CuuDuongThanCong.com Lecture Notes in Computational Vision and Biomechanics Editors João Manuel R.S Tavares R.M Natal Jorge Address: Faculdade de Engenharia Universidade Porto Rua Dr Roberto Frias, s/n 4200-465 Porto Portugal tavares@fe.up.pt, rnatal@fe.up.pt Editorial Advisory Board Alejandro Frangi, University of Sheffield, Sheffield, UK Chandrajit Bajaj, University of Texas at Austin, Austin, USA Eugenio Oñate, Universitat Politécnica de Catalunya, Barcelona, Spain Francisco Perales, Balearic Islands University, Palma de Mallorca, Spain Gerhard A Holzapfel, Royal Institute of Technology, Stockholm, Sweden J Paulo Vilas-Boas, University of Porto, Porto, Portugal Jeffrey A Weiss, University of Utah, Salt Lake City, USA John Middleton, Cardiff University, Cardiff, UK Jose M García Aznar, University of Zaragoza, Zaragoza, Spain Perumal Nithiarasu, Swansea University, Swansea, UK Kumar K Tamma, University of Minnesota, Minneapolis, USA Laurent Cohen, Université Paris Dauphine, Paris, France Manuel Doblaré, Universidad de Zaragoza, Zaragoza, Spain Patrick J Prendergast, University of Dublin, Dublin, Ireland Rainald Löhner, George Mason University, Fairfax, USA Roger Kamm, Massachusetts Institute of Technology, Cambridge, USA Thomas J.R Hughes, University of Texas, Austin, USA Yongjie Zhang, Carnegie Mellon University, Pittsburgh, USA Yubo Fan, Beihang University, Beijing, China For further volumes: http://www.springer.com/series/8910 CuuDuongThanCong.com Lecture Notes in Computational Vision and Biomechanics Volume The research related to the analysis of living structures (Biomechanics) has been a source of recent research in several distinct areas of science, for example, Mathematics, Mechanical Engineering, Physics, Informatics, Medicine and Sport However, for its successful achievement, numerous research topics should be considered, such as image processing and analysis, geometric and numerical modelling, biomechanics, experimental analysis, mechanobiology and enhanced visualization, and their application to real cases must be developed and more investigation is needed Additionally, enhanced hardware solutions and less invasive devices are demanded On the other hand, Image Analysis (Computational Vision) is used for the extraction of high level information from static images or dynamic image sequences Examples of applications involving image analysis can be the study of motion of structures from image sequences, shape reconstruction from images and medical diagnosis As a multidisciplinary area, Computational Vision considers techniques and methods from other disciplines, such as Artificial Intelligence, Signal Processing, Mathematics, Physics and Informatics Despite the many research projects in this area, more robust and efficient methods of Computational Imaging are still demanded in many application domains in Medicine, and their validation in real scenarios is matter of urgency These two important and predominant branches of Science are increasingly considered to be strongly connected and related Hence, the main goal of the LNCV&B book series consists of the provision of a comprehensive forum for discussion on the current state-of-the-art in these fields by emphasizing their connection The book series covers (but is not limited to): • Applications of Computational Vision and Biomechanics • Biometrics and Biomedical Pattern Analysis • Cellular Imaging and Cellular Mechanics • Clinical Biomechanics • Computational Bioimaging and Visualization • Computational Biology in Biomedical Imaging • Development of Biomechanical Devices • Device and Technique Development for Biomedical Imaging • Digital Geometry Algorithms for Computational Vision and Visualization • Experimental Biomechanics • Gait & Posture Mechanics • Multiscale Analysis in Biomechanics • Neuromuscular Biomechanics • Numerical Methods for Living Tissues • Numerical Simulation • Software Development on Computational Vision and Biomechanics CuuDuongThanCong.com • Grid and High Performance Computing for Computational Vision and Biomechanics • Image-based Geometric Modeling and Mesh Generation • Image Processing and Analysis • Image Processing and Visualization in Biofluids • Image Understanding • Material Models • Mechanobiology • Medical Image Analysis • Molecular Mechanics • Multi-modal Image Systems • Multiscale Biosensors in Biomedical Imaging • Multiscale Devices and Biomems for Biomedical Imaging • Musculoskeletal Biomechanics • Sport Biomechanics • Virtual Reality in Biomechanics • Vision Systems Valentin E Brimkov r Reneta P Barneva Editors Digital Geometry Algorithms Theoretical Foundations and Applications to Computational Imaging CuuDuongThanCong.com Editors Valentin E Brimkov