TOPOLOGICAL ANALYSIS OF SCALAR FUNCTIONS ABSTRACT FOR SCIENTIFIC DATA VISUALIZATION TOPOLOGICAL ANALYSIS OF SCALAR FUNCTIONS by FOR SCIENTIFIC DATA VISUALIZATION Vijay Natarajan by Department of Computer Science Vijay Natarajan Duke University Department of Computer Science Duke University Date: Approved: Date: Prof. Herbert Edelsbrunner, Supervisor Approved: Prof. Lars Arge Prof. Herbert Edelsbrunner, Supervisor Prof. John Harer Prof. Lars Arge Prof. Xiaobai Sun Prof. John Harer Prof. Xiaobai Sun Dissertation submitted in partial fulfillment of the An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy requirements for the degree of Doctor of Philosophy in the Department of Computer Science in the Department of Computer Science in the Graduate School of in the Graduate School of Duke University Duke University 2004 2004 Abstract • A new comparison measure between k functions defined on a common d-manifold. For the case d = k = 2, we give alternative formulations of the definition based Scientists attempt to understand physical phenomena by studying various quanti- on a Morse theoretic point of view. We also develop visualization software that ties measured over the region of interest. A majority of these quantities are scalar performs local comparison between pairs of functions in datasets containing (real-valued) functions. These functions are typically studied using traditional visual- multiple and sometimes time-varying functions. ization techniques like isosurface extraction, volume rendering etc. As the data grows We apply our methods to data from medical imaging, electron microscopy, and x-ray in size and becomes increasingly complex, these techniques are no longer effective. crystallography. The results of these experiments provide evidence of the usability of State of the art visualization methods attempt to automatically extract features and our methods. annotate a display of the data with a visualization of its features. In this thesis, we study and extract the topological features of the data and use them for visualization. We have three results: • An algorithm that simplifies a scalar function defined over a tetrahedral mesh. In addition to minimizing the error introduced by the approximation of the function, the algorithm improves the mesh quality and preserves the topology of the domain. We perform an extensive set of experiments to study the effect of requiring better mesh quality on the approximation error and the level of sim- plification possible. We also study the effect of simplification on the topological features of the data. • An extension of three-dimensional Morse-Smale complexes to piecewise linear 3-manifolds and an efficient algorithm to compute its combinatorial analog. Morse-Smale complexes partition the domain into regions with similar gradient flows. Letting n be the number of vertices in the input mesh, the running time of the algorithm is proportional to n log(n) plus the total size of the input mesh plus the total size of the output. We develop a visualization tool that displays different substructures of the Morse-Smale complex. iii iv Acknowledgements I owe whatever I have achieved to my parents. They have been there for me always, supporting my decisions and believing in me. I have learned a lot from them, most importantly my sense of values. I will treasure this for the rest of my life. My advisor, Prof. Herbert Edelsbrunner, has been a major influence in my re- search career. I got introduced to the field of computational topology when I was still To my parents an undergrad and Prof. Subir Ghosh at T.I.F.R. handed me the introductory paper that Herbert had written on the subject. I had recently taken courses in topology and algebra and was excited that these subjects had practical relevance. This excitement has grown by leaps and bounds while working with Herbert. Discussion sessions with him helped me refine my reasoning. His meticulous attention to detail and construc- tive feedback while working on manuscripts helped me improve my writing skills. I will be forever grateful to him for his support and guidance during the past five years. I have had the pleasure of working with two wonderful people: Prof. John Harer and Dr. Valerio Pascucci. I learned many mathematical ideas from John. He ex- plains them in a way that makes it all seem so simple. I can only hope that other mathematicians that I collaborate with in future are like him. Valerio introduced me to the wonderful area of visualization and shared the insights that he had gained from his experiences while doing