Low-Level Image Processing with the Structure Multivector
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Low-Level Image Processing with the Structure Multivector Michael Felsberg Bericht Nr. 0202 Institut f¨ur Informatik und Praktische Mathematik der Christian-Albrechts-Universitat¨ zu Kiel Olshausenstr. 40 D – 24098 Kiel e-mail: [email protected] 12. Marz¨ 2002 Dieser Bericht enthalt¨ die Dissertation des Verfassers 1. Gutachter Prof. G. Sommer (Kiel) 2. Gutachter Prof. U. Heute (Kiel) 3. Gutachter Prof. J. J. Koenderink (Utrecht) Datum der mundlichen¨ Prufung:¨ 12.2.2002 To Regina ABSTRACT The present thesis deals with two-dimensional signal processing for computer vi- sion. The main topic is the development of a sophisticated generalization of the one-dimensional analytic signal to two dimensions. Motivated by the fundamental property of the latter, the invariance – equivariance constraint, and by its relation to complex analysis and potential theory, a two-dimensional approach is derived. This method is called the monogenic signal and it is based on the Riesz transform instead of the Hilbert transform. By means of this linear approach it is possible to estimate the local orientation and the local phase of signals which are projections of one-dimensional functions to two dimensions. For general two-dimensional signals, however, the monogenic signal has to be further extended, yielding the structure multivector. The latter approach combines the ideas of the structure tensor and the quaternionic analytic signal. A rich feature set can be extracted from the structure multivector, which contains measures for local amplitudes, the local anisotropy, the local orientation, and two local phases. Both, the monogenic signal and the struc- ture multivector are combined with an appropriate scale-space approach, resulting in generalized quadrature ﬁlters. Same as the monogenic signal, the applied scale- space approach is derived from the three-dimensional Laplace equation instead of the diffusion equation. Hence, the two-dimensional generalization of the analytic signal turns out to provide a whole new framework for low-level vision. Several applications are presented to show the efﬁciency and power of the theoretic con- siderations. Among these are methods for orientation estimation, edge and corner detection, stereo correspondence and disparity estimation, and adaptive smoothing. vi Abstract ACKNOWLEDGMENT First of all, I appreciate my supervisors Prof. Gerald Sommer and Prof. Ulrich Heute for guiding me through the interesting ﬁelds of computer vision and signal theory. They allowed me to work as independently as was necessary to obtain sub- stantial new results. However, without the countless, intensive discussions with Prof. Gerald Sommer and without his scientiﬁc intuition I would not have been able to develop the presented ideas and to write this thesis as it is. He supported me whenever it was necessary. Furthermore, I acknowledge the support and help of all our former and present members of the cognitive systems group in Kiel. Especially the fruitful discussions with Norbert Kruger,¨ Thomas Bulow,¨ and Hongbo Li have had a great inﬂuence on my work. I thank Gerd Diesner and Henrik Schmidt for their technical support and Franc¸oise Maillard for her help in administrative matters. I am grateful for all the discussions with other scientists and for the refer- ences they have given to me, especially Kieran Larkin, Robin Owens, Terence Tao, Dieter Betten, Joachim Weickert, Christoph Schnorr,¨ Bernd Jahne,¨ Hanno Scharr, Wolfgang Forstner,¨ Gosta¨ Granlund, Josef Bigun,¨ Moshe Porat, Nir Sochen, Tony Lindeberg, and Jan Koenderink. I acknowledge Hongbo Li, Jim Byrnes, Norbert Kruger,¨ and Christian Perwass for proofreading parts of this thesis. I am also grateful to the German National Merit Foundation (Studienstiftung des deutschen Volkes) and the DFG Graduiertenkolleg No. 357 for their ﬁnancial support during my PhD studies. Last, but not least, I thank my family for mental support, my parents Hans and Beate, my sister Susanne, my parents-in-law Otfried and Rosemarie, my brother- in-law Sebastian, my daughter Lisa-Marie and especially my wife Regina. viii Acknowledgment CONTENTS 1. Introduction ..................................... 1 1.1 Terms and Motivation . 1 1.2 Known Approaches for the 2D Analytic Signal . 8 1.3 Outline of this Thesis . 12 2. Geometric Algebra ................................ 15 2.1 Geometric Algebra of IR2 ........................... 17 2.1.1 Deﬁnition of the Algebra . 17 2.1.2 Group Representation in IR2 ..................... 21 2.2 Geometric Algebra of IR3 ........................... 23 2.2.1 Deﬁnition of the Algebra . 23 2.2.2 Group Representation in IR3 ..................... 28 2.3 Complex Analysis and 2D Harmonic Fields . 32 2.3.1 The Cauchy-Riemann Equations . 32 2.3.2 Solutions of the 2D Laplace Equation . 33 2.4 3D Harmonic Fields . 34 2.4.1 The Generalized Cauchy-Riemann Equations . 