of Electrical Engineering and Computer Science, University of Florida, 454 New Engineering Building 33, Center Drive PO Box 116130, Gainesville, FL 32611-6130, U.S., anke@alpha.ee.ufl.ed
Trang 4Pattern Recognition
Editors Bernd Jähne
Interdisciplinary Center for Scientific ComputingUniversity of Heidelberg, Heidelberg, Germany
andScripps Institution of Oceanography
University of California, San Diego
Horst Haußecker Peter Geißler
Interdisciplinary Center for Scientific ComputingUniversity of Heidelberg, Heidelberg, Germany
ACADEMIC PRESS
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Trang 5Copyright © 1999 by Academic Press.
All rights reserved.
No part of this publication may be reproduced or transmitted in any form or
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The appearance of code at the bottom of the first page of a chapter in this book indicates the Publisher’s consent that copies of the chapter may be made for personal or internal use of specific clients This consent is given on the con- dition, however, that the copier pay the stated per-copy fee through the Copy- right Clearance Center, Inc (222 Rosewood Drive, Danvers, Massachusetts 01923), for copying beyond that permitted by Sections 107 or 108 of the U.S Copyright Law This consent does not extend to other kinds of copying, such
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Library of Congress Cataloging-In-Publication Data
Handbook of computer vision and applications / edited by Bernd Jähne, Horst Haussecker, Peter Geissler.
p cm.
Includes bibliographical references and indexes.
Contents: v 1 Sensors and imaging — v 2 Signal processing and
pattern recognition — v 3 Systems and applications.
ISBN 0–12–379770–5 (set) — ISBN 0–12–379771-3 (v 1)
ISBN 0–12–379772–1 (v 2) — ISBN 0–12–379773-X (v 3)
1 Computer vision — Handbooks, manuals etc I Jähne, Bernd
1953– II Haussecker, Horst, 1968– III Geissler, Peter, 1966– TA1634.H36 1999
CIP Printed in the United States of America
99 00 01 02 03 DS 9 8 7 6 5 4 3 2 1
Trang 6I Signal Representation
B Jähne
2.1 Introduction 10
2.2 Continuous signals 10
2.3 Discrete signals 13
2.4 Relation between continuous and discrete signals 23
2.5 Quantization 30
2.6 References 34
3 Spatial and Fourier Domain 35 B Jähne 3.1 Vector spaces and unitary transforms 35
3.2 Continuous Fourier transform (FT) 41
3.3 The discrete Fourier transform (DFT) 51
3.4 Fast Fourier transform algorithms (FFT) 57
3.5 References 66
4 Multiresolutional Signal Representation 67 B Jähne 4.1 Scale in signal processing 67
4.2 Scale filters 70
4.3 Scale space and diffusion 76
4.4 Multigrid representations 84
4.5 References 90
v
Trang 7II Elementary Spatial Processing
B Jähne
5.1 Introduction 94
5.2 Basics 94
5.3 Linear shift-invariant filters 98
5.4 Recursive filters 106
5.5 Classes of nonlinear filters 113
5.6 Efficient neighborhood operations 116
5.7 References 124
6 Principles of Filter Design 125 B Jähne, H Scharr, and S Körkel 6.1 Introduction 125
6.2 Filter design criteria 126
6.3 Windowing techniques 128
6.4 Filter cascading 132
6.5 Filter design as an optimization problem 133
6.6 Design of steerable filters and filter families 143
6.7 References 151
7 Local Averaging 153 B Jähne 7.1 Introduction 153
7.2 Basic features 154
7.3 Box filters 158
7.4 Binomial filters 163
7.5 Cascaded averaging 167
7.6 Weighted averaging 173
7.7 References 174
8 Interpolation 175 B Jähne 8.1 Introduction 175
8.2 Basics 176
8.3 Interpolation in Fourier space 180
8.4 Polynomial interpolation 182
8.5 Spline-based interpolation 187
8.6 Optimized interpolation 190
8.7 References 192
9 Image Warping 193 B Jähne 9.1 Introduction 193
9.2 Forward and inverse mapping 194
9.3 Basic geometric transforms 195
9.4 Fast algorithms for geometric transforms 199
9.5 References 206
Trang 811.3 A general scale-selection principle 247
11.4 Feature detection with automatic scale selection 251
11.5 Feature localization with automatic scale selection 262
11.6 Stereo matching with automatic scale selection 265
11.7 Summary and conclusions 269
11.8 References 270
12 Texture Analysis 275 T Wagner 12.1 Importance of texture 276
12.2 Feature sets for texture analysis 278
12.3 Assessment of textural features 299
12.4 Automatic design of texture analysis systems 306
12.5 References 307
13 Motion 309 H Haußecker and H Spies 13.1 Introduction 310
13.2 Basics: flow and correspondence 312
13.3 Optical flow-based motion estimation 321
13.4 Quadrature filter techniques 345
13.5 Correlation and matching 353
13.6 Modeling of flow fields 356
13.7 Confidence measures and error propagation 369
13.8 Comparative analysis 373
13.9 References 392
14 Bayesian Multiscale Differential Optical Flow 397 E P Simoncelli 14.1 Introduction 397
14.2 Differential formulation 398
14.3 Uncertainty model 400
14.4 Coarse-to-fine estimation 404
14.5 Implementation issues 410
14.6 Examples 414
14.7 Conclusion 419
14.8 References 420
Trang 915 Nonlinear Diffusion Filtering 423
J Weickert
15.1 Introduction 424
15.2 Filter design 425
15.3 Continuous theory 433
15.4 Algorithmic details 436
15.5 Discrete theory 439
15.6 Parameter selection 441
15.7 Generalizations 444
15.8 Summary 446
15.9 References 446
16 Variational Methods 451 C Schnörr 16.1 Introduction 451
16.2 Processing of two- and three-dimensional images 455
16.3 Processing of vector-valued images 471
16.4 Processing of image sequences 476
16.5 References 481
17 Stereopsis - Geometrical and Global Aspects 485 H A Mallot 17.1 Introduction 485
17.2 Stereo geometry 487
17.3 Global stereopsis 499
17.4 References 502
18 Stereo Terrain Reconstruction by Dynamic Programming 505 G Gimel’farb 18.1 Introduction 505
18.2 Statistical decisions in terrain reconstruction 509
18.3 Probability models of epipolar profiles 514
18.4 Dynamic programming reconstruction 520
18.5 Experimental results 524
18.6 References 528
19 Reflectance-Based Shape Recovery 531 R Klette, R Kozera, and K Schlüns 19.1 Introduction 532
19.2 Reflection and gradients 539
19.3 Three light sources 552
19.4 Two light sources 559
19.5 Theoretical framework for shape from shading 571
19.6 Shape from shading 574
19.7 Concluding remarks 586
19.8 References 587
20 Depth-from-Focus 591 P Geißler and T Dierig 20.1 Introduction 592
20.2 Basic concepts 593
20.3 Principles of depth-from-focus algorithms 595
Trang 1022 Fuzzy Image Processing 683
H Haußecker and H R Tizhoosh
22.1 Introduction 684
22.2 Why fuzzy image processing? 691
22.3 Fuzzy image understanding 692
22.4 Fuzzy image processing systems 699
22.5 Theoretical components of fuzzy image processing 702
22.6 Selected application examples 714
