d) OAF (e) Orientation (f) Curvature

Figure 5.6: Using adaptive filters (b-d) to improve the SNR of a curved oriented texture (a).The orientation and curvature estimate used to control the filters are shown in (e,f).One instance of each is filter is indicated with a white drawing superimposed on thefilter result.

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5.2 Non-parametric adaptive filtering 91

It(x, y; 0) is known and (max(I) = 1, min(I) = −1), we compute the SNR using

SNR = 20 log10

(

2

σn

)

dB. (5.5)

The standard deviation σn of the remaining noise in the filtered images filter(It), is esti-mated by

σn = STDDEV[filter(It) − I]. (5.6)

The resulting global signal-to-noise ratios and standard deviations estimates of the testimage after noise reduction are given in table 5.1. In the right column we have shown thesignal-to-noise ratios estimated in the high curvature area, i.e. for r < 20.

Filter σn SNR (dB) SNR1 (dB)Ideal 0.147 22.7 21.7NPAF∗ 0.147 22.7 21.7NPAF 0.152 22.4 21.2COAF 0.162 21.8 18.5OAF 0.238 18.5 14.5Isotropic 0.559 11.1 11.1None 0.710 9.0 9.1

Table 5.1: A quantitative comparison of noise reduction filters applied to a circular orientedtexture. The quantities σn and SNR are computed globally, SNR1 is computed forthe high curvature area (r < 20).

All the filters of table 5.1 use a Gaussian window, and the unfiltered result ‘None’ is addedto indicate the SNR of the original test image. The isotropic filter is applied at scaleσ =

√5 and all the other filters at scale σ1 = 51. The scale of the isotropic filter is chosen

such that the area of the corresponding window is equal to the area of the window of theadaptive filters. The ideal filter result indicates the highest attainable SNR at this filterscale. The ideal filter shape for this test image is a circular line piece, and the ideal filteris controlled by the true parameter values of the noise-free image.

From the SNR values in table 5.1 it is clear that the NPAF is the adaptive filter that yieldsthe highest SNR, and that this SNR is close that of the ideal filter. In figure 5.6 we sawthat the non-parametric filter is able to match the circular structure. To show that thisfilter can mimic the circular filter, we have applied it using the true parameter values ofthe noise-free image as control parameters (NPAF∗). The experimental results show thatthere is no measurable difference between the SNR of the NPAF∗ and the ideal filter. Onewould expect that the differences between the filters are the biggest in the high curvaturearea. Therefore we added an extra column to table 5.1 with the SNR1 estimated in thehigh curvature area, r < 20. It is clear that the SNR of the OAF changes the most, andthis filter performs even worse than the isotropic filter in this area. The NPAF still yieldsthe best results, close to the performance of the ideal filter.

1 The suffix 1 in σ1 indicates that the filter is one dimensional as described in chapter 2

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92 Chapter 5 Structural analysis using a non-parametric description

5.3 Non-parametric confidence estimation

In the previous section we computed a Gaussian weighted average of isotropic filter re-sponses along tracked paths to create an adaptive filter. These paths could be used in thesame way, to combine the analysis windows of the structure tensor. The advantage of alarger tensor window is that the continuity of the structure that is being analyzed, has moreinfluence on the confidence value. This allows for the segmentation between continuousstructures and fragmented structures. Straightforward application of the non-parametricadaptive filters described above to the averaging of the structure tensor elements, resultsin an unnatural description of the tracked structure. It describes the structure inside thenon-parametric window with a single parametric model, see figure 5.7a. The natural ex-tension of the structure tensor would be a piece-wise description as depicted in figure 5.7b.Both the presented linear and quadratic model can be used as the basis of this piece-wisemodel. For the remainder of this section, we will use the quadratic model.

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�������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

e1

e2

(a) Single model

e1

e2 e2

e2

e1

e1

(b) piece-wise model

Figure 5.7: The combination of structure tensor analysis windows along a tracked path. Aver-aging of the elements of the GST using a non-parametric window assumes a linearstructure inside this window (a). A piece-wise model (B) gives a better descriptionof a tracked structure.

Since the piece-wise model gives a non-parametric description of the data, it does notyield new (geometric) parameter estimates. However, the confidence value of the piece-wise model can be computed by averaging the eigenvalues of the parametric models itis composed of. First, the curvature corrected structure tensor is applied to the imageusing isotropic windows. This yields the eigenvalues of the tensor and an estimate of theorientation and curvature of the local structures. The orientation and curvature estimatesdefine, for each point in the image, a path along the local structure that can be foundby tracking. The tracking is performed similar to the tracking for the non-parametricfilter, and is schematically depicted in figure 5.4. If P0 is the starting point and t0 is theestimated tangent at this point, then the tracking algorithm is initiated in both the t0

direction and in the −t0 direction. The tracking is stopped after a fixed number of stepsn and no additional stop criteria are used. The eigenvalues of the non-parametric model(λi)NP are found by averaging the eigenvalues (λi)m of the N parametric models m that

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5.3 Non-parametric confidence estimation 93

correspond to the points of the tracked path

(λi)NP =1

N

N∑

m=0

(λi)m , N = 2n + 1. (5.7)

These averaged eigenvalues can be used to compute the confidence value of the non-parametric model. In the two-dimensional case one can compute the confidence valueusing

(C)NP =(λ1)NP − (λ2)NP

(λ1)NP + (λ2)NP

. (5.8)

This confidence value indicates how well a structure can be modeled as piece-wise quadratic.