Department of Mathematics Buffalo State College State University of New York Buffalo, NY USA Reneta P Barneva Department of Computer and Information Sciences State University of New York at Fredonia Fredonia, NY USA ISSN 2212-9391 ISSN 2212-9413 (electronic) Lecture Notes in Computational Vision and Biomechanics ISBN 978-94-007-4173-7 ISBN 978-94-007-4174-4 (eBook) DOI 10.1007/978-94-007-4174-4 Springer Dordrecht Heidelberg New York London Library of Congress Control Number: 2012939722 © Springer Science+Business Media Dordrecht 2012 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer Permissions for use may be obtained through RightsLink at the Copyright Clearance Center Violations are liable to prosecution under the respective Copyright Law The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made The publisher makes no warranty, express or implied, with respect to the material contained herein Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) CuuDuongThanCong.com Preface Digital geometry is a modern mathematical discipline studying the geometric properties of digital objects (usually modeled by sets of points with integer coordinates) and providing methods for solving various problems defined on such objects Digital geometry is developed with the explicit goal to provide rigorous mathematical foundations and basic algorithms for applied disciplines such as computer graphics, medical imaging, pattern recognition, image analysis and processing, computer vision, image understanding, and biometrics These are in turn applicable to important and societally sensitive areas like medicine, defense, and security Although digital geometry has its roots in several classical disciplines (such as graph theory, topology, number theory, and Euclidean and analytic geometry), it was established as an independent subject only in the last few decades Several researchers have played a pioneering role in setting the foundations of digital geometry Notable among these is the late Azriel Rosenfeld and his seminal works from the late 60’s and early 70’s of the last century Some authors of chapters of the present book are also among the founders of the area or its prominent promoters The last two decades feature an increasing number of active contributors throughout the world A number of excellent monographs and hundreds of research papers have been devoted to the subject One can legitimately say that at present digital geometry is an independent subject with its own history, vibrant international community, regular scientific meetings and events, and, most importantly, serious scientific achievements This contributed book contains thirteen chapters devoted to different (although interrelated) important problems of digital geometry, algorithms for their solution, and various applications All authors are well-recognized researchers, as some of them are world leaders in the field As a general framework, each chapter presents a research topic of considerable importance, provides a review of fundamental results and algorithms for the considered problems, presents new unpublished results, as well as a discussion on related applications, current developments and perspectives By its structure and content, this publication does not appear to be an exhaustive source of information for all branches of digital geometry Rather, the book is aimed at attracting readers’ attention to central digital geometry tasks and related v CuuDuongThanCong.com vi Preface applications, as diverse as creating image-based metrology, proposing new tools for processing multidimensional images, studying topological transformations for image processing, and developing algorithms for shape analysis An advantage of the chosen contributed book framework is that all chapters provide enough complete presentations written by leading experts on the considered specific matters The chapters are self-contained and can be studied in succession dictated by the readers’ interests and preferences We believe that this publication would be a useful source of information for researchers in digital geometry as well as for practitioners in related applied disciplines It can also be used as a supplementary material or a text for graduate or upper level undergraduate courses We would like to thank all those who made this publication possible We are indebted to João Manuel R.S Tavares and Renato Manuel Natal Jorge, editors of the Springer’s series “Lecture Notes in Computational Vision and Biomechanics,” for inviting us to organize and edit a volume of the series We are thankful to Springer’s Office and particularly to Ms Nathalie Jacobs, Senior Publishing Editor, and Dr D Merkle, Editorial Director, for reviewing