22.7 Conclusions 721
22.8 References 722
23 Neural Net Computing for Image Processing 729 A Meyer-Bäse 23.1 Introduction 729
23.2 Multilayer perceptron (MLP) 730
23.3 Self-organizing neural networks 736
23.4 Radial-basis neural networks (RBNN) 740
23.5 Transformation radial-basis networks (TRBNN) 743
23.6 Hopfield neural networks 747
23.7 References 751
24 Graph Theoretical Concepts for Computer Vision 753 D Willersinn et al. 24.1 Introduction 754
24.2 Basic definitions 754
24.3 Graph representation of two-dimensional digital images 760
24.4 Voronoi diagrams and Delaunay graphs 762
24.5 Matching 775
24.6 Graph grammars 780
24.7 References 786
25 Shape Reconstruction from Volumetric Data 791 R Eils and K Sätzler 25.1 Introduction 791
25.2 Incremental approach 794
Trang 1125.3 Three-dimensional shape reconstruction from contour lines 797
25.4 Volumetric shape reconstruction 802
25.5 Summary 811
25.6 References 813
26 Probabilistic Modeling in Computer Vision 817 J Hornegger, D Paulus, and H Niemann 26.1 Introduction 817
26.2 Why probabilistic models? 819
26.3 Object recognition: classification and regression 821
26.4 Parametric families of model densities 826
26.5 Automatic model generation 844
26.6 Practical issues 850
26.7 Summary, conclusions, and discussion 852
26.8 References 852
27 Knowledge-Based Interpretation of Images 855 H Niemann 27.1 Introduction 855
27.2 Model of the task domain 859
27.3 Interpretation by optimization 864
27.4 Control by graph search 865
27.5 Control by combinatorial optimization 868
27.6 Judgment function 870
27.7 Extensions and remarks 872
27.8 References 872
28 Visualization of Volume Data 875 J Hesser and C Poliwoda 28.1 Selected visualization techniques 876
28.2 Basic concepts and notation for visualization 880
28.3 Surface rendering algorithms and OpenGL 881
28.4 Volume rendering 884
28.5 The graphics library VGL 890
28.6 How to use volume rendering 898
28.7 Volume rendering 901
28.8 Acknowledgments 905
28.9 References 905
29 Databases for Microscopes and Microscopical Images 907 N Salmon, S Lindek, and E H K Stelzer 29.1 Introduction 908
29.2 Towards a better system for information management 909
29.3 From flat files to database systems 911
29.4 Database structure and content 912
29.5 Database system requirements 917
29.6 Data flow—how it looks in practice 918
29.7 Future prospects 921
29.8 References 925
Trang 12ern physical sciences and the many novel techniques to acquire images.The second is between basic research and applications When a readerwith a background in one of the fields related to computer vision feels
he has learned something from one of the many other facets of puter vision, the handbook will have fulfilled its purpose
com-The handbook comprises three volumes com-The first volume, Sensors and Imaging, covers image formation and acquisition The second vol- ume, Signal Processing and Pattern Recognition, focuses on processing
of the spatial and spatiotemporal signal acquired by imaging sensors
The third volume, Systems and Applications, describes how computer
vision is integrated into systems and applications
Prerequisites
It is assumed that the reader is familiar with elementary mathematicalconcepts commonly used in computer vision and in many other areas
of natural sciences and technical disciplines This includes the basics
of set theory, matrix algebra, differential and integral equations, plex numbers, Fourier transform, probability, random variables, andgraphing Wherever possible, mathematical topics are described intu-itively In this respect it is very helpful that complex mathematicalrelations can often be visualized intuitively by images For a more for-
com-xi
Trang 13mal treatment of the corresponding subject including proofs, suitablereferences are given.
How to use this handbook
The handbook has been designed to cover the different needs of its
readership First, it is suitable for sequential reading In this way the
reader gets an up-to-date account of the state of computer vision It ispresented in a way that makes it accessible for readers with differentbackgrounds Second, the reader can look up specific topics of inter-est The individual chapters are written in a self-consistent way withextensive cross-referencing to other chapters of the handbook and ex-ternal references The CD that accompanies each volume of the hand-book contains the complete text of the handbook in the Adobe Acrobatportable document file format (PDF) This format can be read on allmajor platforms Free Acrobat reader version 3.01 for all major com-puting platforms is included on the CDs The texts are hyperlinked inmultiple ways Thus the reader can collect the information of interestwith ease Third, the reader can delve more deeply into a subject withthe material on the CDs They contain additional reference material,interactive software components, code examples, image material, andreferences to sources on the Internet For more details see the readmefile on the CDs
Acknowledgments
Writing a handbook on computer vision with this breadth of topics is
a major undertaking that can succeed only in a coordinated effort thatinvolves many co-workers Thus the editors would like to thank firstall contributors who were willing to participate in this effort Theircooperation with the constrained time schedule made it possible thatthe three-volume handbook could be published in such a short periodfollowing the call for contributions in December 1997 The editors aredeeply grateful for the dedicated and professional work of the staff atAEON Verlag & Studio who did most of the editorial work We alsoexpress our sincere thanks to Academic Press for the opportunity towrite this handbook and for all professional advice
Last but not least, we encourage the reader to send us any hints
on errors, omissions, typing errors, or any other shortcomings of thehandbook Actual information about the handbook can be found at theeditors homepagehttp://klimt.iwr.uni-heidelberg.de
Heidelberg, Germany and La Jolla, California, December 1998
Bernd Jähne, Horst Haußecker, Peter Geißler
Trang 14nology in 1990 and 1995, respectively From 1995 to
1996 she was a postdoctoral fellow with the Federal tute of Neurobiology, Magdeburg, Germany Since 1996 she was a visiting assistant professor with the Dept of Electrical Engineering, University of Florida, Gainesville, USA She received the Max-Kade award in Neuroengineer- ing in 1996 and the Lise-Meitner prize in 1997 Her re- search interests include neural networks, image process- ing, biomedicine, speech recognition, and theory of non- linear systems.
