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Constructing Discrete Morse Functions Thomas Lewiner Constructing Discrete Morse Functions MsC Thesis DEPARTMENT OF MATHEMATICS Post{graduate program in Applied Mathematics Rio de Janeiro July 2002 Thomas Lewiner Constructing Discrete Morse Functions MsC Thesis Thesis presented to the Post{graduate Program in Applied Mathematics of the Mathematics Department, PUC{Rio as partial ful¯llment of the requirements for the degree of Master in Applied Mathematics Advisor : Prof. H¶elioC^ortesVieira Lopes Co{Advisor: Prof. Geovan Tavares dos Santos Rio de Janeiro July 2002 Thomas Lewiner Constructing Discrete Morse Functions Thesis presented to the Post{graduate Program in Applied Mathematics of the Mathematics Department, PUC{Rio as partial ful¯llment of the requirements for the degree of Mas- ter in Applied Mathematics. Approved by the Examinating Comission: Prof. H¶elioC^ortesVieira Lopes Advisor Department of Mathematics | PUC{Rio Prof. Geovan Tavares dos Santos Co{Advisor Department of Mathematics | PUC{Rio Prof. Luis Carlos Pacheco R. Velho IMPA Prof. Jorge Stol¯ UNICAMP Prof. Ney Augusto Dumont Coordinator of the Science and Engineering Center | PUC{Rio Rio de Janeiro, 10 de July de 2002 All rights reserved. Thomas Lewiner He graduated from the Ecole Polytechnique (Paris, France) in Algebra and Computer Science, and in Theoretical Phy- sics. He specialized himself at the Ecole Sup¶erieure des T¶el¶ecommunications (Paris, France) in Signal and Image Pro- cessing, and in Project Management. He worked for Inventel in wireless telecommunication systems based on BlueTooth technology. During his master preparation, he developed with his advisors tools oc computational topology. Ficha Catalogr¶a¯ca Lewiner, Thomas Constructing Discrete Morse Functions/ Thomas Lewi- ner; advisor: H¶elioC^ortesVieira Lopes; co{advisor: Geovan Tavares dos Santos. | Rio de Janeiro : PUC{Rio, Depart- ment of Mathematics, 2002. v., 89 f: il. ; 29,7 cm 1. MsC Thesis - Pontif¶³ciaUniversidade Cat¶olicado Rio de Janeiro, Department of Mathematics. Bibliography included. 1. Mathematics { Thesis. 2. Morse Theory. 3. Forman Theory. 4. Computacional Topology. 5. Computacional Ge- ometry. 6. Geometric Modelling. 6. Discrete Mathematics. I. Lopes, H¶elioC^ortesVieira. II. Santos, Geovan Tavares dos. III. Pontif¶³cia Universidade Cat¶olicado Rio de Ja- neiro. Department of Mathematics. IV. Title. CDD: 510 Acknowledgments To my advisors Professors H¶elio Lopes and Geovan Tavares for the support, the everyday kindness and the incentive for the realisation of this work. To the CNPq and the PUC{Rio, for the ¯nancial support, without which this work would not have been realized. To my grand{mothers, who su®ered the most of my expatriation. To my mother and father, sisters and all my family. To my colleagues of the PUC{Rio, who have me loved this place. To the professors Marcos da Silvera, Jean{Marie Nicolas and Anne Germa who o®ered me the opportunity of this cooperation. To the people of the Mathematic departament for the constant help, particularly to Ana Cristina, Creuza and Sinesio. Abstract Lewiner, Thomas; Lopes, H¶elioC^ortesVieira; Santos, Geovan Ta- vares dos. Constructing Discrete Morse Functions. Rio de Janeiro, 2002. 89p. MsC Thesis | Department of Mathematics, Pontif¶³ciaUniversidade Cat¶olicado Rio de Janeiro. Morse theory has been considered a powerful tool in its applications to computational topology, computer graphics and geometric modeling. It was originally formulated for smooth manifolds. Recently, Robin Forman formulated a version of this theory for discrete structures such as cell complexes. It opens up several categories of interesting objects (particularly meshes) to applications of Morse theory. Once a Morse function has been de¯ned on a manifold, then information about its topology can be deduced from its critical elements. The purpose of this work is to design an algorithm to de¯ne optimal discrete Morse functions on general cell complex, where optimality entails having the least number of critical elements. This problem is proven here to be MAX{ SNP hard. However, we provide a linear algorithm that, for the case of 2{manifolds, always reaches optimality. Moreover, we proved various results on the structure of a discrete Morse function. In particular, we provide an equivalent representation by hyperfo- rests. From this point of view, we designed a construction of discrete Morse functions for general cell complexes of arbitrary ¯nite dimension. The re- sulting algorithm is quadratic in time and, although not guaranteed to be optimal, gives optimal answers in most of the practical cases. Keywords Morse Theory; Forman