Mathematics and Visualization Series Editors Gerald Farin Hans-Christian Hege David Hoffman Christopher R. Johnson Konrad Polthier Martin Rumpf For further volumes: http://www.springer.com/series/4562 Michael Breuß ! Alfred Bruckstein Petros Maragos Editors Innovations for Shape Analysis Models and Algorithms With 227 Figures, 170 in color 123 Editors Michael Breuß Alfred Bruckstein Inst. for Appl. Math. and Scient. Comp Department of Computer Science Brandenburg Technical University Technion-Israel Institute of Technology Cottbus, Germany Haifa, Israel Petros Maragos School of Electrical and Computer Engineering National Technical University of Athens Athens, Greece ISBN 978-3-642-34140-3 ISBN 978-3-642-34141-0 (eBook) DOI 10.1007/978-3-642-34141-0 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2013934951 Mathematical Subject Classification (2010): 68U10 c Springer-Verlag Berlin Heidelberg 2013 This! work is subject to copyright. 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To Doris, Sonja, Johannes, Christian, Jonathan and Dominik, Christa and Gerhard To Rita and Ariel with love To Rena, Monica and Sevasti, Agelis and Sevasti Preface Shape understanding remains one of the most intriguing problems in computer vision and human perception. This book is a collection of chapters on shape analysis, by experts in the field, highlighting several viewpoints, including modeling and algorithms, in both discrete and continuous domains. It is a summary of research presentations and discussions on these topics at a Dagstuhl workshop in April 2011. The content is grouped into three main areas: Part I –DiscreteShapeAnalysis Part II –PartialDifferentialEquationsforShapeAnalysis Part III –OptimizationMethodsforShapeAnalysis The chapters contain both new results and tutorial sections that survey various areas of research. It was a pleasure for us to have had the opportunity to collaborate and exchange scientific ideas with our colleagues who participated in the Dagstuhl Workshop on Shape Analysis and subsequently contributed to this collection. We hope that this book will promote new research and further collaborations. Cottbus, Haifa and Athens Michael Breuß Alfred Bruckstein Petros Maragos vii Acknowledgments We would like to express our thanks to the many people who supported the publication of this book. First of all, we would like to thank the staff of Schloss Dagstuhl for their professional help in all aspects. The breathtaking Dagstuhl atmosphere was the basis that made our workshop such a unique and successful meeting. This book would never have attained its high level of quality without a rigorous peer-review process. Each chapter has been reviewed by at least two researchers in one or more stages. We would like to thank Alexander M. Bronstein, Oliver Demetz, Jean-Denis Durou, Laurent Hoeltgen, Yong Chul Ju, Margret Keuper, Reinhard Klette, Jan Lellmann, JoseAlbertoIglesiasMart´ ´ınez, Pascal Peter, Luis Pizarro, Nilanjan Ray, Christian Rossl,¨ Christian Schmaltz, Simon Setzer, Sibel Tari, Michael Wand, Martin Welk, Benedikt Wirth, and Laurent Younes for their dedicated and constructive help in this work. Moreover, we would like to thank the editors of the board of the Springer series Mathematics and Visualization for the opportunity to publish this book at an ideal position in the scientific literature. We are also grateful to Ruth Allewelt from Springer-Verlag for her practical and very patient support. Finally, we would like to thank Anastasia Dubrovinafor producing the nice cover image for the book. ix Contents Part I Discrete Shape Analysis 1ModelingThree-DimensionalMorseandMorse-Smale Complexes ................................................................... 3 Lidija Comiˇ c,´ Leila De Floriani, and Federico Iuricich 1.1 Introduction........................................................... 3 1.2 Background Notions ................................................. 5 1.3 Related Work ......................................................... 9 1.4 Representing Three-Dimensional Morse and Morse- Smale Complexes .................................................... 10 1.4.1 A Dimension-Independent Compact Representation for Morse Complexes ..................... 10 1.4.2 A Dimension-Specific Representation for 3D Morse-Smale Complexes................................... 13 1.4.3 Comparison ................................................. 14 1.5 Algorithms for Building 3D Morse and Morse-Smale Complexes ............................................................ 15 1.5.1 A Watershed-Based Approach for Building the Morse Incidence Graph ................................ 15 1.5.2 A Boundary-Based Algorithm ............................. 18 1.5.3 A Watershed-Based Labeling Algorithm .................. 19 1.5.4 A Region-Growing Algorithm ............................. 20 1.5.5 An Algorithm Based on Forman Theory .................. 21 1.5.6 A Forman-Based Approach for Cubical Complexes ...... 22 1.5.7 A Forman-Based Approach for Simplicial Complexes ... 24 1.5.8 Analysis and Comparison .................................. 25 1.6 Simplification of 3D Morse and Morse-Smale Complexes ........ 26 1.6.1 Cancellation in 3D .......................................... 27 1.6.2 Removal and Contraction Operators....................... 29 1.7 Concluding Remarks ................................................. 31 References.................................................................... 32 xi xii Contents 2GeodesicRegressionandItsApplicationtoShapeAnalysis........... 35 P. Thomas Fletcher 2.1 Introduction........................................................... 35 2.2 Multiple Linear Regression .......................................... 36 2.3 Geodesic Regression ................................................. 37 2.3.1 Least Squares Estimation................................... 38 2.3.2 R2 Statistics and Hypothesis Testing ...................... 40 2.4 Testing the Geodesic Fit ............................................. 41 2.4.1 Review of Univariate Kernel Regression .................. 43 2.4.2 Nonparametric Kernel Regression on Manifolds ......... 44 2.4.3 Bandwidth Selection ....................................... 44 2.5 Results: Regression of 3D Rotations ................................ 45 2.5.1 Overview of Unit Quaternions ............................. 45 2.5.2 Geodesic Regression of Simulated Rotation Data ........ 45 2.6 Results: Regression in Shape Spaces ................................ 46 2.6.1 Overview of Kendall’s Shape Space ....................... 47 2.6.2 Application to Corpus Callosum Aging ................... 48 2.7 Conclusion ............................................................ 51 References.................................................................... 51 3SegmentationandSkeletonization on Arbitrary Graphs Using Multiscale Morphology and Active Contours .................... 53 Petros Maragos and Kimon Drakopoulos 3.1 Introduction........................................................... 53 3.2 Multiscale Morphology on Graphs .................................. 55 3.2.1 Background on Lattice and Multiscale Morphology...... 55 3.2.2 Background on Graph Morphology........................ 57 3.2.3 Multiscale Morphology on Graphs ........................ 60 3.3 Geodesic Active Contours on Graphs ............................... 61 3.3.1 Constant-Velocity Active Contours on Graphs............ 63 3.3.2 Direction of the Gradient on Graphs....................... 64 3.3.3 Curvature Calculation on Graphs .......................... 67 3.3.4 Convolution on Graphs ..................................... 68 3.3.5 Active Contours on Graphs: The Algorithm .............. 69 3.4 Multiscale Skeletonization on Graphs............................... 70 3.5 Conclusions..........................................................