Insti-Dr Anke Meyer-Bäse, Dept of Electrical Engineering and Computer Science, University of Florida, 454 New Engineering Building 33, Center Drive
PO Box 116130, Gainesville, FL 32611-6130, U.S., anke@alpha.ee.ufl.edu
Tobias Dierig graduated in 1997 from the University of
Heidelberg with a master degree in physics and is now pursuing his PhD at the Interdisciplinary Center for Sci- entific Computing at Heidelberg university He is con- cerned mainly with depth from focus algorithms, image fusion, and industrial applications of computer vision within the OpenEye project.
Tobias Dierig, Forschungsgruppe Bildverarbeitung, IWR Universität Heidelberg, Im Neuenheimer Feld 368 D-69120 Heidelberg, Germany
Tobias.Dierig@iwr.uni-heidelberg.de
http://klimt.iwr.uni-heidelberg.de/˜tdierig
xiii
Trang 15Roland Eils studied mathematics and computer science
in Aachen, where he received his diploma in 1990 After
a two year stay in Indonesia for language studies he joint the Graduiertenkolleg “Modeling and Scientific Comput- ing in Mathematics and Sciences” at the Interdisciplinary Center for Scientific Computing (IWR), University of Hei- delberg, where he received his doctoral degree in 1995 Since 1996 he has been leading the biocomputing group,
Structures in Molecular Biology His research interests
include computer vision, in particular computational ometry, and application of image processing techniques
ge-in science and biotechnology.
Dr Roland Eils, Biocomputing-Gruppe, IWR, Universität Heidelberg
Im Neuenheimer Feld 368, D-69120 Heidelberg, Germany
eils@iwr.uni-heidelberg.de
http://www.iwr.uni-heidelberg.de/iwr/bioinf
Peter Geißler studied physics in Heidelberg He received
his diploma and doctoral degree from Heidelberg versity in 1994 and 1998, respectively His research in- terests include computer vision, especially depth-from- focus, adaptive filtering, and flow visualization as well as the application of image processing in physical sciences and oceanography.
Uni-Dr Peter Geißler Forschungsgruppe Bildverarbeitung, IWR Universität Heidelberg, Im Neuenheimer Feld 368 D-69120 Heidelberg, Germany
Peter.Geissler@iwr.uni-heidelberg.de
http://klimt.iwr.uni-heidelberg.de
Georgy Gimel’farb received his PhD degree from the
Ukrainian Academy of Sciences in 1969 and his Doctor of Science (the habilitation) degree from the Higher Certify- ing Commission of the USSR in 1991 In 1962, he began working in the Pattern Recognition, Robotics, and Image Recognition Departments of the Institute of Cybernetics (Ukraine) In 1994–1997 he was an invited researcher in Hungary, the USA, Germany, and France Since 1997, he has been a senior lecturer in computer vision and digital
TV at the University of Auckland, New Zealand His search interests include analysis of multiband space and aerial images, computational stereo, and image texture analysis.
re-Dr Georgy Gimel’farb, Centre for Image Technology and Robotics,
Department of Computer Science, Tamaki Campus
The University of Auckland, Private Bag 92019, Auckland 1, New Zealand g.gimelfarb@auckland.ac.nz , http://www.tcs.auckland.ac.nz/˜georgy
Trang 16Jürgen Hesser is assistant professor at the Lehrstuhl für
Informatik V, University of Mannheim, Germany He heads the groups on computer graphics, bioinformat- ics, and optimization His research interests are real- time volume rendering, computer architectures, compu- tational chemistry, and evolutionary algorithms In addi- tion, he is co-founder of Volume Graphics GmbH, Heidel- berg Hesser received his PhD and his diploma in physics
at the University of Heidelberg, Germany.
Jürgen Hesser, Lehrstuhl für Informatik V Universität Mannheim
B6, 26, D-68131 Mannheim, Germany jhesser@rumms.uni-mannheim.de ,
Joachim Hornegger graduated in 1992 and received his
PhD degree in computer science in 1996 from the versität Erlangen-Nürnberg, Germany, for his work on statistical object recognition Joachim Hornegger was research and teaching associate at Universität Erlangen- Nürnberg, a visiting scientist at the Technion, Israel, and
Uni-at the Massachusetts Institute of Technology, U.S He
is currently a research scholar and teaching associate
at Stanford University, U.S Joachim Hornegger is the author of 30 technical papers in computer vision and speech processing and three books His research inter- ests include 3-D computer vision, 3-D object recognition, and statistical meth- ods applied to image analysis problems.
Dr Joachim Hornegger, Stanford University, Robotics Laboratory
Gates Building 1A, Stanford, CA 94305-9010, U.S.
jh@robotics.stanford.edu , http://www.robotics.stanford.edu/˜jh
Trang 17Bernd Jähne studied physics in Saarbrücken and
Hei-delberg He received his diploma, doctoral degree, and habilitation degree from Heidelberg University in 1977,
1980, and 1985, respectively, and a habilitation gree in applied computer science from the University of Hamburg-Harburg in 1992 Since 1988 he has been a Ma- rine Research Physicist at Scripps Institution of Oceanog- raphy, University of California, and, since 1994, he has been professor of physics at the Interdisciplinary Center
de-of Scientific Computing He leads the research group on image processing His research interests include com- puter vision, especially filter design and image sequence analysis, the application of image processing techniques
in science and industry, and small-scale air-sea interaction processes Prof Dr Bernd Jähne, Forschungsgruppe Bildverarbeitung, IWR
Universität Heidelberg, Im Neuenheimer Feld 368, D-69120 Heidelberg Bernd.Jaehne@iwr.uni-heidelberg.de
http://klimt.iwr.uni-heidelberg.de
Reinhard Klette studied mathematics at Halle University,
received his master degree and doctor of natural science degree in mathematics at Jena University, became a do- cent in computer science, and was a professor of com- puter vision at Berlin Technical University Since June
1996 he has been professor of information technology
in the Department of Computer Science at the University
of Auckland His research interests include theoretical and applied topics in image processing, pattern recogni- tion, image analysis, and image understanding He has published books about image processing and shape reconstruction and was chairman of several international conferences and workshops on computer vision Recently, his research interests have been directed at 3-D biomedical image analysis with digital geometry and computational geometry as major subjects.