35 2.4.2 The 3D Laplace Equation and its Solutions . 36 2.5 Summary of Chapter 2 . 37 3. Signal Theoretic Fundamentals ......................... 39 3.1 The 1D Fourier Transform and LSI Operators . 40 3.1.1 Deﬁnitions . 40 x Contents 3.1.2 Basic Theorems . 42 3.2 The Phase in 1D . 44 3.2.1 The Analytic Signal and the Local Phase . 45 3.2.2 Properties of the 1D Analytic Signal . 47 3.3 The 2D Fourier Transform and Linear Operators . 50 3.3.1 Deﬁnitions . 50 3.3.2 Basic Theorems . 53 3.3.3 Plancherel Theorem and 2D Uncertainty . 56 3.4 The Structure Tensor . 58 3.4.1 The Structure Tensor and the Intrinsic Dimension . 59 3.4.2 Interpretation of the Structure Tensor . 62 3.5 Spherical Harmonics and Steerable Filters . 65 3.5.1 Fourier Series . 65 3.5.2 Applying Steerable Filters . 66 3.6 Summary of Chapter 3 . 67 4. The Isotropic 2D Generalization of the Analytic Signal ............ 69 4.1 Derivation of the Hilbert Transform . 69 4.1.1 The 1D Signal Model . 70 4.1.2 Relation to the Laplace Transform . 72 4.2 The Riesz Transform . 73 4.2.1 The 2D Signal Model . 73 4.2.2 Relation to the Fourier Kernel and a Short Survey . 75 4.3 The Monogenic Signal . 78 4.3.1 Deﬁnition and Properties . 78 4.3.2 The Deﬁnition of the Local Phase . 81 4.4 The Interpretation of the Local Phase . 83 4.4.1 Transferring the 1D Phase to 2D . 83 4.4.2 Decomposition of the Phase Vector . 86 Contents xi 4.5 A New Scale-Space Approach . 91 4.5.1 The Scale-Space Axiomatic of Iijima . 91 4.5.2 Comparison to Gaussian Scale-Space . 93 4.6 The Spherical Quadrature Filter . 95 4.6.1 The DOP Wavelet . 95 4.6.2 The Deﬁnition of Spherical Quadrature Filter . 97 4.6.3 The Intrinsic Scale . 98 4.7 Summary of Chapter 4 . 100 5. Adaption of 2D Quadrature Filters to the Local Intrinsic Dimension .... 101 5.1 Introducing Local Anisotropies . 102 5.1.1 The Metric Tensor . 102 5.1.2 The Laplace Equation in the Deformed Space . 104 5.1.3 Interpretation in the Fourier Domain . 106 5.2 Approximation of the Local Metric . 109 5.2.1 Taylor Expansion of the Poisson Kernel . 109 5.2.2 Taylor Expansion of the Riesz Transform . 111 5.2.3 Isotropy and Coherence . 112 5.3 Steering of the Approximation . 115 5.3.1 Steering the Poisson Kernel Approximation . 115 5.3.2 Steering the Conjugate Poisson Kernel Approximation . 117 5.4 The Isotropic Analytic Signal for i2D Signals . 119 5.4.1 The 2D Signal Model for i2D Signals . 119 5.4.2 The Error of the Monogenic Signal for i2D Signals . 122 5.4.3 Spectral Decomposition According to Symmetries . 123 5.4.4 Rotation Invariant 2D Decomposition . 125 5.5 The Structure Multivector . 129 5.5.1 Steering by the Local Orientation . 129 5.5.2 Amplitude and Phase Decomposition . 131 xii Contents 5.5.3 The Properties of the Structure Multivector . 136 5.6 Summary of Chapter 5 . 138 6. Applications ..................................... 141 6.1 Orientation Estimation . 141 6.1.1 Optimal Local Orientation Estimation . 142 6.1.2 Orientation Estimation by the Riesz Transform . 144 6.1.3 Orientation Estimation with Spherical Harmonics . 148 6.2 Edge Detection . 151 6.2.1 Phase Congruency . 151 6.2.2 Derivation of the Filter Set . 152 6.2.3 Experiments and Comparisons . 154 6.3 Stereo Correspondence and Disparity Estimation . 158 6.3.1 Stereo Correspondence by Matching on the Epipolar Line . 158 6.3.2 Disparity from Monogenic Phase . 160 6.4 Adaptive Smoothing . 163 6.4.1 The Adaptive Smoothing Algorithm . 163 6.4.2 Experiments and Comparison . 164 6.5 Corner Detection and Curvature Estimation . 166 6.5.1 The Local Isotropy and Curvature . 167 6.5.2 Corner Detection Experiments and Comparisons . 169 6.6 Summary of Chapter 6 . 171 7. Conclusion ...................................... 173 7.1 Summary of the Results . 173 7.2 Further Extensions, Open Problems, and Future Work . 174 A. Fourier Transform Pairs and Uncertainties ................... 177 A.1 1D Fourier Transform Pairs . 177 A.2 2D Fourier Transform Pairs . 179 Contents xiii A.3 Uncertainties . 183 A.4 Simpliﬁed Transfer Function of the Monogenic Signal . 186 Bibliography ..................................... 187 Index ......................................... 199 xiv Contents Chapter 1 INTRODUCTION ‘This question is best answered by making use of one quite singular improper rotation, namely reﬂection in O; it carries any point P into its antipode P 0 ...’ Hermann Weyl (1885-1955) The quotation above1 which states that the answer is a reﬂection in the origin can be considered as the central point of my thesis. However, what is the question to this answer? This questions could be: ‘Is there a sophisticated 2D generalization of the Hilbert transform which can be used for image processing and if there is, what does it look like?’ As it can be concluded from the question above, this thesis is about a topic which is related to three different ﬁelds, as there are mathematics (harmonic analysis), engineering (signal theory), and computer science (image processing).