Table Of ContentShape in Picture
Mathematical Description of Shape in Grey-level Images
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Series F: Computer and Systems Sciences Vol. 126
Shape in Picture
Mathematical Description of Shape
in Grey-level Images
Edited by
Ying-Lie 0
Centre for Mathematics and Computer Science (CWI)
Kruislaan 413, 1098 SJ Amsterdam, The Netherlands
Alexander Toet
Netherlands Organization for Applied Scientific Research
Institute for Human Factors (TNO-IZF)
Kampweg 5, 3769 DE Soesterberg, The Netherlands
David Foster
University of Keele
Department of Communication and Neuroscience
Keele, Staffordshire ST5 5BG, UK
Henk J. A.M. Heijmans
Centre for Mathematics and Computer Science (CWI)
Kruislaan 413, 1098 SJ Amsterdam, The Netherlands
Peter Meer
Rutgers University
Department of Electrical and Computer Engineering
Piscataway, NJ 08855-0909, USA
Springer-Verlag Berlin Heidelberg GmbH
Proceedings ofthe NATO Advanced Research Workshop "Shape in Picture", held
at Driebergen, The Netherlands, September 7-11, 1992
CR Subject Classification (1991): 1.3-5
ISBN 978-3-642-08188-0 ISBN 978-3-662-03039-4 (eBook)
DOI 10.1007/978-3-662-03039-4
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© Springer-Verlag Berlin Heidelberg 1994
Originally published by Springer-Verlag Berlin Heidelberg New York in 1994
Softcover reprint of the hardcover 1s t edition 1994
Typesetting: Camera ready by authors
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Preface
The fields of image analysis, computer vision, and artificial intelligence all make
use of descriptions of shape in grey-level images. Most existing algorithms for
the automatic recognition and classification of particular shapes have been devel
oped for specific purposes, with the result that these methods are often restricted
in their application. The use of advanced and theoretically well-founded math
ematical methods should lead to the construction of robust shape descriptors
having more general application.
Shape description can be regarded as a meeting point of vision research,
mathematics, computing science, and the application fields of image analy
sis, computer vision, and artificial intelligence. The NATO Advanced Research
Workshop "Shape in Picture" was organised with a twofold objective: first, it
should provide all participants with an overview of relevant developments in
these different disciplines; second, it should stimulate researchers to exchange
original results and ideas across the boundaries of these disciplines.
This book comprises a widely drawn selection of papers presented at the
workshop, and many contributions have been revised to reflect further progress
in the field. The focus of this collection is on mathematical approaches to the
construction of shape descriptions from grey-level images. The book is divided
into five parts, each devoted to a different discipline. Each part contains papers
that have tutorial sections; these are intended to assist the reader in becoming
acquainted with the variety of approaches to the problem. It is hoped that the
collection may thus be useful as a reference work and as a graduate text.
The editors would like to thank all who contributed to the production of
the proceedings and to the workshop. More specifically, the editors are grateful
to the authors for their essential contributions, to the numerous reviewers for
their constructive comments, to the English-language editorial assistant for her
precise corrections to the text, and to the participants for making the workshop
successful. In particular, the editors wish to express their sincere gratitude to
the NATO Scientific Affairs Division and The United States Air Force European
Office of Aerospace Research and Development (EOARD) for their generous
support, and to the co-sponsors for making this undertaking possible.
VI Preface
The editors have, as far as possible, tried to minimize errors and omissions in
this mathematically oriented work. Nevertheless, they recognize the magnitude
of the problem and apologize in advance for any failings.
October 1993 0 Ying-Lie
Alexander Toet
David H. Foster
Henk J.A.M. Heijmans
Peter Meer
Director
Alexander Toet
Organizing Committee
0 Ying-Lie
David H. Foster
Henk J.A.M. Heijmans
Peter Meer
Sponsors
NATO Scientific Affairs Division
The United States Air Force European Office of Aerospace Research and Devel
opment (EOARD)
Local co-sponsors
Netherlands Organisation for Applied Scientific Research, Institute for Human
Factors (TNO-IZF)
Centre for Mathematics and Computer Science ( CWI)
The Foundation for Computer Science in The Netherlands (SION)
Local Organizing Committee
Peter N acken
Frank Kooi
Antoine de Reus
English-language Editorial Assistant
Kay B. Merifield
Correspondence should be addressed to Alexander Toet.