5.3.1 Application to channel detection

One of the practical goals of this thesis is to develop tools for the automatic detection ofchannels and related sedimentary structures. In chapter 4 we used the confidence value(C)QGST of the structure tensor based on the quadratic curve model, for the detection ofchannels in a seismic volume. The channels give rise to a higher confidence value than thebackground, but there are still a lot of other structures, in the data with a confidence valuecomparable to that of the channels. The visually most distinguishing feature between thechannels and the other structures is the larger spatial continuity of the channels. Thiscontinuity is not exploited by the local analysis of the structure tensor. The quadraticparametric model did however give a confident local description on most parts of thechannels. Earlier in this chapter, we utilized the corresponding parameter estimates forthe tracking of channels. This tracking method was able to follow the channels, except forthe location where a channel makes a radical turn. The non-parametric model describedabove uses these tracked paths to increase the analysis scale. Therefore we expect that theconfidence value of this model (C)NP , is more suitable for the detection of channels thanthe confidence value (C)QGST .

To experimentally verify this hypothesis we use both the confidence values (C)QGST and(C)NP for the detection of the channels in a 2D cross-section (512*256 pixels) of a seismicvariance attribute volume. This cross-section shown in figure 5.8a and is identical to the2D image in figure 5.3a, that was used for the tracking of channels. The manual channeldetection is shown in figure 5.8b, and will be used for the evaluation of the automaticdetection. We applied the curvature corrected structure tensor at scale (σg = 2, σT = 4) tothe 2D cross-section and the confidence estimate is shown in figure 5.8c. The confidencevalue (C)NP of the non-parametric model is computed using eq.(5.8). The eigenvalue ofthe structure tensor are averaged along the tracked paths defined by the orientation andcurvature estimate. The number of steps n of the tracking algorithm was set to 50, andthe step size to one pixel. The result of the (C)NP estimation is shown in figure 5.8e.

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94 Chapter 5 Structural analysis using a non-parametric description

(a) Channel image (b) Manual detection

(c) (C)QGST (d) Detection using (C)QGST

(e) (C)NP (f) Detection using (C)NP

Figure 5.8: Channel detection based on the confidence estimate of the quadratic structure tensor(C)QGST (d) and the non-parametric model (C)NP (f). The visual comparison ofthese detections with the manual detection (b) shows that (C)NP yields a betterdetection.

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5.3 Non-parametric confidence estimation 95

The confidence values do not give a segmentation of the image, but an estimate of the‘channelness’ of each point in the image. For a segmentation or classification into theclasses ‘channel’ and ‘non-channel’ based on the confidence estimation, a decision boundaryor threshold value should be introduced. All the points with a confidence value above thethreshold value, are classified as ‘channel’, and all other points as ‘non-channel’. Thesegmentation of the confidence values (C)QGST , (C)NP using a threshold value Ctr = 0.75are shown in resp. figure 5.8d,f. The white regions form the channel class and the blackregion the non-channel class.

The threshold value Ctr = 0.75 was chosen manually and optimized by visual comparisonwith result of the manual detection. This optimization is clearly very subjective. A moreobjective method to optimize the threshold value is to ‘learn’ it from examples. Manysupervised statistical classifiers are developed in the pattern recognition community [Bis95].classifiers do however require a set of examples of both classes. Since the ground truth inthis detection problem is not known, these examples have to be selected manually by thehuman expert.

Comparison of the channel detection in figures 5.8d and f, shows that the detection basedon confidence value of the non-parametric model is less fragmented. The three channelsof the manual detection are partly found in both cases, but in figure 5.8f they form threecontinuous regions. Furthermore, the number of false positives is reduced significantlydue to the incorporation of the continuity of structures in the confidence estimation. Weconclude that the hypothesis that (C)NP is more suitable for the detection of channelsthan the confidence value (C)QGST , is correct. For the determination of the false positivedetections we have used the manual detection as the ground truth. The largest falsepositive in the top-left corner of figure 5.8f could however be a correct detection. This isnot clear from this cross-section. Inspection of the 3D seismic data showed that in fact apart of another channel was found.

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6. Coherency estimation

Sedimentary rock consists of petrified layers of sediments that were deposited on top of eachother in previous geological times. An individual layer forms a surface that approximatelycorresponds to the earth’s surface where the deposits were made upon. Therefore the layersusually form continuous1 surfaces. Discontinuities in the layered structure of sedimentaryrock often reveal geologically interesting processes. Discontinuities can for example becreated by faults1 or incisions made by channels. The continuity of the sedimental layersis often called the bedding continuity.

The continuity that can be measured in a seismic image is the reflection continuity. There-fore we need to link the bedding continuity to the reflection continuity. The rock propertythat is measured by the seismic reflection method is the acoustic impedance. If we couldmeasure this property directly we would obtain a 3D data volume with in each point theacoustic impedance. During the seismic acquisition an acoustic wave package is generatedat the surface. This package travels down the subsurface that acts like an acoustic filterand partly reflects where the acoustic impedance changes. The reflected wave packagesare recorded at the surface. The amplitude of this wave package can be used for the de-termination reflectivity and the travel time for the determination of the depth where thepackage is reflected. The conversion of an acoustic impedance volume to seismic amplitudevolume consist of two steps. First the reflectivity is determined by computing the verticalderivative. Next, the filtering effect of the subsurface is taken into account by verticallyconvolving the reflectivity with the seismic wavelet that was measured at the surface. Thetwo vertical operations of the conversion change the intensity values, but the geometry ofthe layered structure remains unchanged. The reflection continuity therefore correspondsdirectly to the bedding continuity.

Coherency is a measure for the reflection continuity in a seismic image. It was first in-troduced in literature as a seismic attribute by Bahorich and Farmer [BF95]. Inspired byhorizon tracking algorithms they used the cross-correlation between vertical lines of datafor the estimation of coherency. Bahorich et al. showed that by making the continuity ex-plicit, the delineation of both faults and channels becomes easier, especially in time slices.Time slices of seismic data are difficult to interpret because they often cut through differentreflectors. Traditionally this is solved by tracking of the reflector of interest and creating2D map view of the intensity values on created horizon. The drawback of horizon tracking

1 Here we refer to continuity in the mathematical sense. Consider a surface in 3D, z = f(x, y) = f(x).The surface is said to be continuous if limx→x0

f(x) = f(x0) for all x0.

