Image Structure

277 Pages·1997·8.51 MB·English

by  Luc Florack (auth.)

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