Prof Dr Reinhard Klette, Centre for Image Technology and Robotics,
Computer Science Department, Tamaki Campus
The Auckland University, Private Bag 92019, Auckland, New Zealand
r.klette@auckland.ac.nz , http://citr.auckland.ac.nz/˜rklette
Christoph Klauck received his diploma in computer
sci-ence and mathematics from the University of slautern, Germany, in 1990 From 1990 to 1994 he worked as research scientist at the German Research Center for Artificial Intelligence Inc (DFKI GmbH) at Kaiserslautern In 1994 he finished his dissertation in computer science Since then he has been involved in the IRIS project at the University of Bremen (Artificial Intelligence Group) His primary research interests in- clude graph grammars and rewriting systems in general, knowledge representation, and ontologies.
Trang 18http://www.iwr.uni-heidelberg.de/˜Stefan.Koerkel/
Ryszard Kozera received his M.Sc degree in pure
mathe-matics in 1985 from Warsaw University, Poland, his PhD degree in computer science in 1991 from Flinders Uni- versity, Australia, and finally his PhD degree in mathe- matics in 1992 from Warsaw University, Poland He is currently employed as a senior lecturer at the University
of Western Australia Between July 1995 and February
1997, Dr Kozera was at the Technical University of Berlin and at Warsaw University as an Alexander von Humboldt Foundation research fellow His current research inter- ests include applied mathematics with special emphasis
on partial differential equations, computer vision, and numerical analysis.
Dr Ryszard Kozera, Department of Computer Science, The University of ern Australia, Nedlands, WA 6907, Australia, ryszard@cs.uwa.edu.au
West-http://www.cs.uwa.edu.au/people/info/ryszard.html
Tony Lindeberg received his M.Sc degree in
engineer-ing physics and applied mathematics from KTH (Royal Institute of Technology), Stockholm, Sweden in 1987, and his PhD degree in computing science in 1991 He
is currently an associate professor at the Department
of Numerical Analysis and Computing Science at KTH His main research interests are in computer vision and relate to multiscale representations, focus-of-attention, and shape He has contributed to the foundations of continuous and discrete scale-space theory, as well as
to the application of these theories to computer vision problems Specifically, he has developed principles for automatic scale selection, methodologies for extracting salient image struc- tures, and theories for multiscale shape estimation He is author of the book
“Scale-Space Theory in Computer Vision.”
Trang 19Tony Lindeberg, Department of Numerical Analysis and Computing Science KTH, S-100 44 Stockholm, Sweden.
tony@nada.kth.se , http://www.nada.kth.se/˜tony
Steffen Lindek studied physics at the RWTH Aachen,
Ger-many, the EPF Lausanne, Switzerland, and the sity of Heidelberg, Germany He did his diploma and PhD theses in the Light Microscopy Group at the Euro- pean Molecular Biology Laboratory (EMBL), Heidelberg, Germany, developing high-resolution light-microscopy techniques Since December 1996 he has been a post- doctoral fellow with the BioImage project at EMBL He currently works on the design and implementation of the image database, and he is responsible for the administra- tion of EMBL’s contribution to the project.
Univer-Dr Steffen Lindek, European Molecular Biology Laboratory (EMBL)
Postfach 10 22 09, D-69120 Heidelberg, Germany
lindek@EMBL-Heidelberg.de
Hanspeter A Mallot studied biology and mathematics at
the University of Mainz where he also received his toral degree in 1986 He was a postdoctoral fellow at the Massachusetts Institute of Technology in 1986/87 and held research positions at Mainz University and the Ruhr-Universität-Bochum In 1993, he joined the Max- Planck-Institut für biologische Kybernetik in Tübingen.
doc-In 1996/97, he was a fellow at the doc-Institute of Advanced Studies in Berlin His research interests include the per- ception of shape and space in humans and machines, cognitive maps, as well as neural network models of the cerebral cortex.
Dr Hanspeter A Mallot, Max-Planck-Institut für biologische Kybernetik Spemannstr 38, 72076 Tübingen, Germany
Hanspeter.Mallot@tuebingen.mpg.de
http://www.kyb.tuebingen.mpg.de/bu/
Heinrich Niemann obtained the degree of Dipl.-Ing in
electrical engineering and Dr.-Ing at Technical sity Hannover in 1966 and 1969, respectively From
Univer-1967 to 1972 he was with Fraunhofer Institut für formationsverarbeitung in Technik und Biologie, Karls- ruhe Since 1975 he has been professor of computer sci- ence at the University of Erlangen-Nürnberg and since
In-1988 he has also served as head of the research group, Knowledge Processing, at the Bavarian Research Institute for Knowledge-Based Systems (FORWISS) His fields of research are speech and image understanding and the application of artificial intelligence techniques in these fields He is the author or co-author of 6 books and approximately 250 jour- nal and conference contributions.
Trang 20Universität Erlangen-Nürnberg, Martensstr 3, 91058 Erlangen, Germany paulus@informatik.uni-erlangen.de
http://www5.informatik.uni-erlangen.de
Christoph Poliwoda is PhD student at the Lehrstuhl für
Informatik V, University of Mannheim, and leader of the development section of Volume Graphics GmbH His re- search interests are real-time volume and polygon ray- tracing, 3-D image processing, 3-D segmentation, com- puter architectures and parallel computing Poliwoda received his diploma in physics at the University of Hei- delberg, Germany.
Christoph Poliwoda Lehrstuhl für Informatik V Universität Mannheim B6, 26, D-68131 Mannheim, Germany poliwoda@mp-sun1.informatik.uni-mannheim.de
Nicholas J Salmon received the master of engineering
degree from the Department of Electrical and Electronic Engineering at Bath University, England, in 1990 Then
he worked as a software development engineer for coni Radar Systems Ltd., England, helping to create a vastly parallel signal-processing machine for radar appli- cations Since 1992 he has worked as software engineer
Mar-in the Light Microscopy Group at the European lar Biology Laboratory, Germany, where he is concerned with creating innovative software systems for the con- trol of confocal microscopes, and image processing Nicholas J Salmon, Light Microscopy Group,
Molecu-European Molecular Biology Laboratory (EMBL)
Postfach 10 22 09, D-69120 Heidelberg, Germany
salmon@EMBL-Heidelberg.de ,
Trang 21Kurt Sätzler studied physics at the University of
Hei-delberg, where he received his diploma in 1995 Since then he has been working as a PhD student at the Max- Planck-Institute of Medical Research in Heidelberg His research interests are mainly computational geometry applied to problems in biomedicine, architecture and computer graphics, image processing and tilted view mi- croscopy.
Kurt Sätzler, IWR, Universität Heidelberg
Im Neuenheimer Feld 368, D-69120 Heidelberg or
Max-Planck-Institute for Medical Research, Department of Cell Physiology Jahnstr 29, D-69120 Heidelberg, Germany
Kurt.Saetzler@iwr.uni-heidelberg.de
Hanno Scharr studied physics at the University of
Hei-delberg, Germany and did his diploma thesis on ture analysis at the Interdisciplinary Center for Scien- tific Computing in Heidelberg Currently, he is pursu- ing his PhD on motion estimation His research interests include filter optimization and motion estimation in dis-
tex-crete time series of n-D images.
