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Image Structure PDF

277 Pages·1997·8.51 MB·English
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Image Structure Computational Imaging and Vision Managing Editor MAX A. VIERGEVER Utrecht University, Utrecht, The Netherlands Editorial Board RUZENA BAJCSY, University ofP ennsylvania, Philadelphia, USA MIKE BRADY, Oxford University, Oxford, UK OLIVIER D. FAUGERAS,INRIA, Sophia-Antipolis, France JAN J. KOENDERINK, Utrecht University, Utrecht, The Netherlands STEPHEN M. PIZER, University ofN orth Carolina, Chapel Hill, USA SABURO TSUJI, Wakayama University, Wakayama, Japan STEVEN W. ZUCKER, McGill University, Montreal, Canada Volume 10 Image Structure by Luc Florack Department o/Computer Science, Utrecht University, Utrecht, The Netherlands SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-4937-7 ISBN 978-94-015-8845-4 (eBook) DOI 10.1007/978-94-015-8845-4 Printed on acid-free paper All Rights reserved © 1997 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1997 Softcover reprint of the hardcover 1st edition 1997 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. @}ebanfen o~ne 3n~alt nnb leer, !nfd,lauungen o~ne ~egriffe nnb ~linb. -IMMANUEL KANT Contents Foreword xi Preface xiii 1 Introduction 1 1.1 Scalar Images in Practice . . . . . . . . . . . . . . . . . . . . . . .. 1 1.2 Syntax versus Semantics ........................ 5 1.3 Synthesis versus Analysis ....................... 6 1.4 Image Analysis a Science? ....................... 8 1.5 An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10 2 Basic Concepts 13 2.1 A Conventional Representation of Images. . . . . . . . . . . . . .. 13 2.2 Towards an Improved Representation ................. 16 2.2.1 Device Space as the Dual of State Space. . . . . . . . . .. 17 2.2.2 State Space as the Dual of Device Space: Distributions. .. 21 * 2.2.2.1 * ~ d=ef 1) ( 0 ) ...................... 23 2.2.2.2 ~ ~ £(0) ...................... 26 * 2.2.2.3 ~ dg S(IRn) .. . . . . . . . . . . . . . . . . . .. 26 2.3 More on the Theory of Schwartz .................... 27 2.4 Summary.................................. 35 Problems .................................... 36 3 Local Samples and Images 39 3.1 Local Samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40 3.2 Covariance versus Invariance . . . . . . . . . . . . . . . . . . . . .. 42 3.3 Linearit*y ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45 3.3.1 Linearisation from an Abstract Viewpoint . . . . . . . . .. 46 3.4 Images................................... 47 3.5 Raw Images. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50 3.6 Static versus Dynamic Representations . . . . . . . . . . . . . . .. 51 3.7 The Newtonian Spacetime Model . . . . . . . . . . . . . . . . . . .. 52 3.8 Image Processing. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 54 3.9 The Point Operator . . . . . . . . . . . . . . . . . . . . . . . . . . .. 57 viii Contents 3.10 Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . .. 63 3.11 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 65 3.12 Discretisation Schemes. . . . . . . . . . . . . . . . . . . . . . . . .. 68 3.13 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . .. 72 Problems .................................... 83 4 The Scale-Space Paradigm 89 4.1 The Co*nc ept of Scale and Some Analogies. . . . . . . . . . . . .. 89 4.1.1 * Scale and Brownian Motion: Einstein's Argument. . . .. 92 4.1.2 * Scale and Brownian Motion: Functional Intergration. . .. 94 4.1.3 * Scale and Regularisation . . . . . . . . . . . . . . . . . .. 98 4.1.4 Scale and Entropy . . . . . . . . . . . . . . . . . . . . . .. 99 4.2 The Multiscale Local Jet . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.3 Temporal Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.3.1 Manifest Causality . . . . . . . . . . . . . . . . . . . . . . . . 109 4.3.2 The "Specious Present": Real-Time Sampling ........ 114 4.3.3 Relation to "Classical" Scale-Space ... . . . . . . . . . . . 118 4.4 Summary and Discussion ........ . . . . . . . . . . . . . . . . 118 Problems .................................... 127 5 Local Image Structure 133 5.1 Groups and Invariants .......................... 134 5.2 Tensor Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.2.1 The Euclidean Metric. . . . . . . . . . . . . . . . . . . . . . . 137 5.2.2 General Tensors ......................... 139 5.2.3 Tensors on a Riemannian Manifold . . . . . . . . . . . . . . . 141 5.2.4 *Co variant Derivatives ....................... 144 5.2.5 Tensors on a Curved Manifold . . . . . . . . . . . . . . . . 147 5.2.6 The Levi-Civita Tensor . . . . . . . . . . . . . . . . . . . . . . 148 5.2.7 Relative Tensors and Pseudo Tensors . . . . . . . . . . . . . 149 5.3 Differential Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.3.1 Construction of Differential Invariants ............. 151 5.3.2 Complete Irreducible Invariants ................. 156 5.3.3 Gauge Coordinates. . . . . . . . . . . . . . . . . . . . . . . . 159 5.3.4 Geometric or Grey-Scale Invariants ...... . . . . . . . . 161 Problems .................................... 171 6 Multiscale Optic Flow 175 6.1 Towards an Operational Definition of Optic Flow ........... 176 6.1.1 The "Aperture Problem" ..................... 177 6.1.2 Computational Problems ..................... 179 6.2 The Optic Flow Constraint Equation .................. 179 6.3 Computational Model for Solving the OFCE . . . . . . . . . . . . . . 183 6.4 Examples.................................. 186 6.4.1 Zeroth, First, and Second Order Systems ........... 186 6.4.2 Simulation and Verification ................... 186 Contents ix 6.4.2.1 Density Gaussian .. . . . . . . . . . . . . . . . . . 187 6.4.2.2 Scalar Gaussian . . . . . . . . . . . . . . . . . . . . 188 6.4.2.3 *Nu merical Test . . . . . . . . . . . . . . . . . . . . . 189 6.4.2.4 Conceptual Comparison with Similar Methods . 190 6.5 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 192 Problems .................................... 202 A Geometry and Tensor Calculus 205 A.1 Literature .................................. 205 A.2 Geometric Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 A.2.1 Preliminaries ........................... 