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98 Chapter 6 Coherency estimation

is that this process is not yet fully automated, and therefore it remains time consuming.Due to the ever increasing computing power and size of computer memory, the computa-tion of 3D seismic attributes at each point of 3D seismic data volume became feasible. Itappeared that the interpretation of a time slice of a coherency attribute volume is muchless hampered by the fact that it cuts through different reflectors. This can be explainedby the fact that each point in the attribute volume is computed using a neighborhood inthe seismic image around the point. The computation of a 3D attribute volume can bedone ‘off-line’, i.e. without the intervention of the interpreter, and is therefore not timeconsuming for the interpreter.

As mentioned before, reflectors are generally not parallel to the xy-planes or time slices ofthe seismic data. The estimation of coherency can therefore not be performed accuratelywithout the estimation of the orientation of the reflector. This orientation estimationwas incorporated into the cross-correlation algorithm by searching for the maximum cross-correlation of a user-defined set of lagged or skewed vertical data line pairs. The drawbackof this algorithm based on the cross-correlation between individual lines of data, is that itsestimates for coherency and orientation are not robust with respect to noise.

The estimation can be stabilized by increasing the size of the analysis window. Coherencycan be defined as the resemblance of the local data to a planar reflector. Examples ofmathematical resemblance measures used for the estimation of coherency are semblance[MKF98] and correlation [GM99]. A seismic image of a single constant planar reflector isjust a stack of isophote planes, and it therefore has a plane-like linear structure. Since theconfidence value Cplane of the GST is a measure for the resemblance of an image structureto a plane-like linear structure, it can be used as an estimate of coherency as well.

In this chapter we will compare the confidence estimate of the GST with the coherencyestimate based on the eigenstructure of the covariance matrix described in [GM99]. Westart with the mathematical definition of both methods. Next, we will show the effectof structural dip on the coherency estimation. Two methods for the compensation ofstructural dip are discussed and compared. We finish this chapter with the experimentalcomparison of the coherency estimates applied to the detection of faults.

6.1 Coherency based on the eigenstructure of the

covariance matrix

The first coherency measure was based on the cross-correlation between neighboring win-dowed traces, the one-dimensional time signals of a seismic image I(x = x0, y = y0, t).If we compute this coherency measure using a time window of N samples, we need tocorrelate two data vectors d1 and d2, with N samples each. This yields the off-diagonal

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6.1 Coherency based on the eigenstructure of the covariance matrix 99

element C12 of a 2 by 2 covariance matrix

C =

[

C11 C12

C21 C22

]

=1

N

[∑N

n=1d2

n1

∑Nn=1

dn1dn2∑N

n=1dn1dn2

∑Nn=1

d2n2

]

, (6.1)

where C11 and C22 are the auto-covariances of the data vectors d1 and d2. The word tracecan either refer to the sum of the diagonal elements of a matrix or a seismic time signal.Since the only data vector we use in this section are those corresponding to windowedseismic traces, we use ‘data vector’ as synonym for ‘windowed seismic trace’.

For the generalization to the covariance matrix of J data vectors, we define the data matrixD. This JxN matrix is given by

D =

d11 d12 . . . d1J

d21 d22 . . . d2J...

.... . .

...dN1 dN2 . . . dNJ

, (6.2)

where each single column represents a single data vector with N samples. If we assumethat each data vector has a zero mean, then the JxJ data covariance matrix is given by

C =DT · D

N, (6.3)

where the dot denotes the matrix product. Since the definition is based on the correlationbetween data vectors, which are one-dimensional signals, it can be applied to both 2D and3D seismic images without change. Applied in 3D using a rectangular analysis windowwith sizes (wx, wy, wt), we have N = wt and J = wx · wy.

If all J data vectors in matrix D are identical, which is the case for a flat planar reflectorwithout noise, then the rank of C is one. This means that C has only one non-zeroeigenvalue. An increase in the variability of the traces, e.g. due to a discontinuity, causesan increase in the number of non-zero eigenvalues. The resemblance of the local data toa flat continuous planar reflector can therefore be measured by the fraction of the totalsignal energy that is captured by the largest eigenvalue of C. It can easily be checked thatthe total signal energy E, is equal to the sum of the diagonal elements of C, known as thetrace Tr(C). From linear algebra [GvL96], we know that the trace of a matrix is invariantunder orthonormal transformations. Therefore we have

E =

J∑

j=1

N∑

n=1

d2

nj =

J∑

j=1

cjj = Tr(C) =

J∑

j=1

λj, (6.4)

with dnj, cjj the elements of D,C and λj the eigenvalues of C. If λ1 is the largest eigenvaluethen the coherency estimate based on the eigenstructure of the covariance matrix is givenby

ccov =λ1

Tr(C). (6.5)

This measure was presented as an estimate of seismic coherency in [GM99].

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100 Chapter 6 Coherency estimation

6.2 Coherency estimation using the GST

An image of a single planar reflector can be described as a stack of isophote planes. Theconfidence value Cplane of the gradient structure tensor (GST) can therefore be used as ameasure of the resemblance of the local data to a planar reflector. The GST was presentedand discussed in detail in chapter 2. In this section we will discuss the GST only inrelation with coherency estimation. The gradient structure tensor T is defined as theaveraged dyadic product of the gradients g

T = ggT . (6.6)

For most practical purposes, the tensor can be treated as a NxN matrix, with N thedimensionality of the data. The eigenvalues of this tensor indicate the gradient energyin the orientations defined by the corresponding eigenvectors. In the case of a planarreflector the tensor has only one non-zero eigenvalue, and the corresponding eigenvectoris the normal vector of the reflector. Any deviation of the data from a constant planarreflector leads to an increase of the gradient energy in the lateral direction. The coherencyof the GST could therefore be estimated by

cgst =λ1

Tr(T)(6.7)

Although the definitions of ccov and cgst seem identical, they are not. The eigenvalues ofthe covariance matrix represent the correlation between seismic traces and the eigenvaluesthe GST represent the gradient energies of a geometrically ordered set of traces. Thismeans that the reflector continuity is measured with the correlation between traces by thefirst method, while the second method uses the gradient energy in the lateral direction asa measurement for continuity. For simplicity, we will now consider only two traces (t1, t2),both with a zero mean value

i

t1i = 0 ,∑

i

t2i = 0 (6.8)

The normalized correlation C between the traces is defined as

C =t1 · t2

‖t1‖‖t2‖. (6.9)

The gradient in the lateral direction can be defined as the difference between the traces.The corresponding lateral gradient energy El is in that case given by

El = ‖t1 − t2‖2 (6.10)

If the gradient energy El is zero then the correlation C is one. However, if the normalizedcorrelation between the traces is one, then the gradient energy is not necessarily zero. Inpractice this means that for a perfectly flat reflector with an increasing amplitude ccov = 1,but cgst < 1.