our proposal and giving us the opportunity to publish this work with Springer, as well as for the pleasant cooperation throughout the editorial process Lastly and most importantly, our thanks go to all authors who contributed excellent chapters to this book Fredonia and Buffalo, NY, USA CuuDuongThanCong.com Valentin E Brimkov Reneta P Barneva Contents Part I General Digital Geometry in Image-Based Metrology Alfred M Bruckstein Provably Robust Simplification of Component Trees of Multidimensional Images Gabor T Herman, T Yung Kong, and Lucas M Oliveira Part II 27 Topology, Transformations Discrete Topological Transformations for Image Processing Michel Couprie and Gilles Bertrand 73 Modeling and Manipulating Cell Complexes in Two, Three and Higher Dimensions 109 ˇ Lidija Comi´ c and Leila De Floriani Binarization of Gray-Level Images Based on Skeleton Region Growing 145 Xiang Bai, Quannan Li, Tianyang Ma, Wenyu Liu, and Longin Jan Latecki Topology Preserving Parallel 3D Thinning Algorithms 165 Kálmán Palágyi, Gábor Németh, and Péter Kardos Separable Distance Transformation and Its Applications 189 David Coeurjolly and Antoine Vacavant Separability and Tight Enclosure of Point Sets 215 Peter Veelaert vii CuuDuongThanCong.com viii Contents Part III Image and Shape Analysis Digital Straightness, Circularity, and Their Applications to Image Analysis 247 Partha Bhowmick and Bhargab B Bhattacharya 10 Shape Analysis with Geometric Primitives 301 Fabien Feschet 11 Shape from Silhouettes in Discrete Space 323 Atsushi Imiya and Kosuke Sato 12 Combinatorial Maps for 2D and 3D Image Segmentation 359 Guillaume Damiand and Alexandre Dupas 13 Multigrid Convergence of Discrete Geometric Estimators 395 David Coeurjolly, Jacques-Olivier Lachaud, and Tristan Roussillon Index 425 CuuDuongThanCong.com Contributors Xiang Bai Huazhong University of Science and Technology, Wuhan, China Gilles Bertrand Laboratoire d’Informatique Gaspard-Monge, Équipe A3SI, Université Paris-Est, ESIEE Paris, Marne-la-Vallée, France Bhargab B Bhattacharya Advanced Computing and Microelectronics Unit, Indian Statistical Institute, Kolkata, India Partha Bhowmick Department of Computer Science and Engineering, Indian Institute of Technology, Kharagpur, India Alfred M Bruckstein Ollendorff Professor of Science, Computer Science Department, Technion, IIT, Haifa, Israel David Coeurjolly LIRIS, UMR CNRS 5205, Université de Lyon, Villeurbanne, France ˇ Lidija Comi´ c Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia Michel Couprie Laboratoire d’Informatique Gaspard-Monge, Équipe A3SI, Université Paris-Est, ESIEE Paris, Marne-la-Vallée, France Guillaume Damiand LIRIS, UMR5205, Université de Lyon, CNRS, Lyon, France Leila De Floriani Department of Computer Science, University of Genoa, Genoa, Italy Alexandre Dupas Unit 698, Inserm, Paris, France Fabien Feschet IGCNC - EA 2782, Clermont Université, Université d’Auvergne, Clermont-Ferrand, France Gabor T Herman Computer Science Ph.D Program, Graduate Center, City University of New York, New York, NY, USA ix CuuDuongThanCong.com 414 D Coeurjolly et al • O(h0.32 ) for the flower (but note that the error graph of the BC estimator is not straight and further experiments should be done at smaller grid steps to get the convergence speed) Eventually the MDCA, GMC, BC estimators appear to be experimentally multigrid-convergent, but there is no correct theoretical convergence results for curvature estimation as far as we know, contrary to the case of tangent estimation (Sect 13.4) 13.6 Implementation In this section, we discuss about implementation details of both the geometrical estimators presented in the previous sections, and the experimental evaluation framework All the estimators described in this chapter have been implemented in DGtal [18] DGtal is an open-source C++ library focusing on the implementation of digital geometry objects and concepts For short, it allows to represent images and objects in n-dimensional digital spaces equipped with both geometrical and topological tools In the context of this chapter, we will only consider the representation and the analysis of shape in dimension As discussed in the introduction, the input digital object can be obtained either from an explicit description, from a segmentation process of an image (iso-level, ), or as the digitization D(X, h) of a continuous shape X ∈ X For the first two cases, DGtal provides mechanisms to construct such digital sets either explicitly or from a contour tracking process For the last case, DGtal implements various implicit and parametric continuous shapes for which some global and local geometrical quantities are known All such shape implementations are model of a concept of