Table of Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Part 1 Mathematical Background
Topology and Geometry
The Khalimsky Line as a Foundation for Digital Topology . 3
Ralph K opperman
Topological Foundations of Shape Analysis 21
Vladimir A. K ovalevsky
A New Concept for Digital Geometry 37
Vladimir A. K ovalevsky
Theoretical Approaches to N-Dimensional Digital Objects . 53
Klaus Voss
On Boundaries and Boundary Crack-Codes of Multidimensional Digital
Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
T. Yung Kong
Studying Shape Through Size Functions 81
Claudio Uras and Alessandro Verri
Categorical Shape Theory
Introduction to Categorical Shape Theory, with Applications in
Mathematical Morphology . . . . . . . . . . . . . . . . . . . 91
Mirek Husek
Shape Theory: an ANR-Sequence Approach . . . . . . . . . . . . . . . . . 111
Jack Segal
Can Categorical Shape Theory Handle Grey-level Images? . . . . . . . . . 127
Timothy Porter
viii Table of Contents
Part 2 Local Extraction
Mathematical Morphology
Mathematical Morphology as a Tool for Shape Description . . . . . . . . 147
Henk J.A.M. Heijmans
On Information Contained in the Erosion Curve . . . . . . . . . . . . . . 177
Juliette Mattioli and Michel Schmitt
Morphological Area Openings and Closings for Grey-scale Images . . . . . 197
Luc Vincent
Manifold Shape: from Differential Geometry to Mathematical
Morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Jos B. T.M. Roerdink
On Negative Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Pijush K. Ghosh
Wavelets
An Overview of the Theory and Applications of Wavelets . . . . . . . . . 249
Bjorn Jawerth and Wim Sweldens
Fractal Surfaces, Multiresolution Analyses, and Wavelet Transforms . . . 275
JeffreyS. Geronimo, Douglas P. Hardin, and Peter R. Massopust
Interpolation in Multiscale Representations ..... 291
Charles H. Anderson and Subrata Rakshit
Part 3 Theory of Shape
Keynote Address
Discrete Stochastic Growth Models for Two-Dimensional Shapes . . . . . 301
Scott Thompson and Azriel Rosenfeld
Differential Geometry
Classical and Fuzzy Differential Methods in Shape Analysis . . . . . . . . 319
David H. Foster
Elements of a Fuzzy Geometry for Visual Space . . . . . . . . . . . . . . . 333
Mario Ferraro and David H. Foster
On the Relationship Between Surface Covariance and Differential
Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
Jens Berkmann and Terry Caelli
Image Representation Using Affine Covariant Coordinates . . . . . . . . . 353
Jun Zhang
Table of Contents ix
Theory of Shape Perception
Equivariant Dynamical Systems: a Formal Model for the Generation of
Arbitrary Shapes . . . . . . . 363
William C. Hoffman
Neural Processing of Overlapping Shapes . . . . . . . . . . . . . . . . . . 383
Andre J. N oest
Contour Texture and Frame Curves for the Recognition of Non-Rigid
Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
J. Brian Subirana-Vilanova
Part 4 Symbolic Representation
Shape Primitives
Conic Primitives for Projectively Invariant Representation of Planar
Curves ................................... 403
Stefan Carlsson
Blind Approximation of Planar Convex Shapes . . . . . . . . . . . . . . . 415
Michael Lindenbaum and Alfred M. Bruckstein
Recognition of Affine Planar Curves Using Geometric Properties . . . . . 423
Craig Gotsman and Michael Werman
Recognizing 3-D Curves from a Stereo Pair of Images: a Semi-differential
Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
Theo Moons, Eric J. Pauwels, Luc J. Van Goo!, Michael H. Brill,
and Eamon B. Barrett
Statistical Shape Methodology in Image Analysis . . . . . . . . . . . . . . 443
John T. Kent and Kanti V. Mardia
Recognition of Shapes from a Finite Series of Plane Figures . . . . . . . . 453
Nikolai M. Sirakov
Polygonal Harmonic Shape Characterization . . . . . . . . . . . . . . . . 463
Anthony J. Maeder, Andrew J. Davison, and Nigel N. Clark
Shape Description and Classification Using the Interrelationship of
Structures at Multiple Scales . . . . . . . . . . . . . . . . . . . . . . . 473
Gregory Dudek
Learning Shape Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
Stanley M. Dunn and K yugon Cho
Hierarchical Representation
Inference of Stochastic Graph Models for 2-D and 3-D Shapes . . . . . . . 493
Jakub Segen
x Table of Contents
Hierarchical Shape Analysis in Grey-level Images . . . . . . . 511
Annick Montanvert, Peter Meer, and Pascal Bertolino
Irregular Curve Pyramids . . . . . . . . . . . . . . . . . . . . . . . 525
Walter G. Kropatsch and Dieter Willersinn
Multiresolution Shape Description by Corners . . . . . . . . . . . . . . . . 539
Cornelia Fermuller and Walter G. Kropatsch
Model-based Bottom-Up Grouping of Geometric Image Primitives . . . . 549
Peter Nacken and Alexander Toet
Hierarchical Shape Representation for Image Analysis . . . . . . . . . . . 559
0 Ying-Lie
Part 5 Evolutionary Systems
Evolutionary Representation
Scale-Space for N-dimensional Discrete Signals . . . . . . . . . . . . . . . 571
Tony Lindeberg
Scale-Space Behaviour and Invariance Properties of Differential
Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591
Tony Lindeberg
Exploring the Shape Manifold: the Role of Conservation Laws . . . . . 601
Benjamin B. Kimia, Allen R. Tannenbaum, and Steven W. Zucker
Performance in Noise of a Diffusion-based Shape Descriptor . . . . 621
Murray H. Loew and Sheng-Yuan Hwang
Towards a Morphological Scale-Space Theory . . . . . . . . . . . . . . . . 631
Rein van den Boomgaard and Arnold W.M. Smeulders
Geometry-based Image Segmentation Using Anisotropic Diffusion. . . . . 641
Ross T. Whitaker and Stephen M. Pizer
Multiscale Description
Images: Regular Tempered Distributions . . . . . . . . . . . . . . . . . . . 651
Luc M. J. Florack, Bart M. ter Haar Romeny, Jan J. Koenderink,
and Max A. Viergever
Local and Multilocal Scale-Space Description . . . . . . . . . . . . . . . 661
Alfons H. Salden, Bart M. ter Haar Romeny, and Max A. Viergever
List of Authors . 671
Subject Index . 673
