Curvature Detection by Integral Transforms
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DISSERTATION Curvature Detection by Integral Transforms Thomas Fink Dissertation eingereicht an der Fakultät für Informatik und Mathematik der Universität Passau zur Erlangung des Grades eines Doktors der Naturwissenschaften Betreuer: Prof. Dr. Brigitte Forster-Heinlein Prof. Dr. Uwe Kähler Passau, Mai 2019 iii Abstract In various fields of image analysis, determining the precise geometry of occurrent edges, e.g. the contour of an object, is a crucial task. Especially the curvature of an edge is of great practi- cal relevance. In this thesis, we develop different methods to detect a variety of edge features, among them the curvature. We first examine the properties of the parabolic Radon transform and show that it can be used to detect the edge curvature, as the smoothness of the parabolic Radon transform changes when the parabola is tangential to an edge and also, when additionally the curvature of the parabola coincides with the edge curvature. By subsequently introducing a parabolic Fourier transform and establishing a precise relation between the smoothness of a certain class of functions with the decay of the Fourier transform, we show that the smoothness result for the parabolic Radon transform can be translated into a change of the decay rate of the parabolic Fourier transform. Furthermore, we introduce an extension of the continuous shearlet transform which addition- ally utilizes shears of higher order. This extension, called the Taylorlet transform, allows for a detection of the position and orientation, as well as the curvature and other higher order ge- ometric information of edges. We introduce novel vanishing moment conditions of the form R g( t k )t mdt 0 which enable a more robust detection of the geometric edge features and ex- R § Æ amine two different constructions for Taylorlets. Lastly, we translate the results of the Taylorlet transform in R2 into R3 and thereby allow for the analysis of the geometry of object surfaces. v Danksagung An dieser Stelle möchte ich mich bei allen Menschen bedanken, die mich während meiner Dok- torandenzeit in Passau unterstützt, begleitet und geleitet haben. Zu allererst möchte ich meiner Betreuerin Brigitte Forster danken - für ihre Geduld, ihre exzel- lenten Ratschläge und die zahlreichen Diskussion, die stets auf Augenhöhe stattfanden. Sie hat immer ein offenes Ohr für mathematische und menschliche Fragen und hat mir in der Forschung große Freiheiten gewährt, sodass ich genau die Themen bearbeiten konnte, die mich brennend interessierten. Für ihre großartige Betreuung, ihre Offenheit und die Einblicke, die sie mir in ihre Arbeit gegeben hat, möchte ich ein ganz herzliches Dankeschön aussprechen. Many thanks go to Paula Cerejeiras and Uwe Kähler for their outstanding Portuguese hospitality and the joyful and relaxed atmosphere during my stays in Aveiro. I would like to thank Paula Cerejeiras especially for her enormous patience and her willingness to follow my presentation of a large proof focused and patiently for a whole week and thereby contributing hugely to correct the errors and increase the readability. I also wish to express my deep gratitude to Uwe Kähler for agreeing to be my second supervisor, for his contagious enthusiasm for mathematics and the many discussions we had that led to a joint paper. Außerdem möchte ich mich besonders bei meinem langjährigen Bürokollegen Johannes Nagler für das sorgfältige Gegenlesen großer Teile dieser Thesis und die vielen wunderbaren Gespräche über Mathematik, das Leben, das Universum und den ganzen Rest bedanken. Zu guter Letzt möchte ich meiner Familie für ihre ständige Unterstützung und Ermutigung und für das wunderbares Zuhause, zu dem ich immer gerne zurückkehre und das für mich oftmals eine Oase der Ruhe in schwierigen Zeiten war, meinen tiefen Dank aussprechen. vii Contents 1 Introduction 1 1.1 Outline.............................................2 1.2 Publications . .3 1.3 Notation and basic results . .4 1.3.1 Symbols . .4 1.3.2 Curvature .......................................6 1.3.3 Fourier transform . 10 2 Conformal monogenic signal curvature 15 2.1 Underlying concepts . 15 2.1.1 Analytic and monogenic signals . 16 2.1.2 Poisson scale-space . 18 2.1.3 Isophote Curvature . 21 2.2 The conformal monogenic signal . 23 1 2.3 The divergence of the monogenic signal fs ........................ 27 2.4 The non-equality of the conformal monogenic signal curvature and the isophote curvature............................................ 28 3 Edge curvature and the parabolic Fourier transform 35 3.1 Edges and their curvature . 35 3.2 Parabolic Radon transform . 42 3.3 Smoothness results . 49 3.4 Parabolic Fourier transform . 58 3.4.1 Basic properties of the parabolic Fourier transform . 58 viii 3.4.2 Singularities and the decay of the Fourier transform . 63 3.4.3 Decay result for singularities . 