CEuclideanShapes1 (see Fig 13.1 for an illustration of DGtal Euclidean shapes) A digital object is thus obtained from a GaussDigitizer parametrized by a model of CEuclideanShapes and a grid step h CEuclideanShapes models will be crucial for multigrid convergence analysis As discussed above and whatever the way the input digital object is specified, we need to access to its geometrical information in various ways: • As a sequence of grid points, subset of Z2 , e.g for area and moment descriptors; • As a representation of its boundary, e.g for tangent or curvature estimators In the latter case, several options exist to define and represent a shape contour Most of the options depend on the underlying topological model (Kong’s like digital topology or cellular topology) Furthermore, depending on the algorithm used to perform the analysis, one may prefer a sequence of chain codes, a sequence DGtal uses a generic programming paradigm based on concepts and models of concepts If a structure name starts from a capital “C”, we describe a concept CuuDuongThanCong.com 13 Multigrid Convergence of Discrete Geometric Estimators 415 Fig 13.9 Different representations of a Euclidean shape digitization: as a set of pixels (a), as a sequence of 4-connected pixels (b), as a sequence of 1-cell or linels (c), as a sequence of grid point displacements (d) of linels or a sequence of 4-connected grid points to describe the contour (cf Fig 13.9) To obtain a generic and extensible implementation of contour based estimators, we have defined a GridCurve structure constructed upon a topological cellular model which aims to provide several facets of a shape contour More precisely, given the result of the contour tracking process, it provides mechanism (Ranges and Iterators on Ranges) to process the boundary sequence either as a set of grid points or a set of linels Hence, a local geometrical estimator on contour, or more precisely a model of CLocalGeometricalEstimator, have an interface containing at least the two following methods: • void init(double h, ConstIterator & begin, ConstIterator & end, ): initialize the geometrical estimator with grid step h on a contour defined between iterators begin and end; • Quantity eval( ConstIterator & it): evaluate the estimator at the position it of the contour and return a Quantity In our framework, the type ConstIterator is a template parameter chosen in the contour iterator types provided in GridCurve Similarly, we have a concept of CGlobalGeometricalEstimator and models of this concept have an eval() method which returns a unique quantity for a shape (or subset of it) Based on models of CEuclideanShapes, we can obtain expected continuous values using TrueLocalEstimatorOnPoints and TrueGlobalEstimatorOnPoints Since both expected and estimated values are given by estimators with a consistent interface, it makes the multigrid comparison very simple Indeed, it allows to design a generic CompareLocalEstimators which return a statistic on the difference of two estimator values In the following example, we illustrate the multigrid Euclidean shape construction and the comparison of three length estimators (RosenProffitt, DSS and MLP) In this example, we have detailed the overall process: shape construction and digitization, domain and Khalimsky space construction, contour tracking and finally, evaluation of estimators CuuDuongThanCong.com 416 D Coeurjolly et al // / / h and r a d i u s a r e p a r a m e t e r s h e r e // / / Types t y p e d e f Ball2D < Space > Shape ; typedef Space : : P o i n t P o i n t ; typedef Space : : R e a l P o i n t R e a l P o i n t ; typedef Space : : I n t e g e r I n t e g e r ; t y p e d e f HyperRectDomain < Space > Domain ; t y p e d e f KhalimskySpaceND < S p a c e : : d i m e n s i o n , I n t e g e r > KSpace ; t y p e d e f KSpace : : S C e l l S C e l l ; t y p e d e f G r i d C u r v e : : P o i n t s R a n g e P o i n t s R a n g e ; t y p e d e f G r i d C u r v e : : ArrowsRange ArrowsRange ; typedef PointsRange : : C o n s t I t e r a t o r ConstIteratorOnPoints ; / / Euclidean ball Shape a Sh a p e ( P o i n t ( , ) , r a d i u s ) ; / / Gauss D i g i t i z a t i o n G a u s s D i g i t i z e r < Space , Shape > d i g ; d i g a t t a c h ( aShape ) ; / / a t t a c h e s t h e shape d i g i n i t ( a Sh a p e getLowerBound ( ) , a Sh a p e g e t U p p e r b o u n d ( ) , h ) ; / / The domain s i z e i s g i v e n by t h e d i g i t i z e r a c c o r d i n g t o / / t h e window and t h e s t e p Domain domain = d i g getDomain ( ) ; / / Create c e l l u l a r space KSpace K ; b o o l ok = K i n i t ( d i g getLowerBound ( ) , d i g g etUpperBound ( ) , t r u e ) ; i f ( ! ok ) { s t d : : c e r r