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6.3 The presence of structural dip 101

6.3 The presence of structural dip

The coherency estimate ccov is based on the correlation of windowed traces. The optimalcorrelation measurement is obtained when the correlation is computed between 1D subsetsperpendicular to the reflector surface. The estimate ccov is therefore optimal if the reflectorbeing analyzed is parallel to the xy-plane, and becomes less accurate in the presence ofstructural dip. Here structural means, consistent at a larger scale not determined byrandom fluctuations. The compensation of structural dip requires the estimation of thisdip and the coherency estimate ccov should be computed after transforming the data insuch a way that the reflector becomes horizontal. In figure 6.1 a tilted reflector is drawn.Window A in this figure is the xt-aligned window of ccov, and B is the effective windowafter dip compensation.

BA

t

x

Figure 6.1: Coherency estimation in the presence of structural dip in an xt-aligned analysiswindow (A) yields an sub-optimal result. The window can be adapted (B), usingan estimation of the structural dip.

To demonstrate the negative effect of structural dip on the estimate ccov, we applied it toa 2D seismic image (128*128 pixel) with a lot of faults, see figure 6.2. We next estimatethe orientation of the reflectors using the GST at scale (σg = 1, σt = 5). We then use thisorientation estimate to locally rotate the data in such a way that the reflectors becomehorizontal before estimating the coherency. The result of this adaptive version of ccov isshown in figure 6.2e. A 5 by 11 pixel coherency window was used in both cases. For thenon-adaptive windows this means that the coherency was computed over 5 traces using atime window of 11 pixels, and the widths of the adaptive windows were 5 pixels laterallyand 11 pixels in the perpendicular direction. The results show a clear improvement due tothe dip compensation.

To show the difference between ccov and the semblance based coherency estimate presentedin [MKF98], we have repeated the experiment using the semblance based coherency. Thesemblance based coherency estimate is given by

csem =u · C · uTr(C)

, u =1√J

11...1

, (6.11)

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102 Chapter 6 Coherency estimation

(a) 2D seismic image (b) Covariance (c) Semblance

(d) GST Orientation (e) Adaptive Covariance (f) Adaptive Semblance

Figure 6.2: Coherency estimation of a 2D seismic image using the eigenstructure of the co-variance matrix (b) and semblance (c) in an xt-aligned window. The orientationestimate (d) of the GST is used to compensate for the structural dip (e,f).

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6.3 The presence of structural dip 103

with C the J by J covariance matrix defined in (6.3). The results of csem and its orientationadaptive version are shown in figures 6.2c and f. The results show that the faults highlightedby ccov are highlighted by csem as well. However, the eigenstructure based coherencyestimate has a narrower fault response. A more detailed experimental comparison is givenin [MSG+99] and a theoretical comparison is given in [GM99].

The choice of the window size for the coherency and dip estimation, is a trade-off due to theuncertainty principle in filtering [WG84, Dau85]. A small window yields an accurate, highresolution measurement, but is sensitive to noise. A larger window yields a more robustmeasurement with respect to noise, but with a lower resolution. The optimal windowsize for the coherency measurement is not necessarily the optimal window size for the dipmeasurement.

6.3.1 Dip estimation by sampling the dip dependency

We have demonstrated that compensating for structural dip significantly improves thecoherency estimation. We used the GST to estimate the orientation of the reflector, androtated the data before estimating coherency. The dip compensation for the semblancebased method [MKF98] was done by a straight forward dip search. The idea of this dipsearch is to compute the coherency for all possible dips, and the trial dip that yields thehighest coherency value corresponds to the actual dip of the reflector. The coherency as afunction of both the position (x, y, t) and the apparent dips (p, q) is defined as

c(x, y, t, p, q) = c(x, y, t − px − qy). (6.12)

The estimate of the reflector dip d is given by

d =√

p2 + q2 , (p, q) = arg maxp,q

c(x, y, t, p, q). (6.13)

The sample spacing of the apparent dip parameters ∆p and ∆q, depends on the sizes(wx, wy) of the coherency window and the highest temporal frequency component fmax

contained in the seismic data. The Nyquist sampling criterion gives the following expres-sions

∆p =1

2wxfmax

, ∆q =1

2wyfmax

. (6.14)

In [MKF98] they assume that the interpreter is able to estimate the maximum true dip,dmax, to reduce the range of the dip search. Using this maximum dip, c(x, y, t, p, q) has tobe computed over np · nq discrete dip pairs (p, q), where

np =2dmax

∆p+ 1 , nq =

2dmax

∆q+ 1. (6.15)

For example, if the image is critically sampled 1/fmax ≈ 2 pixels, the maximum dipdmax = 1 pixel (45 degrees), and the window sizes are wx = wy = 5, then c(x, y, t, p, q)needs to computed over 121 dips.

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104 Chapter 6 Coherency estimation

In [MSG+99], Marfurt et al. applied this dip compensation to the coherency estimatebased on the eigenstructure of the covariance matrix. They found that the high faultdiscrimination and resolution of this algorithm reverted to that of the semblance basedmethod. For high resolution fault highlighting a small lateral window size is needed, e.g.wx = wy = 5. Estimating the dip near a fault in such a small window does not yield thedesired result. The apparent dips that give the maximum coherency value appeared tocompensate the offset of the fault rather then the structural dip. The structural dip needsto be estimated at a larger scale.