73 4 Taylorlet transform 85 4.1 Continuous shearlet transform . 86 4.2 Basic definitions and properties of the Taylorlet transform . 88 4.3 Construction of a Taylorlet . 94 4.3.1 Derivative-based construction . 94 4.3.2 Construction based on q-calculus . 100 4.4 Detection result . 120 4.5 Numerical examples . 137 4.5.1 Detection procedure . 137 4.5.2 Derivative-based construction . 138 4.5.3 Construction based on q-calculus . 139 4.5.4 Images . 139 5 Extension of the Taylorlet transform to three dimensions 145 5.1 Basic definitions and notation . 146 5.2 Detection results . 150 5.3 Proof of the detection results . 153 5.4 Construction of a three-dimensional Taylorlet . 179 5.5 Detection algorithm for higher-dimensional edges . 183 6 Conclusion and outlook 187 1 CHAPTER 1 Introduction The automatization of processes plays a major role for the technological progress of today’s society. For this venture, it is often necessary to process information in a fast and stable fashion. In the field of autonomous driving, for instance, the recognition of road marks, traffic signs and other road users is of pivotal importance. In medical imaging, it is desirable to design systems that are capable of autonomous detection of cancerous tissue. But also many applications in other fields show the importance of object recognition and thus, good and stable detectors are crucial. Since the concept of an edge, i.e., the occurrence of a rapid change of colour along a curve, already plays a prominent role in the mammalian visual system [HW62, HW68], it is no sur- prise that the detection of edges also is a major part in most object recognition algorithms, e.g. [Mar76, MH80]. Many secondary features of edges are also often used in the field of com- puter vision. Among them the curvature of an edge is of special interest, see e.g. [DZMÅ07, MEO11,FB14]. The great significance of the edge curvature to the human capability of recog- nizing objects has already been discovered in the 1950s in psychological studies [Att54,AA56]. There is also a mathematical reason for the usefulness of curvature, as every planar C 2-curve is uniquely determined by its curvature profile up to translations and rotations. Thus, the curva- ture provides a natural and meaningful measure to categorize shapes and object contours. The structure of an edge curvature estimator is usually divided into two parts. In the first step, an edge detector is applied and subsequently, a curvature estimator is employed on the newly found edge. For the purpose of detecting edges there exists a plethora of different methods, e.g. the classical Canny edge detector [Can87] and multiscale approaches such as the method of wavelet modulus maxima [MH92] or shearlet-based algorithms [YLEK09]. The numerical computation of the curvature could be handled by naively using the curvature formula for a C 2-curve γ : I R2 on an interval I: ! ¡ ¢ det γ0(t),γ00(t) ·γ(t) for all t I. Æ γ (t) 3 2 k 0 k2 As a calculation based on this formula is numerically unstable due to the presence of derivatives 2 Chapter 1. Introduction and a division, usually other methods are utilized. Either a purely discrete approach is applied by defining a discrete curvature, e.g. [CMT01], or the discretely given curve is artificially made continuous, e.g. by scale-space methods [MB03] or by interpolation [FJ89]. A drawback of the approach to first detect edges and then estimate the curvature is its numerical instability, as the deviation of the detected edge from the actual edge can increase the error of the curvature estimation step immensely. The main goal of this thesis is to overcome this prob- lem by a localization of the procedure. We have developed methods that are able to determine the local orientation and curvature of the edge without knowledge of the complete edge. Such an approach has already been successfully used, e.g. in [LPS16]. In this thesis, we will present two different methods that follow the idea of localization. The first one is described in chapter 3 and utilizes the parabolic Radon transform, i.e., the integral of a function over parabolae, and the thereof derived parabolic Fourier transform. The second method is called the Taylorlet transform and chapters 4 and 5 are devoted to this approach. It is based on the continuous shearlet transform and additionally uses shears of higher order to allow for a detec- tion of the curvature. Both methods yield detection results that are based on differences of the decay rate of the respective integral transform. The main challenge of our approach is to find conditions for the analyzed function and in case of the Taylorlet transform also for the analyzing function, the Taylorlet, that enforce the differences in the decay rate we need for detecting the edge orientation and curvature. Similar results on the decay rate already exist for the continuous wavelet transform, e.g. [MH92], and the continuous shearlet transform in 2D [GL09,Gro11,KP15] and in 3D [GL11].