Hanno Scharr Interdisciplinary Center for Scientific Computing
Im Neuenheimer Feld 368, 69120 Heidelberg, Germany Hanno.Scharr@iwr.uni-heidelberg.de
http://klimt.iwr.uni-heidelberg.de/˜hscharr/
Karsten Schlüns studied computer science in Berlin He
received his diploma and doctoral degree from the nical University of Berlin in 1991 and 1996 From 1991 to
Tech-1996 he was research assistant in the Computer Vision Group, Technical University of Berlin, and from 1997
to 1998 he was a postdoctoral research fellow in puting and information technology, University of Auck- land Since 1998 he has been a scientist in the image processing group at the Institute of Pathology, Univer- sity Hospital Charité in Berlin His research interests include pattern recognition and computer vision, espe- cially three-dimensional shape recovery, performance analysis of reconstruc- tion algorithms, and teaching of computer vision.
com-Dr Karsten Schlüns, Institute of Pathology,
University Hospital Charité, Schumannstr 20/21, D-10098 Berlin, Germany Karsten.Schluens@charite.de , http://amba.charite.de/˜ksch
Trang 22include pattern recognition, machine vision, and related aspects of computer graphics, machine learning, and applied mathematics.
Prof Dr Christoph Schnörr, University of Mannheim
Dept of Math & Computer Science, D-68131 Mannheim, Germany
schnoerr@ti.uni-mannheim.de , http://www.ti.uni-mannheim.de
Eero Simoncelli started his higher education with a
bach-elor’s degree in physics from Harvard University, went
to Cambridge University on a fellowship to study matics for a year and a half, and then returned to the USA
mathe-to pursue a docmathe-torate in Electrical Engineering and puter Science at MIT He received his PhD in 1993, and joined the faculty of the Computer and Information Sci- ence Department at the University of Pennsylvania that same year In September of 1996, he joined the faculty
Com-of the Center for Neural Science and the Courant tute of Mathematical Sciences at New York University He received an NSF Faculty Early Career Development (CA- REER) grant in September 1996, for teaching and research in “Visual Informa- tion Processing”, and a Sloan Research Fellowship in February 1998.
Insti-Dr Eero Simoncelli, 4 Washington Place, RM 809, New York, NY 10003-6603 eero.simoncelli@nyu.edu , http://www.cns.nyu.edu/˜eero
Pierre Soille received the engineering degree from the
Université catholique de Louvain, Belgium, in 1988 He gained the doctorate degree in 1992 at the same univer- sity and in collaboration with the Centre de Morphologie Mathématique of the Ecole des Mines de Paris He then pursued research on image analysis at the CSIRO Math- ematical and Information Sciences Division, Sydney, the Centre de Morphologie Mathématique of the Ecole des Mines de Paris, and the Abteilung Mustererkennung of the Fraunhofer-Institut IPK, Berlin During the period 1995-1998 he was lecturer and research scientist at the Ecole des Mines d’Alès and EERIE, Nîmes, France Now he is a senior research scientist at the Silsoe Research Institute, England He worked on many ap-
Trang 23plied projects, taught tutorials during international conferences, co-organized the second International Symposium on Mathematical Morphology, wrote and edited three books, and contributed to over 50 scientific publications Prof Pierre Soille, Silsoe Research Institute, Wrest Park
Silsoe, Bedfordshire, MK45 4HS, United Kingdom
Pierre.Soille@bbsrc.ac.uk , http://www.bbsrc.ac.uk
Hagen Spies graduated in January 1998 from the
Univer-sity of Heidelberg with a master degree in physics He also received an MS in computing and information tech- nology from the University of Dundee, Scotland in 1995.
In 1998/1999 he spent one year as a visiting scientist at the University of Western Ontario, Canada Currently he works as a researcher at the Interdisciplinary Center for Scientific Computing at the University of Heidelberg His interests concern the measurement of optical and range flow and their use in scientific applications.
Hagen Spies, Forschungsgruppe Bildverarbeitung, IWR Universität Heidelberg, Im Neuenheimer Feld 368
D-69120 Heidelberg, Germany, Hagen.Spies@iwr.uni-heidelberg.de
http://klimt.iwr.uni-heidelberg.de/˜hspies
E H K Stelzer studied physics in Frankfurt am Main and
in Heidelberg, Germany During his Diploma thesis at the Max-Planck-Institut für Biophysik he worked on the physical chemistry of phospholipid vesicles, which he characterized by photon correlation spectroscopy Since
1983 he has worked at the European Molecular ogy Laboratory (EMBL) He has contributed extensively
Biol-to the development of confocal fluorescence microscopy and its application in life sciences His group works
on the development and application of high-resolution techniques in light microscopy, video microscopy, con- focal microscopy, optical tweezers, single particle analy- sis, and the documentation of relevant parameters with biological data Prof Dr E H K Stelzer, Light Microscopy Group,
European Molecular Biology Laboratory (EMBL), Postfach 10 22 09
D-69120 Heidelberg, Germany, stelzer@EMBL-Heidelberg.de,
Hamid R Tizhoosh received the M.S degree in electrical
engineering from University of Technology, Aachen, many, in 1995 From 1993 to 1996, he worked at Man- agement of Intelligent Technologies Ltd (MIT GmbH), Aachen, Germany, in the area of industrial image pro- cessing He is currently a PhD candidate, Dept of Tech- nical Computer Science of Otto-von-Guericke-University, Magdeburg, Germany His research encompasses fuzzy logic and computer vision His recent research efforts include medical and fuzzy image processing He is cur- rently involved in the European Union project INFOCUS, and is researching enhancement of medical images in radiation therapy.
Ger-H R Tizhoosh, University of Magdeburg (IPE)
Trang 24of industrial image processing systems.
Dr.-Ing Thomas Wagner, Fraunhofer Institut für Intregrierte Schaltungen
Am Weichselgarten 3, D-91058 Erlangen, Germany
wag@iis.fhg.de , http://www.iis.fhg.de
Joachim Weickert obtained a M.Sc in industrial
math-ematics in 1991 and a PhD in mathmath-ematics in 1996, both from Kaiserslautern University, Germany After re- ceiving the PhD degree, he worked as post-doctoral re- searcher at the Image Sciences Institute of Utrecht Uni- versity, The Netherlands In April 1997 he joined the computer vision group of the Department of Computer Science at Copenhagen University His current research interests include all aspects of partial differential equa- tions and scale-space theory in image analysis He was awarded the Wacker Memorial Prize and authored the book “Anisotropic Diffusion in Image Processing.”