1

t

x

2

Figure 6.3: Dip estimation near a fault at two scales (1,2), indicated by the gray discs. Thewhite rectangles indicate the effective windows that yield the maximum coherencyestimate c(x, y, t, p, q).

A schematic representation of dip estimation at two scales is depicted in figure 6.3. The graydiscs indicate the desired scales for the coherency estimation (1), and the dip estimation (2).The white rectangles indicates the effective windows that yield the maximum coherencyestimate c(x, y, t, p, q).

6.3.2 Comparison between the dip estimates of the GST and

dip search

In this section we will compare the dip estimates of the GST dgst and of the dip search

method dds. The GST does not yield an estimation for dip directly. The first eigenvector,belonging to the largest eigenvalue, of the GST gives an estimate of the normal of thereflector plane. The dip can be derived from this vector by

φ = arccose1 · ez

‖e1‖‖ez‖, dgst = tan(

π

2− φ), (6.16)

where e1 is the first eigenvector of the GST, ez is a vector along the z-axis, and φ is theangle between e1 and ez. The differences in the key features of the GST and dip searchanalysis are summarized in table 6.1, and will be discussed in descending order.

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6.3 The presence of structural dip 105

GST Dip search

uni-modal multi-modalrotation invariant skew invariantgradient based covariance based

Table 6.1: Key features of the dip analysis of the GST and dip search.

For the determination of the modality of the methods, we can interpret both methods asthe measurement of the resemblance R(d) between the local data and a planar reflector asa function of the dip2. The dip search method samples the dip parameter di, and measuresR for all instances i in the search range. In this way the distribution of R as a function ofd is measured allowing multiple peaks or maxima as a function of the parameter at a singleposition in the image. This is per definition a multi-modal parameter estimation. Theorientation analysis of the GST on the other hand, is uni-modal. It implicitly assumes thatthere is only one orientation at a single position in the image, see chapter 2. In figure 6.4we have drawn three reflector configurations: a single reflector (a), two touching reflectors(b), and two intersecting reflectors (c). A multi-modal analysis is able to represent all threeconfigurations, whereas an uni-modal analysis is limited to the case of the single reflector.However, in chapter 2 we introduced the generalized Kuwahara filter to avoid filteringacross borders. Combined with the GST, this filter yields a correct orientation estimate oftouching reflectors as well. Since intersecting reflectors have no geological meaning, it isnot necessary to deploy a multi-modal analysis for the estimation of reflector orientationor dip.

(a) single (b) touching (c) intersecting

Figure 6.4: Three different reflector configurations.

The methods also differ in their invariance. The GST is rotation invariant and the dipsearch is skew invariant. The parameter estimate of the GST, orientation, indicates therotation needed to level the reflector. The dip estimate is the amount of skewing necessaryto level the reflector. The difference between a rotated and a skewed analysis windowincreases with the dip. In figure 6.5, we have drawn a skewed and a rotated window ontop of each other for two different angles: 20 and 45 degrees. The advantage of skewingis that if wx = wy < wt the spatial and temporal window sizes do not dependent onthe amount of skewing. Furthermore, transformation of the data based on skewing onlyrequires a one-dimensional interpolation along the t-axis. Rotation on the other hand

2 For simplicity we assume that the azimuth is known. In practice the estimation of the reflector diprequires the simultaneous estimation of two parameters.

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106 Chapter 6 Coherency estimation

treats all dimensions on equal footing. The advantage of rotation is that the shape of thewindow is unaffected. If an isotropic window is used, such as the Gaussian window used forthe computation of the GST, then the part of the image data inside the analysis windowis identical for all orientations.

Figure 6.5: The difference between rotated and skewed analysis windows, for 20 (left) and 45(right) degree angles.

The third difference between the two dip estimators is the resemblance measure. Thedifference of gradient versus covariance is already discussed in section 6.2. The experi-mental comparison between the GST and the covariance based coherency measures will beperformed in the next section.

To see the practical implications of the differences between the dip-estimators, we haveapplied both the GST and the dip search using ccov to the dip estimation on a 2D seismicimage. We then used the two dip estimates for the computation of the dip-compensatedcoherency based on the covariance eigenstructure and the results are shown in figure 6.6.The dip search and all coherency measurements are performed in a wx = 5 by wt = 11window. The GST is applied at scale (σg = 1, σT = 5). The dip search estimate dds andits corresponding coherency measurement are shown in figures 6.6a and d. To increasethe effective scale of the dip estimate dds, we filtered it using a isotropic Gaussian filter(σ = 5), see figures 6.6b. Contrary to orientation, dip is not periodic and may thereforebe smoothed directly without the need of a mapping.

If we analyze the results in figure 6.6, we see that the dip estimate dds computed at asmall scale (a), shows rapid changes near the faults. As depicted in figure 6.3, the throwof the fault influences the dip estimate. Smoothing of the dip estimate dds (b) practicallyremoves this influence. A comparison of the corresponding coherency estimates (d,e) showsthat increasing the scale of the dip estimation improves both the discrimination and theresolution of the fault-highlighting. The coherency estimate using GST dip dgst (f), doesnot visibly differ from the result using the smoothed dip (e). In (h) we have shown theresult of rotating the data to compensate the reflector dip using GST orientation (i). If wecompare (f) with (h), we see that for the two faults in the right half of the image, rotationyields a more continuous fault-highlighting. The discrimination of the two faults in thebottom-left corner is less in (h).

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6.3 The presence of structural dip 107

(a) Dip search (b) Smoothed (c) GST dip

(d) (e) (f)

(g) 2D seismic image (h) OA Coherency (i) GST Orientation

Figure 6.6: Dip estimation on a 2D seismic image (g) using both the dip search (a) as theGST (c). A Gaussian smoothed (σ = 5) version of (a) is shown in (b). Dip adaptivecoherency estimates using the dip estimates (a,b,c) are shown in (d,e,f) respectively.The GST orientation and corresponding adaptive coherency are shown in (h,i) forcomparison.