Theory; Computational Topology; Computatio- nal Geometry; Solid Modeling; Discrete Mathematics. Contents Foreword 12 I Introduction 14 I.1 Motivations and applications 14 I.2 Results 16 I.3 Outline 17 II Preliminaries 18 II.1 Discrete structures 18 II.2 Basic concepts of algebraic topology 23 III Forman's discrete Morse theory 31 III.1 Discrete gradient vector ¯eld 31 III.2 Discrete Morse functions 35 III.3 Topological properties 39 III.4 Optimality in discrete Morse functions 42 IV Optimal discrete Morse functions on surfaces 45 IV.1 Optimality conditions 45 IV.2 Algorithm 47 IV.3 First extension to non{manifolds 50 IV.4 Proof and analysis 51 IV.5 Mixing with geometry 52 IV.6 Results 53 V Structure of a discrete Morse function 55 V.1 Layers of a combinatorial vector ¯eld 55 V.2 Discrete Morse functions, acyclic matchings and hyperforests 58 V.3 Discrete Morse numbers as topological invariants for 3{manifolds 63 VI Constructing discrete Morse functions 65 VI.1 Data structure 65 VI.2 Algorithms for constructing discrete Morse functions out of hyper- forests 66 VI.3 Optimality considerations 69 VI.4 Di®erent heuristics 71 VI.5 Results 76 VII Future Works 85 Bibliography 86 List of ¯gures II.1 An 8-squares cylinder model. 18 II.2 The 0/1 graph of an 8-squares cylinder (¯gure II.1): nodes represents vertices, links represents edges. 19 II.3 A tree extracted from the graph of ¯gure II.2. 19 II.4 A bipartite representation of the graph of ¯gure II.2. 20 II.5 A matching on the graph of ¯gure II.4. 20 II.6 The dual graph of ¯gure II.1. 20 II.7 A tree extracted from the graph of ¯gure II.6. 20 II.8 A hypergraph example. 21 II.9 The dual hypergraph of the graph of ¯gure II.2. 21 II.10 A triangulated torus. 24 II.11 A construction of a torus with 4 cells. 24 II.12 The Hasse diagram of a simple cell complex. 25 II.13 The Hasse diagram of a non{PL torus. 26 II.14 A non{manifold example: non{regular edge. 26 II.15 Another non{manifold example: the neighborhood of the points of the dangling edge is homeomorphic to R. 26 II.16 A part of a triangulation and its dual. 27 II.17 Cycles in a 1{cell complex are circuits. 28 II.18 The cycles of a map are around seas (holes), where lands are faces and frontiers are edges. 28 II.19 An elementary collapse. 30 II.20 The collapse of a tetrahedra onto a point. 30 III.1 An example of a combinatorial vector ¯eld. 32 III.2 The closed V {path of the combinatorial vector ¯eld of ¯gure III.1 (in red). 32 III.3 Examples of discrete gradient vector ¯eld. 33 III.4 Hasse diagram of the examples of ¯gure III.3. 33 III.5 The Hasse diagram of the combinatorial vector ¯eld of ¯gure III.1, and the circuit of its closed V {path (in red). 34 III.6 Canceling critical cells 34 III.7 Examples of discrete Morse functions 36 III.8 A discrete gradient vector ¯eld and its corresponding discrete Morse function. 37 III.9 Re¯ning a discrete gradient vector ¯eld: all the new vertices, edges and faces are paired easily. 37 III.10The Hasse diagram of an optimal discrete gradient vector ¯eld on a Segment and on a Triangle. 38 III.11The Hasse diagram of the cartesian product of the discrete gradient vector ¯elds of ¯gure III.10. 38 III.12Steps of the collapse of a simple cell complex. 39 III.13The critical steps of the extension of a torus. 41 III.14The number of matchings in the Hasse diagram is ¯nite. 42 III.15A \gadget" used in E¸gecio¸gluand Gonzalez proof. 43 IV.1 Examples of surfaces without boundary. 45 IV.2 Examples of surfaces with a non{empty boundary. 46 IV.3 EdgeBreaker mesh compression codes on a triangulated torus. 48 IV.4 The resulting face{spanning tree T and its complement graph G. 48 IV.5 Loop added at step 2 to a face{spanning tree T of ¯gure II.1 (no critical cell) 48 IV.6 The discrete Morse function on the spanning tree T of ¯gure IV.3 (1 critical face). 48 IV.7 The complement graph G of the cylinder model and its discrete Morse function (1 critical vertex and 1 critical edge). 49 IV.8 The discrete Morse function on the complement graph G of ¯gure IV.3 (1 critical vertex and 2 critical edges). 49 IV.9 A non{manifold complex with a dangling edge (on the left) and its critical cells (on the right): 1 critical vertex and 2 critical faces. 50 IV.10A non{manifold complex with a corner vertex (on the left) and its critical cells (on the right): 1 critical vertex and 2 critical faces. 50 IV.11A non{manifold complex with a non{regular edge (on the left) and its critical cells (on the right): 1 critical vertex and 2 critical faces. 50 IV.12Two discrete gradient vector ¯elds on a 2{sphere, both with 2 critical cells. 52 IV.13Execution time vs. the number of cells of the cell complex K: the complexity is linear.
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