Dr Joachim Weickert, Department of Computer Science, University of hagen, Universitetsparken 1, DK-2100 Copenhagen, Denmark
Copen-joachim@diku.dk , http://www.diku.dk/users/joachim/
Dieter Willersinn received his diploma in electrical
en-gineering from Technical University Darmstadt in 1988 From 1988 to 1992 he was with Vitronic Image Process- ing Systems in Wiesbaden, working on industrial appli- cations of robot vision and quality control He then took
a research position at the Technical University in Vienna, Austria, from which he received his PhD degree in 1995.
In 1995, he joined the Fraunhofer Institute for tion and Data Processing (IITB) in Karlsruhe, where he initially worked on obstacle detection for driver assis- tance applications Since 1997, Dr Willersinn has been the head of the group, Assessment of Computer Vision Systems, Department for Recognition and Diagnosis Systems.
Informa-Dr Dieter Willersinn, Fraunhofer Institut IITB, Fraunhoferstr 1
D-76131 Karlsruhe, Germany, wil@iitb.fhg.de
Trang 26signals processed in computer vision originate from the radiance of an
object that is collected by an optical system (Volume 1, Chapter5) The
irradiance received by a single photosensor or a 2-D array of
photosen-sors through the optical system is converted into an electrical signaland finally into arrays of digital numbers (Volume 2, Chapter2) Thewhole chain of image formation from the illumination and interaction
of radiation with the object of interest up to the arrays of digital bers stored in the computer is the topic of Volume 1 of this handbook
num-(subtitled Sensors and Imaging).
This volume deals with the processing of the signals generated byimaging sensors and this introduction covers four general topics Sec-tion1.1discusses in which aspects the processing of higher-dimension-
al signals differs from the processing of 1-D time series We also orate on the task of signal processing for computer vision Patternrecognition (Section1.2) plays a central role in computer vision because
elab-it uses the features extracted by lowlevel signal processing to classifyand recognize objects
Given the vast amount of data generated by imaging sensors thequestion of the computational complexity and of efficient algorithms is
of utmost importance (Section1.3) Finally, the performance evaluation
of computer vision algorithms (Section1.4) is a subject that has beenneglected in the past Consequently, a vast number of algorithms existfor which the performance characteristics are not sufficiently known
1Handbook of Computer Vision and Applications Copyright © 1999 by Academic Press
Trang 27This constitutes a major obstacle for progress of applications usingcomputer vision techniques.
1.1 Signal processing for computer vision
One-dimensional linear signal processing and system theory is a
stan-dard topic in electrical engineering and is covered by many stanstan-dardtextbooks, for example, [1,2] There is a clear trend that the classicalsignal processing community is moving into multidimensional signals,
as indicated, for example, by the new annual international IEEE ence on image processing (ICIP) This can also be seen from some re-cently published handbooks on this subject The digital signal process-ing handbook by Madisetti and Williams [3] includes several chaptersthat deal with image processing Likewise the transforms and applica-tions handbook by Poularikas [4] is not restricted to one-dimensionaltransforms
confer-There are, however, only a few monographs that treat signal cessing specifically for computer vision and image processing Themonograph of Lim [5] deals with 2-D signal and image processing andtries to transfer the classical techniques for the analysis of time series
pro-to 2-D spatial data Granlund and Knutsson [6] were the first to publish
a monograph on signal processing for computer vision and elaborate on
a number of novel ideas such as tensorial image processing and malized convolution that did not have their origin in classical signalprocessing
nor-Time series are 1-D, signals in computer vision are of higher mension They are not restricted to digital images, that is, 2-D spatialsignals (Chapter2) Volumetric sampling, image sequences and hyper- spectral imaging all result in 3-D signals, a combination of any of these
di-techniques in even higher-dimensional signals
How much more complex does signal processing become with creasing dimension? First, there is the explosion in the number of datapoints Already a medium resolution volumetric image with 5123vox-els requires 128 MB if one voxel carries just one byte Storage of evenhigher-dimensional data at comparable resolution is thus beyond thecapabilities of today’s computers Moreover, many applications requirethe handling of a huge number of images This is also why appropriatedatabases including images are of importance An example is discussed
in-in Chapter29
Higher dimensional signals pose another problem While we do nothave difficulty in grasping 2-D data, it is already significantly more de-manding to visualize 3-D data because the human visual system is builtonly to see surfaces in 3-D but not volumetric 3-D data The more di-
mensions are processed, the more important it is that computer
Trang 28graph-Chapter10is an important basis The selection of the proper scale for
image processing has recently come into the focus of attention ter 11) As signals processed in computer vision come from dynam-
(Chap-ical 3-D scenes, important features also include motion (Chapters13
and 14) and various techniques to infer the depth in scenes
includ-ing stereo (Chapters17and18), shape from shading and photometricstereo (Chapter19), and depth from focus (Chapter20)
There is little doubt that nonlinear techniques are crucial for
fea-ture extraction in computer vision However, compared to linear filtertechniques, these techniques are still in their infancy There is also nosingle nonlinear technique but there are a host of such techniques oftenspecifically adapted to a certain purpose [7] In this volume, a rathergeneral class of nonlinear filters by combination of linear convolutionand nonlinear point operations (Chapter10), and nonlinear diffusionfiltering (Chapter15) are discussed
1.2 Pattern recognition for computer vision
In principle, pattern classification is nothing complex Take some
ap-propriate features and partition the feature space into classes Why is
it then so difficult for a computer vision system to recognize objects?The basic trouble is related to the fact that the dimensionality of the in-put space is so large In principle, it would be possible to use the imageitself as the input for a classification task, but no real-world classifi-cation technique—be it statistical, neuronal, or fuzzy—would be able
to handle such high-dimensional feature spaces Therefore, the needarises to extract features and to use them for classification
Unfortunately, techniques for feature selection have widely been glected in computer vision They have not been developed to the samedegree of sophistication as classification where it is meanwhile well un-
Trang 29ne-derstood that the different techniques, especially statistical and neuraltechniques, can been considered under a unified view [8].