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108 Chapter 6 Coherency estimation

6.4 An experimental comparison for fault detection

In the sections above we discussed the coherency estimation ccov based on the eigenstructureof the covariance matrix [GM99] and we showed that the eigenvalues of the GST can beused for the estimation of coherency as well. In chapter 2 we described the application ofthe GST to the detection of faults. In section 2.4, we showed that a combination of theeigenvalues of the GST, Cfault, is able to highlight faults. We used this highlighting asthe basis for the detection of faults in 3D seismic data. In this section we will compareCfault with the coherency estimate ccov. Furthermore, we will compare the automatic faultdetection based on these to measures, by applying the same post-processing steps to ccov aswe applied in the fault detection application. In the previous section we have addressed theeffects of structural dip on the coherency estimation. We have shown that the orientationestimate of the GST can be used to compensate for this dip, and we will use this dipcompensation for all coherency measurements in this section.

We perform the fault analysis on the same 3D seismic image (1283 voxels) as we used insection 2.4. We start by highlighting the faults using both Cfault and ccov. The results onthe 3D image are visualized in figure 6.7 by three perpendicular 2D cross-sections, namelya xt, a yt, and a xy-slice or time-slice. To allow easy visual comparison between ccov andCfault, we have displayed (1 − Cfault). The coherency estimate ccov was computed in a(21∗5∗5) window, and Cfault was computed at scale (σ1 = 6, σ2 = σ3 = 2). The elongatedwindows are steered perpendicular to the reflector planes using the orientation of the firsteigenvector of the GST at scale (σg = 1, σT = 9). Visual comparison of the results infigure 6.7, shows that the GST based highlighting is smoother than the covariance basedhighlighting. This could mean that the resolution of ccov is higher. With the term resolutionwe refer to the ability to resolve faults that are in close proximity of each other. A lowerresolution does not necessarily result in a worse localization of the faults. The two closestfaults visible in figure 6.7 in the bottom-left quadrant of the time slice (g,h,i), are stillresolved by the GST based highlighting. Therefore we are not able to confirm the lowerresolution. With respect to the discrimination between faults and continuous reflectors,both methods perform equally well. In some cases Cfault performs better, in some casesccov.

We next use the fault highlighting shown in figure 6.7 for the segmentation faults in theseismic image as described in 2.4. The ‘faultiness’ of a position in the image is given byCfault and (1−ccov). First we apply a non-maximum suppresion. The candidate fault-pointsare reduced to the local maxima in the faultiness images. These maxima are computedin a one-dimensional window (5 ∗ 1 ∗ 1) along the gradient direction. We next take thevalue of the faultiness of the candidate points into account using two threshold values,Tl = 0.1, Th = 0.4. Points with a faultiness below the low threshold Tl value are rejected.The final segmentation is obtained by accepting only those connected segments whosemaximum value is above the high threshold value Th. The resulting fault detections basedon both the GST and the covariance analysis are shown in figure 6.8. The covariance based

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6.4 An experimental comparison for fault detection 109

detection is more sensitive to small variations in the data than the GST based detection.This is consistent with the smoother fault highlighting of the GST.

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110 Chapter 6 Coherency estimation

(a) Slice y = 34 (b) GST (c) Covariance

(d) Slice x = 113 (e) GST (f) Covariance

(g) Slice t = 64 (h) GST (i) Covariance

Figure 6.7: Coherency estimation on a 3D seismic image containing many faults, visualized inthree 2D cross-sections, a xt (a),a yt (d), and a xy-slice or time-slice (g). The GSTbased coherency was computed at scale (σ1 = 6, σ2 = σ3 = 2), and the covariancebased coherency in a window with sizes (w1 = 21, w2 = w3 = 5). The elongatedwindows are steered perpendicular to the reflector planes using the first eigenvectorof the GST applied at scale (σg = 1, σT = 9).

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6.4 An experimental comparison for fault detection 111

(a) Slice y = 34 (b) GST (c) Covariance

(d) Slice x = 113 (e) GST (f) Covariance

(g) Slice t = 64 (h) GST (i) Covariance

Figure 6.8: Fault detection based on the fault highlighting shown in figure 6.7.

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7. Conclusions

Is it possible to improve the efficiency of the seismic interpretation process by automatingcertain interpretation tasks? In this thesis an image processing approach to this question ispresented. We limited the broad range of seismic interpretation tasks to horizon tracking,fault detection, and channel detection in 2D and 3D migrated seismic reflection images.The global structure of a seismic image of sedimentary rock is determined by reflectorsurfaces stacked on top of each other. The interpretation tasks mentioned above all requirean estimation of the global structure. We found that the faults and the channels manifestthemselves as deviations from the global structure. Futhermore, automatic horizon trackingcould be improved by enhancing the global structure before tracking. The first step towardsthe automation of the interpretation tasks therefore is to model the global structure in thedata and to find or create image processing algorithms for the estimation of the model-parameters.

In addition to the inherent difficulty of finding the most effective structure model, weface a practical problem regarding the vast size of seismic datasets1. This demands for aminimization of the computational complexity of the image processing algorithms used.

7.1 Image processing approach

The image processing approach chosen in this thesis is the local analysis of the imagestructures. We have presented several geometric structure models in both two and three-dimensions.

We started the structural analysis of images by assuming local shift invariance and wecalled an image structure with shift invariance in one or more orientations, a linear struc-ture. The gradient structure tensor (GST) locally models an image as a linear structure.This is equivalent to describing the image structure as locally straight (line-like structures)or planar (plane-like structures). The GST yields estimates of both the parameters, orien-tation, and the confidence of the linear structure model. The confidence estimate decreasesat image locations where the neighborhood deviates from the linear structure model. An

1 A seismic dataset can take up to a couple of Terabytes of computer storage space. During the timethis thesis was written, an average workstation contains a storage capacity smaller than 0.1 Terabyte andapproximately 2 Gigabytes of memory.

