Shape in Picture: Mathematical Description of Shape in Grey-level Images PDF
Preview Shape in Picture: Mathematical Description of Shape in Grey-level Images
Shape in Picture Mathematical Description of Shape in Grey-level Images NATO ASI Series Advanced Science Institutes Series A series presenting the results of activities sponsored by the NATO Science Committee, which aims at the dissemination of advanced scientific and technological knowledge, with a view to strengthening links between scientific communities. The Series is published by an international board of publishers in conjunction with the NATO Scientific Affairs Division A Life Sciences Plenum Publishing Corporation B Physics London and NewYork C Mathematical and Kluwer Academic Publishers Physical Sciences Dordrecht, Boston and London D Behavioural and Social Sciences E Applied Sciences F Computer and Springer-Verlag Systems Sciences Berlin Heidelberg New York G Ecological Sciences London Paris Tokyo Hong Kong H Cell Biology Barcelona Budapest I Global Environmental Change NATO-PCO DATABASE The electronic index to the NATO ASI Series provides full bibliographical references (with keywords and/or abstracts) to more than 30000 contributions from international scientists published in all sections of the NATO ASI Series. 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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 CIP data applied lor. This work is subject to copyright. AII rights are reserved, whether the whole or part 01 the material is concerned, specilicallythe rights oftranslation. reprinting. reuse 01 iIIustrations. recitation. broadcast ing, reproduction on microlilms or in any other way. and storage in data banks. Duplication 01 this publication or parts thereol is permitted only under the provisions 01 the German Copyright Law 01 September 9, 1965. in its current version. and permission lor use must always be obtained Irom Springer-Verlag Berlin Heidelberg GmbH . Violations are liable lor prosecution under the German Copyright Law. © 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 45/3140 -5432 1 0-Printed on acid-Iree paper 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
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