Thus part IV of this volume focuses in part on some more advancedfeature-extraction techniques An important role in this aspect is played
by morphological operators (Chapter21) because they manipulate theshape of objects in images Fuzzy image processing (Chapter22) con-tributes a tool to handle vague data and information
The remainder of part IV focuses on another major area in puter vision Object recognition can be performed only if it is possible
com-to represent the knowledge in an appropriate way In simple cases theknowledge can just be rested in simple models Probabilistic model-ing in computer vision is discussed in Chapter26 In more complexcases this is not sufficient The graph theoretical concepts presented
in Chapter24are one of the bases for knowledge-based interpretation
of images as presented in Chapter27
1.3 Computational complexity and fast algorithms
The processing of huge amounts of data in computer vision becomes aserious challenge if the number of computations increases more than
linear with the number of data points, M = N D (D is the dimension
of the signal) Already an algorithm that is of order O(M2) may be prohibitively slow Thus it is an important goal to achieve O(M) or at least O(M ld M) performance of all pixel-based algorithms in computer
vision Much effort has been devoted to the design of fast algorithms,that is, performance of a given task with a given computer system in aminimum amount of time This does not mean merely minimizing thenumber of computations Often it is equally or even more important
to minimize the number of memory accesses
Point operations are of linear order and take cM operations Thus they do not pose a problem Neighborhood operations are still of lin- ear order in the number of pixels but the constant c may become quite
large, especially for signals with high dimensions This is why there isalready a need to develop fast neighborhood operations Brute forceimplementations of global transforms such as the Fourier transform re-
quire cM2operations and can thus only be used at all if fast algorithmsare available Such algorithms are discussed in Section3.4 Many otheralgorithms in computer vision, such as correlation, correspondenceanalysis, and graph search algorithms are also of polynomial order,some of them even of exponential order
A general breakthrough in the performance of more complex
al-gorithms in computer vision was the introduction of multiresolutional data structures that are discussed in Chapters4and14 All chapters
Trang 30is even harder to perform a sophisticated error analysis On the otherhand, the computer vision community has ignored the fact to a largeextent that any algorithm is only as good as its objective and solidevaluation and verification.
Fortunately, this misconception has been recognized in the time and there are serious efforts underway to establish generally ac-
mean-cepted rules for the performance analysis of computer vision algorithms.
We give here just a brief summary and refer for details to Haralick et al.[9] and for a practical example to Volume 3, Chapter7 The three majorcriteria for the performance of computer vision algorithms are:
Successful solution of task Any practitioner gives this a top priority.
But also the designer of an algorithm should define precisely forwhich task it is suitable and what the limits are
Accuracy This includes an analysis of the statistical and systematic errors under carefully defined conditions (such as given signal-to- noise ratio (SNR), etc.).
Speed Again this is an important criterion for the applicability of an
algorithm
There are different ways to evaluate algorithms according to the mentioned criteria Ideally this should include three classes of studies:
fore-Analytical studies This is the mathematically most rigorous way to
verify algorithms, check error propagation, and predict catastrophicfailures
Performance tests with computer generated images These tests are
useful as they can be carried out under carefully controlled tions
condi-Performance tests with real-world images This is the final test for
practical applications
Trang 31Much of the material presented in this volume is written in the spirit
of a careful and mathematically well-founded analysis of the methodsthat are described although the performance evaluation techniques arecertainly more advanced in some areas than in others
1.5 References
[1] Oppenheim, A V and Schafer, R W., (1989) Discrete-time Signal
Process-ing Hall Signal Processing Series Englewood Cliffs, NJ:
Prentice-Hall.
[2] Proakis, J G and Manolakis, D G., (1992) Digital Signal Processing
Prin-ciples, Algorithms, and Applications New York: McMillan.
[3] Madisetti, V K and Williams, D B (eds.), (1997) The Digital Signal
Pro-cessing Handbook Boca Raton, FL: CRC Press.
[4] Poularikas, A D (ed.), (1996) The Transforms and Applications Handbook.
Boca Raton, FL: CRC Press.
[5] Lim, J S., (1990) Two-dimensional Signal and Image Processing Englewood
Cliffs, NJ: Prentice-Hall.
[6] Granlund, G H and Knutsson, H., (1995) Signal Processing for Computer
Vision Norwell, MA: Kluwer Academic Publishers.
[7] Pitas, I and Venetsanopoulos, A N., (1990) Nonlinear Digital Filters
Prin-ciples and Applications Norwell, MA: Kluwer Academic Publishers.
[8] Schürmann, J., (1996) Pattern Classification, a Unified View of Statistical
and Neural Approaches New York: John Wiley & Sons.
[9] Haralick, R M., Klette, R., Stiehl, H.-S., and Viergever, M (eds.), (1999)
Eval-uation and Validation of Computer Vision Algorithms Boston: Kluwer.
Trang 342.3.3 Irregular lattices 17 2.3.4 Metric in digital images 17 2.3.5 Neighborhood relations 19 2.3.6 Errors in object position and geometry 20 2.4 Relation between continuous and discrete signals 23 2.4.1 Image formation 24 2.4.2 Sampling theorem 25 2.4.3 Aliasing 28 2.4.4 Reconstruction from samples 28 2.5 Quantization 30 2.5.1 Equidistant quantization 30 2.5.2 Unsigned or signed representation 31 2.5.3 Nonequidistant quantization 32 2.6 References 34
9Handbook of Computer Vision and Applications Copyright © 1999 by Academic Press
Trang 352.1 Introduction
Images are signals with two spatial dimensions This chapter dealswith signals of arbitrary dimensions This generalization is very usefulbecause computer vision is not restricted solely to 2-D signals On theone hand, higher-dimensional signals are encountered Dynamic scenesrequire the analysis of image sequences; the exploration of 3-D spacerequires the acquisition of volumetric images Scientific exploration ofcomplex phenomena is significantly enhanced if images not only of asingle parameter but of many parameters are acquired On the otherhand, signals of lower dimensionality are also of importance when acomputer vision system is integrated into a larger system and imagedata are fused with time series from point measuring sensors
Thus this chapter deals with continuous (Section2.2) and discrete(Section2.3) representations of signals with arbitrary dimensions Whilethe continuous representation is very useful for a solid mathematicalfoundation of signal processing, real-world sensors deliver and digitalcomputers handle only discrete data Given the two representations,the relation between them is of major importance Section 2.4 dis-
cusses the spatial and temporal sampling on signals while Section2.5
treats quantization, the conversion of a continuous signal into digital
numbers
2.2 Continuous signals
2.2.1 Types of signals
An important characteristic of a signal is its dimension A
zero-dimen-sional signal results from the measurement of a single quantity at asingle point in space and time Such a single value can also be averagedover a certain time period and area There are several ways to extend
a zero-dimensional signal into a 1-D signal (Table2.1) A time series records the temporal course of a signal in time, while a profile does the
same in a spatial direction or along a certain path
A 1-D signal is also obtained if certain experimental parameters ofthe measurement are continuously changed and the measured parame-ter is recorded as a function of some control parameters With respect
to optics, the most obvious parameter is the wavelength of the magnetic radiation received by a radiation detector When radiation is
electro-recorded as a function of the wavelength, a spectrum is obtained The
wavelength is only one of the many parameters that could be ered Others could be temperature, pressure, humidity, concentration
consid-of a chemical species, and any other properties that may influence themeasured quantity
Trang 363 Hyperspectral image g(x, y, λ)
4 Volumetric image sequence g(x, y, z, t)
4 Hyperspectral image sequence g(x, y, λ, t)
5 Hyperspectral volumetric image sequence g(x, y, z, λ, t)
With this general approach to multidimensional signal processing,
it is obvious that an image is only one of the many possibilities of a2-D signal Other 2-D signals are, for example, time series of profiles orspectra With increasing dimension, more types of signals are possible
as summarized in Table2.1 A 5-D signal is constituted by a tral volumetric image sequence.
