Gaussian scale-space is one of the best understood multi-resolution techniques available to the computer vision and image analysis community. It is the purpose of this book to guide the reader through some of its main aspects. During an intensive weekend in May 1996 a workshop on Gaussian scale-space theory was held in Copenhagen, which was attended by many of the leading experts in the field. The bulk of this book originates from this workshop. Presently there exist only two books on the subject. In contrast to Lindeberg's monograph (Lindeberg, 1994e) this book collects contributions from several scale space researchers, whereas it complements the book edited by ter Haar Romeny (Haar Romeny, 1994) on non-linear techniques by focusing on linear diffusion. This book is divided into four parts. The reader not so familiar with scale-space will find it instructive to first consider some potential applications described in Part 1. Parts II and III both address fundamental aspects of scale-space. Whereas scale is treated as an essentially arbitrary constant in the former, the latter em phasizes the deep structure, i.e. the structure that is revealed by varying scale. Finally, Part IV is devoted to non-linear extensions, notably non-linear diffusion techniques and morphological scale-spaces, and their relation to the linear case. The Danish National Science Research Council is gratefully acknowledged for providing financial support for the workshop under grant no. 9502164.
Preface xi(2) Contributors xiii(2) Scale in Perspective xv Jan J. Koenderink I Applications 1(42) 1 Applications of Scale-Space Theory 3(18) Bart ter Haar Romeny 1.1 Introduction 3(1) 1.2 Feature detection 4(2) 1.3 Shape Properties 6(2) 1.4 Higher Order Invariants 8(1) 1.5 Deblurring 9(2) 1.6 Ridges and Multimodality Matching 11(1) 1.7 Deep Structure and the Hyperstack 12(1) 1.8 Denoising and Edge-Preserving Smoothing 13(1) 1.9 Discussion 14(7) 2 Enhancement of Fingerprint Images using Shape-Adapted Scale-Space Operators 21(10) Andres Almansa Tony Lindeberg 2.1 Introduction 21(2) 2.2 Shape-adapted smoothing 23(1) 2.3 Enhancement of ridges by shape adaptation 24(2) 2.4 Automatic scale selection 26(3) 2.5 Summary and discussion 29(2) 3 Optic Flow and Stereo 31(12) Wiro J. Niessen Robert Maas 3.1 Introduction 31(1) 3.2 Generalized Brightness Constraint Equation 32(2) 3.3 Optic flow 34(1) 3.4 Binocular stereo 35(2) 3.5 Scale selection 37(2) 3.6 Optic flow results 39(1) 3.7 Stereo results 40(1) 3.8 Summary 41(2) II The Foundation 43(94) 4 On the History of Gaussian Scale-Space Axiomatics 45(16) Joachim Weickert Seiji Ishikawa Atsushi Imiya 4.1 Introduction 45(2) 4.2 Iijimas 1-D Axiomatic (1962) 47(4) 4.2.1 Motivation 47(1) 4.2.2 Axioms 47(1) 4.2.3 Consequences 48(2) 4.2.4 Further Results 50(1) 4.3 Otsus 2-D Axiomatic (1981) 51(2) 4.3.1 Derivation of the Gaussian 51(1) 4.3.2 Further Results 52(1) 4.4 Relation to Other Work 53(5) 4.5 Discussion 58(3) 5 Scale-Space and Measurement Duality 61(14) Luc Florack 5.1 Introduction 61(1) 5.2 Pseudo-Static Image Concept 62(4) 5.3 Kinematic Image Concept 66(7) 5.4 Conclusion 73(2) 6 On the Axiomatic Foundations of Linear Scale-Space 75(24) Tony Lindeberg 6.1 Introduction 75(3) 6.2 Axiomatic formulations of linear scale-space 78(7) 6.2.1 Original formulation 78(1) 6.2.2 Causality 78(1) 6.2.3 Non-creation of local extrema 79(1) 6.2.4 Semi-group structure and continuous scale parameter 80(1) 6.2.5 Non-enhancement and infinitesimal generator 81(2) 6.2.6 Scale invariance 83(2) 6.3 Semi-group and causality: continuous domain 85(6) 6.3.1 Assumptions 86(1) 6.3.2 Definitions 87(2) 6.3.3 Necessity 89(2) 6.3.4 Sufficiency 91(1) 6.3.5 Application to scale invariant semi-groups 91(1) 6.4 Summary and conclusions 91(1) 6.5 Extensions of linear scale-space 92(7) 6.5.1 Relaxing rotational symmetry 92(3) 6.5.2 Non-linear smoothing 95(4) 7 Scale-Space Generators and Functionals 99(16) Mads Nielsen 7.1 Introduction 99(2) 7.2 Linear scale-space generators 101(2) 7.3 Linear regularization 103(6) 7.3.1 Scale-invariant regularization 104(2) 7.3.2 Evolution