206 A.2.2 Vectors............................... 208 A.2.3 Covectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 A.2.4 Dual Bases ............................ 210 A.2.5 Riemannian Metric . . . . . . . . . . . . . . . . . . . . . . . . 211 A.2.6 Tensors............................... 212 A.2.7 Push Forward, Pull Back, Derivative Map . . . . . . . . . . . 217 B The Filters c»Pl" ·PI 219 !'l···!'k C Proof of Proposition 5.4 223 D Proof of Proposition 5.5 225 D.1 Irreducible System for {Lij} ....................... 225 0.2 Irreducible System for {L, Li, Lij} .................... 226 Solutions to Problems 227 Symbols 237 Glossary 239 Bibliography 245 Index 259 Foreword Because of reasons best known to their authors I have already written forewords to various books on the general topic of "scale space": Why yet another one? Well, the present book is different. Most developments in scale space theory and practice have been due to scientists who are either best classified as "computer scientists" or as "(applied) mathematicians". Whereas the mathematician rightly pursues art for art's sake, the computer scientists tend to pursue craft for craft's sake, they are-in the first place-engineers. This places both disciplines at most at the margin of the empirical sciences in the sense that neither is censored much by exposure to the physical world. (The bulk of computer science is of this nature, in this sense it is different from civil engineering: At least bridges may collapse.) The present book has been written from the perspective of the physicist and this shows through at every page. Indeed, it is possible to view scale space theory as essentially a mathematical theory and thus arbitrary from the standpoint of physics (I will write "physics" when "empirical sciences" would serve equally well). This is in fact the common attitude and it explains the many ad hoc "generalizations" and embellishments that appear indeed attractive from the vantage point of the mathematician. De spite the often admirable virtuosity displayed, such exercises often fail to excite me because 1 can't perceive them as being about something. Even the "hard re sults" in computer vision and image processing often leave me in a quandary because we only too often seem to lack the touchstone needed to size up the re sult. Luc Florack bases the theory firmly as a (very general) description of the pro cess of observation. This is much more intricate than it might sound at first blush since when one thinks of "observation" one tends to assume the existence of an entity that is being observed. Such is certainly the case in the context of signal detection and communication theory, i.e., the communication between agents. However, nature is not an agent and whatever it may be that is "being observed" can only be known through the observation itself! Observations are indeed all we have. Florack takes this seriously in that he defines the elementary process of ob servation in terms of the structure of the observer, rather than the structure of the entity being observed. The latter is only (indeed: can only be) implicitly defined in terms of the structure of the observer. (Notice how this is the exact opposite of the standard procedure of signal detection and communication theory and also of most image analysis.) Different observers, when confronted with the same slice xii Foreword of nature, will come up with different observations: Reality is observer-relative and not "God given". Indeed, "man is the measure of all things": This goes so far that even the spatiotemporal framework is seen as a property of the structure of the observer, rather than of the world! Of course this is not really that novel (few concepts are) and Florack finds himself in (among more) the varied company of Kant (philosopher), Mach (physicist) and Whitehead (mathematician). What makes the book fascinating to me is that such general and important concepts underlying human perception (and thus cognition) and empirical sci ence are shown to lead to a very beautiful, finnly conceptually based and com putationally powerful theoretical framework. The mathematics involved is the calculus in its widest sense, involving topology, measure theory and continuous groups. Much of the backbone is Schwartz's notion of "tempered distributions". This theory-sorely neglected by the natural sciences-provides the natural tool for differentiation in the real world. Notice that perceptual or image analysis nec essarily involves the differentiation of observations, or rather, the observation of differences. Here the "infinitesimal domain" becomes tangible. In fact the theory provides rigorous definitions of such geometrical entities as "point" or "tangent vector" that are also fully operational: "Rigorous" will appeal to the mathemati cian, "operational" to the physicist and operational in the guise of "computa tional" to the computer scientist. The conceptual and fonnal groundwork leads to an edifice that contains all of "linear scale space theory", the theory of local image operators and differential invariants, by now generally recognized as the natural tools in the front end of image processing and analysis. Florack shows generalizations to arbitrary spatial dimensions including space-time. Well chosen examples include the deblurring of pictures, multiscale representation in real time, and various image flow esti mators. It is my hope that the book will find a wide audience, including physicists who still are largely unaware of the general importance and power of scale space theory, mathematicians-who will find in it a principled and fonnally tight expo sition of a topic awaiting further development, and computer scientists-who will find here a unified and conceptually well founded framework for many appar ently unrelated and largely historically motivated methods they already know and love. The book is suited for self-study and graduate courses, the carefully fonnulated exercises are designed to get to grips with the subject matter and pre pare the reader for original research. Jan Koenderink Utrecht, July 7th 1997

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