hyperspec-2.2.2 Unified description
Mathematically all these different types of multidimensional signals can
be described in a unified way as continuous scalar functions of multiple
parameters or generalized coordinates q das
g(q) = g(q1, q2, , q D ) with q = [q1, q2, , q D ] T (2.1)
that can be summarized in a D-dimensional parameter vector or
gen-eralized coordinate vector q An element of the vector can be a spatial
direction, the time, or any other parameter
As the signal g represents physical quantities, we can generally
as-sume some properties that make the mathematical handling of the nals much easier
sig-Continuity Real signals do not show any abrupt changes or
discon-tinuities Mathematically this means that signals can generally be garded as arbitrarily often differentiable
Trang 37re-Finite range The physical nature of both the signal and the imaging
sensor ensures that a signal is limited to a finite range Some signalsare restricted to positive values
Finite energy Normally a signal corresponds to the amplitude or the
energy of a physical process (see also Volume 1, Chapter2) As theenergy of any physical system is limited, any signal must be squareintegrable:
Depending on the underlying physical process the observed signalcan be regarded as a stochastic signal More often, however, a signal
is a mixture of a deterministic and a stochastic signal In the simplest
case, the measured signal of a deterministic process g dis corrupted by
additive zero-mean homogeneous noise This leads to the simple signal
model
where n has the variance σ2
n = h n2i In most practical situations, thenoise is not homogeneous but rather depends on the level of the signal.Thus in a more general way
g(q) = g d (q) + n(g) with n(g) = 0,Dn2(g)E= σ2
n (g) (2.4)
A detailed treatment of noise in various types of imaging sensors can
be found in Volume 1, Sections7.5,9.3.1, and10.2.3
2.2.3 Multichannel signals
So far, only scalar signals have been considered If more than one signal
is taken simultaneously, a multichannel signal is obtained In some
cases, for example, taking time series at different spatial positions, themultichannel signal can be considered as just a sampled version of ahigher-dimensional signal In other cases, the individual signals cannot
be regarded as samples This is the case when they are parameters withdifferent units and/or meaning
A multichannel signal provides a vector at each point and is
there-fore sometimes denoted as a vectorial signal and written as
g(q) = [q1(q), q2(q), , q D (q)] T (2.5)
Trang 38square, and hexagonal meshes; N e : number of neighbors with common edge;
N c : number of neighbors with common edge and/or corner; l: basis length l of regular polygon; d: distance d to nearest neighbor; and A: area of cell
Depend-be a higher-order signal, for example, a tensorial signal Such types of
multichannel images are encountered when complex features are tracted from images One example is the tensorial description of localstructure discussed in Chapter10
ex-2.3 Discrete signals
2.3.1 Regular two-dimensional lattices
Computers cannot handle continuous signals but only arrays of
digi-tal numbers Thus it is required to represent signals as D-dimensional
arrays of points We first consider images as 2-D arrays of points A
point on the 2-D grid is called a pixel or pel Both words are viations of picture element A pixel represents the irradiance at the
abbre-corresponding grid position There are two ways to derive 2-D latticesfrom continuous signals
Trang 39Figure 2.2: Elementary cells of regular grids for 2-D digital images: a triangle
grid, b square grid, c hexagonal grid.
First, the continuous 2-D space can be partitioned into space-filling
cells For symmetry reasons, only regular polygons are considered Then there are only three possible tesselations with regular polygons:
triangles, squares, and hexagons as illustrated in Fig.2.1(see also ble2.2) All other regular polygons do not lead to a space-filling ge-ometrical arrangement There are either overlaps or gaps From themesh of regular polygons a 2-D array of points is then formed by thesymmetry centers of the polygons In case of the square mesh, thesepoints lie again on a square grid For the hexagonal mesh, the sym-metry centers of the hexagons form a triangular grid In contrast, thesymmetry centers of the triangular grid form a more complex pattern,where two triangular meshes are interleaved The second mesh is offset
Ta-by a third of the base length l of the triangular mesh.
A second approach to regular lattices starts with a primitive cell A
primitive cell in 2-D is spanned by two not necessarily orthogonal base
vectors b1and b2 Thus, the primitive cell is always a parallelogram cept for square and rectangular lattices (Fig.2.2) Only in the latter case
ex-are the base vectors b1 and b2 orthogonal Translating the primitivecell by multiples of the base vectors of the primitive cell then forms the
lattice Such a translation vector or lattice vector r is therefore given
by
r = n1b1+ n2b2 n1, n2∈Z (2.6)The primitive cells of the square and hexagonal lattices (Fig 2.2band c) contains only one grid located at the origin of the primitive cell.This is not possible for a triangular grid, as the lattice points are not
arranged in regular distances along two directions (Fig. 2.1a) Thus,the construction of the triangular lattice requires a primitive cell withtwo grid points One grid point is located at the origin of the cell, theother is offset by a third of the length of each base vector (Fig.2.2a)The construction scheme to generate the elementary cells of regularshape from the lattice points is illustrated in Fig.2.3 From one latticepoint straight lines are drawn to all other lattice points starting with
Trang 40Figure 2.4: Representation of digital images by orthogonal lattices: a square
lattice for a 2-D image; and b cubic lattice for a volumetric or 3-D image.
the nearest neighbors (dashed lines) Then the smallest cell formed
by the lines perpendicular to these lines and dividing them into twohalves results in the primitive cell For all three lattices, only the nearestneighbors must be considered for this construction scheme
The mathematics behind the formation of regular lattices in twodimensions is the 2-D analog to 3-D lattices used to describe crystals
in solid state physics and mineralogy The primitive cell constructedfrom the lattice points is, for example, known in solid state physics as
the Wigner-Seitz cell.
Although there is a choice of three lattices with regular polygons—and many more if irregular polygons are considered—almost exclu-sively square or rectangular lattices are used for 2-D digital images