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114 Chapter 7 Conclusions

example of such a neighborhood is the area around a border between two oriented textures.The confidence estimate can therefore be used to these highlight borders.

The generalized Kuwahara filter avoids filtering across borders by allowing decentralizedwindows and selecting the decentralized window with the highest confidence value. Theuni-modal orientation estimate of the GST gives near orientation borders a weighed averageof the two orientations on both sides of the border. We have shown that this orientationestimate can be corrected by combining the GST with the generalized Kuwahara filter.Compared to a multi-orientation approach, this method has a low computational complex-ity. The corrected orientation estimate makes it possible to correctly steer an orientationadaptive filter over images composed of several touching oriented textures. An orientationadaptive filter can be made edge or border preserving by combining it with a steered ver-sion of the generalized Kuwahara filter. This edge preserving orientation adaptive filtergives results that are visually comparable to the results of anisotropic diffusion, but is atleast one order of magnitude faster.

As mentioned before, the confidence estimate of the GST decreases at image locationswhere the local structure deviates from the linear structure model. A frequently occurringdeviation in linear structures is curvature. For the description of line-like and plane-likecurvilinear structures, we introduced the curvature corrected GST based on a quadraticcurve model and the quadratic surface model respectively. Both models yield a curvaturecorrected confidence estimate and an estimate of the curvature as a valuable by-product.We have used the orientation and curvature estimate of the corrected GST to controlan adaptive filter for the reduction of noise in curved oriented textures. The curvatureand orientation adaptive filter (COAF) yields both a visibly better result and a highersignal-to-noise ratio improvement than the orientation adaptive filter (OAF).

Describing image structures, such as the meandering structure of a channel, at a ‘larger’scale with a simple geometric model, is often not possible. Instead of introducing a morecomplex parametric model with more parameters, we turned to a non-parametric model.The local parameter estimates obtained from the simple geometric models can be used fora piece-wise description of curvilinear image structures. The shape of the structure can befound by the tracking of this piece-wise description. We have applied the correspondingnon-parametric adaptive filter (NPAF) to the reduction of noise in curved oriented textures.The NPAF yields a signal-to-noise ratio improvement that is higher than that of the COAF,and approaches the improvement of the theoretically predicted ideal filter.

7.2 Application to seismic interpretation

The parameter estimates of the local geometric models can control the window shape of afilter in such a way that it approximates the shape of the local structure. A low-pass filtercontrolled in this way can enhance the structure of a layered texture, and thus the structural

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7.2 Application to seismic interpretation 115

information in a seismic image. Automatic horizon trackers have been around for years andare available in almost every seismic interpretation software package. The problem howeveris that these trackers fail at many points in the image due to small deviations from thelayered structure. Accurate horizon tracking still requires a lot of user interaction in theform of providing many starting points for the auto-tracker. The structural information inseismic images can be enhanced by removing the deviations using an adaptive smoothingfilter controlled by the gradient structure tensor. We have shown that by combining theadaptive smoothing filter with the generalized Kuwahara filter, smoothing across faultscan be avoided.

The confidence estimate of the local straight model can be used for the detection of faults.The automatic detection of faults is presented as a two-step procedure. First an attributeis computed that measures the ‘faultiness’ at each point in the image. In the second stepthis faultiness is used for the segmentation of the faults. We have shown that a faultinessmeasure Cfault can be constructed from the eigenvalues of the GST. We compared thismeasure with the coherency estimate ccov based on the eigenstructure of the covariancematrix [MSG+99]. We found that Cfault yields a smoother fault-highlighting than ccov.For the manual interpretation of the fault-highlighting the more crisp fault responses ofccov are preferred, furthermore the erratic false responses of ccov are easily suppressed bythe human visual system. The automatic detection of faults, on the other hand, is nothampered by a smoother fault-highlighting as long as all the faults are spatially resolved.Moreover, the automatic discrimination between small erratic responses and true faultresponses requires an additional image processing step. This increases the computationaldemands of the detection based on the already more computational complex coherencyestimate ccov.

The automatic detection of channels appeared to be a more difficult task than the automaticdetection of faults. Due to the large natural variability in the shape of channels, it is notpossible to capture their characteristics in a simple geometric model. We have presented achannel detection method based on the confidence estimate of the structure tensor for 3Dline-like curvilinear structures. This method was able to detect most parts of the targetchannels. However, the detection was fragmented and many other sedimentary structureswere falsely detected as channels as well. The visually most distinguishing feature betweenthe channels and the other sedimentary structures is the larger spatial continuity of thechannels. we have shown that the exploitation of this continuity by elongating the analysiswindow along the channel using a piece-wise description of the channel shape, results ina more continuous detection and a significant reduction of the number of false-positivedetections.

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116 Chapter 7 Conclusions

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Summary

In this thesis we have examined the possibility of automating 3-D seismic interpretation,by making use of image processing techniques. We limited the broad range of seismicinterpretation tasks to horizon tracking, fault detection, and channel detection in 2D and3D migrated seismic reflection images. All these tasks require a geometrical descriptionof the earth layers, i.e. a quantitative structural interpretation. Earth strata manifestthemselves in seismic images as layered structures.

For the quantitative description of these structures we have decided to use a local dif-ferential geometric description by means of the structure tensor. The structure tensormodels the data as locally straight and yields both an estimate of the orientation of thestructure as a confidence measure for the model. This confidence measure can be used tohighlight and detect faults that manifest themselves as discontinuities. We have comparedour discontinuity based fault measure both theoretically and experimentally with a seismiccoherency measure from recent literature. The orientation estimate can be used to steer anadaptive filter. This steered filter can enhance the structural information in the image byreducing the noise in seismic image without degrading the layered structure. By combiningthis filter with the generalized Kuwahara filter, it can be made fault preserving as well.Furthermore, the fault detection can be improved by elongating the tensor window alongthe faults.