equations 106(1) 7.3.3 Scale-space interpretation of regularization 107(2) 7.4 Truncated smoothness operators 109(2) 7.5 Implementation issues of truncated filters 111(3) 7.6 Summary 114(1) 8 Invariance Theory 115(14) Alfons Salden 8.1 Introduction 115(1) 8.2 The Equivalence Problem 116(1) 8.3 Intrinsic Symmetries 117(10) 8.3.1 Solution of Similarity Equivalence Problem 118(1) 8.3.2 Solution of Affine and Euclidean Equivalence Problems 118(9) 8.4 Conclusion and Discussion 127(2) 9 Stochastic Analysis of Image Acquisition and Scale-Space Smoothing 129(8) Kalle Astrom Anders Heyden 9.1 Introduction 129(1) 9.2 Image acquisition 130(1) 9.3 Interpolation and smoothing 131(2) 9.4 The random field model 133(2) 9.5 Applications 135(1) 9.6 Conclusions 136(1) III The Structure 137(64) 10 Local Analysis of Image Scale Space 139(8) Peter Johansen 10.1 Introduction 139(4) 10.1.1 Codimension 141(2) 10.2 Analysis of codimension one 143(1) 10.3 Analysis of codimension two 144(2) 10.4 Suggestions for Further Research 146(1) 10.5 Acknowledgement 146(1) 11 Local Morse Theory for Gaussian Blurred Functions 147(18) James Damon 11.1 Classifying Properties of Blurred Images via Equivalences 148(2) 11.2 Understanding Generic Properties 150(2) 11.3 Genericity via Stability 152(1) 11.4 Weighted Homogeneity and the Heat Equation 153(4) 11.5 Generic Properties of Blurred Images 157(1) 11.6 Consequences of the Generic Properties 158(2) 11.7 Genericity via Transversality and Versality 160(2) 11.8 Conclusion 162(3) 12 Critical Point Events in Affine Scale-Space 165(16) Lewis Griffin 12.1 Motivation for Affine Scale Space 165(1) 12.2 A Representation of Affine Scale Space 166(2) 12.3 Local Structure 168(2) 12.4 Mathematical Technique 170(2) 12.5 The Catastrophes 172(9) 12.5.1 Fold Catastrophe: H(u, v; p, q, r) = pu + u(3) + v(2) 172(2) 12.5.2 Cusp Catastrophe: H(u, v; p, q, r) = pu + qu(2) + u(4) XXX v(2) 174(3) 12.5.3 Swallowtail: H(u, v; p, q, r) = pu + qu(2) + ru(3) + u(5) + v(2) 177(1) 12.5.4 Elliptic Umbilic: H(u, v; p, q, r) = pu + qv + r(u(2) + v(2)) + u(3) - 3uv(2) 177(2) 12.5.5 Hyperbolic Umbilic: H(u, v; p, q, r) = pu + qv + r(u(2) - v(2)) + u(3) + 3uv(2) 179(2) 13 Topological Numbers and Singularities 181(10) Stiliyan Kalitzin 13.1 Introduction 181(2) 13.2 Theory. Topological invariants 183(2) 13.3 Examples 185(2) 13.3.1 Second order singular points 185(1) 13.3.2 One dimensional case 186(1) 13.3.3 Two dimensional images 186(1) 13.4 Scale space evolution 187(2) 13.4.1 One dimensional fold 187(1) 13.4.2 Two dimensional umbilic point 188(1) 13.5 Summary and discussion 189(2) 14 Multi-Scale Watershed Segmentation 191(10) Ole Fogh Olsen 14.1 Introduction 191(1) 14.2 Morse functions and catastrophe theory 192(1) 14.3 Watersheds and Catchment basins 192(3) 14.4 Catastrophe theory applied 195(2) 14.5 The segmentation tool 197(3) 14.6 Conclusions 200(1) IV Non-linear Extensions 201(34) 15 The Morphological Equivalent of Gaussian Scale-Space 203(18) Rein van den Boomgaard Leo Dorst 15.1 Introduction 203(2) 15.2 Morphological Operators 205(3) 15.3 Geometrical Interpretation of Morphological Operators 208(1) 15.4 Tangential Dilation and the Slope Transform 209(3) 15.5 Scale-Space from Basic Principles 212(4) 15.6 The Infinitesimal Generator of Scale-Space 216(3) 15.7 Conclusions 219(2) 16 Nonlinear Diffusion Scale-Spaces 221(14) Joachim Weickert 16.1 Introduction 221(1) 16.2 The Perona-Malik model 222(3) 16.2.1 Basic idea 222(1) 16.2.2 Ill-posedness 223(2) 16.3 Regularized nonlinear diffusion filtering 225(3) 16.3.1 Basic idea 225(1) 16.3.2 General well-posedness and scale-space framework 225(3) 16.4 Semidiscrete nonlinear diffusion scale-spaces 228(3) 16.4.1 An example 228(2) 16.4.2 General well-posedness and scale-space framework 230(1) 16.4.3 Application to the example 231(1) 16.5 Fully discrete nonlinear diffusion scale-spaces 231(3) 16.5.1 An example 232(1) 16.5.2 General well-posedness and scale-space framework 232(1) 16.5.3 Application to the example 233(1) 16.5.4 A more efficient discretization 234(1) 16.6 Generalizations 234(1) Bibliography 235(23) Index 258