Since curved structures occur often in practice, we have introduced the curvature correctedstructure tensor. This corrected tensor yields an estimate of the curvature as a valuableby-product. We have described the curvature correction to both line-like and plane-likecurvilinear structures. The confidence measure of the line-like curvilinear model can beused to detect channels. To make better use of the continuity of channels, we have in-troduced a non-parametric structure description method. This method is able to describecomplex structures such as channels on a large scale without the need for additional geo-metric parameters.

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Samenvatting

In dit proefschrift is onderzocht of het mogelijk is om, met behulp van beeldbewerkings-technieken, driedimensionale seismische interpretatie te automatiseren. De interpretatie-taken waar we ons in eerste instantie tot beperkt hebben, zijn het vinden van horizon- enbreukvlakken en het vinden van geulen. Voor deze taken is een geometrische beschrijvingvan de aardlagen nodig, i.e. een structurele interpretatie. De aardlagen manifesteren zichin seismische beelden als gelaagde structuren.

Voor een kwantitatieve beschrijving van deze gelaagde structuren, hebben we gekozen vooreen lokaal differentieel geometrische beschrijving met behulp van de structuurtensor. Destructuurtensor modelleert de data als lokaal recht en levert zowel een schatting van deorientatie van de structuur als een betrouwbaarheidsmaat van dit model. Deze betrouw-baarheidsmaat kan gebruikt worden om breuken die zich manifesteren als discontinuiteitente accentueren en te detecteren. We hebben onze op discontinuiteit gebaseerde breukmaattheoretisch en experimenteel vergeleken met een coherentiemaat uit de recente literatuur.De orientatieschatting kan gebruikt worden om een adaptief filter te sturen. Dit gestuurdefilter is in staat de structurele informatie in het beeld te versterken, door de ruis in seis-mische beelden te reduceren zonder de gelaagde structuur aan te tasten. Door dit filterte combineren met het gegeneraliseerde Kuwahara-filter, blijven ook de breuken intact.Verder kan de breukdetectie verbetert worden door het tensorvenster langs de breuk teverlengen.

Omdat in de praktijk gekromde structuren vaak voorkomen, hebben we de curvatuur-gecorrigeerde structuurtensor geintroduceerd. Deze gecorrigeerde structuurtensor leverteen schatting van de curvatuur als waardevol bijproduct. We hebben de curvatuur correc-tie zowel voor vlak-achtige als lijn-achtige structuren beschreven. De betrouwbaarheids-maat van het lijn-achtige curvilineaire model kan gebruikt worden voor het detecteren vangeulen. Om beter gebruik te maken van de continuiteit van geulen, hebben we een niet-parametrische structuurbeschrijvingsmethode geintroduceerd. Deze methode kan zondertoevoeging van parameters, complexe structuren zoals geulen op een grote schaal beschrij-ven.

Page 128: Image structure analysis for seismic interpretation · Image structure analysis for seismic interpretation Proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit
Page 129: Image structure analysis for seismic interpretation · Image structure analysis for seismic interpretation Proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit

Dankwoord

Het leven als AIO, in Delft omgedoopt tot promovendus, is voor mij een heel interessante”reis” geweest. De eerste drieeneenhalf jaar zijn voor mijn gevoel echt voorbij gevlogen.Toen kwam het onvermijdelijke moment in het leven van een promovendus: het schrijvenvan het proefschrift. De metamorfose van experimentator tot schrijver. In de praktijkbetekent dit vaak een half jaar (of langer...) de laatste loodjes. Ik wil graag iedereenbedanken die direct en indirect heeft bijgedragen aan het voltooien van deze reis.

Het werk wat ten grondslag ligt aan dit proefschrift is uitgevoerd in een samenwerkingsver-band tussen TU Delft, TNO-TPD en Shell. Ik wil Gert van Antwerpen (TNO-TPD), Mar-tin Kraaijveld (Shell), Gijs Fehmers (Shell) en de andere betrokkenen bedanken voor allediscussies.

Speciaal wil ik bedanken mijn begeleiders Piet Verbeek en Lucas van Vliet. Jullie deurstaat altijd open en jullie hebben mij tijdens deze hele reis enorm geholpen met al julliegoede ideeen en adviezen.

Verder wil ik mijn kamergenoten Michael, Judith, en Bernd bedanken voor alle weten-schappelijke en vooral ook de minder wetenschappelijke discussies. Jullie hebben samenmet alle ph-aio’s ervoor gezorgd dat het niet alleen een productieve maar ook een gezelligereis was. De vele TPKV en de ASCI avonden, de ‘vlak na de lunch’ recreatieve aktiviteitenen de squashpartijen hebben hieraan zeker bijgedragen.

Mijn ouders wil ik bedanken voor hun onophoudelijke steun en vertrouwen. Tenslotte, ‘lastbut definitely not least’, Jeanine, bedankt voor alles.

Page 130: Image structure analysis for seismic interpretation · Image structure analysis for seismic interpretation Proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit
Page 131: Image structure analysis for seismic interpretation · Image structure analysis for seismic interpretation Proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit

Curriculum vitae

Peter Bakker was born in Linz on December 8, 1974. In 1993 he obtained the VWO diplomaat the Bonaventura college in Leiden and started a study in physics at the University ofUtrecht. After finishing the propedeuse he continued his study at the RijksuniversiteitLeiden, and obtained his MSc. in the quantum optics group in 1997.

In the same year he started a Ph.D at the Pattern Recognition Group of the TechnicalUniversity Delft, in an IOP Beeldbewerking project called ‘Advanced texture segmentationin 2D and 3D subsurface images’. This work was carried out under supervision of Dr.Piet Verbeek and Prof.Dr.Ir. Lucas J. van Vliet. In 2002 he joined Shell InternationalExploration and Production in Rijswijk as a Research Geophysicist.