interdisciplinary research. These have been very valuable to me due to the nature of my work where I need to talk with scientists from other disciplines. He has also played the role of a mentor, helping me make my career decisions. I was lucky to have great committee members in Prof. Lars Arge and Prof. Xiaobai Sun. I want to thank them for giving valuable comments on my thesis and for being so flexible in setting up a date for my final presentation. I will always be grateful to v vi Prof. Pankaj Agarwal for giving me valuable advice on many academic matters and and I want to thank all of them for their support: Jaishankar, Uma, Sunder and career options. He has often gone out of the way to offer his help or give a word of Subha. Finally, I want to thank my fianc´ee Mala for her patience and unconditional encouragement. The administrative and lab staff members in the department have support. always been very helpful. I would like to particularly thank Diane Riggs: she has This work was supported by NSF under grant NSF-CCR-0086013 and by Lawrence been of great assistance throughout the course of my stay in the department. I want Livermore National Laboratory under sub-contracts with Duke. to thank Celeste Hodges for her kind words of encouragement. Various students and postdocs at Duke and UNC gave much needed professional support, attending my practice talks and giving feedback, discussing ideas, answering questions etc. Besides being colleagues, they are my good friends as well. I want to thank Sathish Govindarajan, Yusu Wang, Ajith Mascarenhas, Gopi Meenakshisundaram, David Cohen-Steiner, Lipyeow Lim, Andrew Ban, and Vicky Choi and wish them a bright future. Going for coffee to the Bryan center with many of them, mostly with Yusu, served as a nice break in the afternoon. During my stay at Duke, I have enjoyed the friendship of many people. I want to thank all of them for making my graduate life memorable. I could not have asked for better housemates than Hari and Mohan. Knowing Vijay Srinivasan has been a rewarding and humbling experience. Friday dinners with Jaidev, Lavanya and us \usual suspects" were awesome. Late night discussions on spirituality, with Vamsee and Pramod arguing endlessly, were thought provoking. Having Srikanth as an officemate in my first year was enlightening because I learned a lot about doing research. With Dmitriy as my officemate in the past few months, there was never a dull moment. I have been fortunate to be able to stay in touch with great friends from college especially through our mailing list diljale. These are some of the most talented and versatile people that I have met. They have and continue to inspire me in many ways. My brothers and sisters-in-law have always encouraged me to pursue my dreams vii viii Contents 2.3.2 Algorithm Overview . 21 2.3.3 Checking Link Conditions . 22 Abstract iii 2.3.4 Contracting Edges . 23 Acknowledgements vi 2.3.5 Reentering Edges . 23 2.3.6 Accumulating Edge Bisectors . 24 List of Tables xiii 2.4 Experiments . 25 List of Figures xiv 2.4.1 Datasets . 25 1 Introduction 1 2.4.2 Tetrahedral Shape Improvement . 26 1.1 Scalar Functions . 1 2.4.3 Approximation Error vs Tetrahedral Shape . 27 1.2 Topological Analysis . 4 2.4.4 Topology Preservation vs Tetrahedral Shape . 27 1.3 Contributions . 5 2.4.5 Density Map Preservation . 28 1.4 Layout of Material . 7 2.4.6 Critical Point Statistics . 29 2 Density Map Simplification 8 2.4.7 Sanity Checks . 31 2.1 Introduction . 8 2.4.8 Inclusion-Exclusion . 33 2.1.1 Motivation . 8 2.5 Discussion . 33 2.1.2 Prior Work . 9 3 3D Morse-Smale Complexes 38 2.1.3 Approach and Results . 10 3.1 Introduction . 38 2.2 Edge contraction . 12 3.1.1 Motivation . 38 2.2.1 Preserving Topology . 13 3.1.2 Related work . 39 2.2.2 Specialized Link Conditions . 14 3.1.3 Approach and Results . 41 2.2.3 Cost of Contraction . 15 3.2 Definition . 42 2.2.4 Optimal Vertex Placement . 18 3.2.1 Smooth 3-Manifolds . 42 2.3 Algorithm . 20 3.2.2 Piecewise Linear 3-Manifolds . 45 2.3.1 Data Structure . 20 ix x 3.3 Data Structures . 46 4.3.2 Alternative Formulations . 99 3.3.1 Triangulation . 47 4.3.3 Alternative Algorithms . 103 3.3.2 Morse-Smale Complex . 48 4.3.4 Local Contributions . 108 3.3.3 Normal Structures . 50 4.3.5 Handling Degeneracies . 108 3.4 Algorithm . 52 4.4 Experiments . 111 3.4.1 Overview. 52 4.4.1 Synthetic Functions . 112 3.4.2 Descending Manifold Construction . 54 4.4.2 Testing Algebraic Properties . 112 3.4.3 Ascending manifold construction . 71 4.4.3 Comparative Visualization . 113 3.5 Experiments . 77 4.4.4 Robustness . 119 3.5.1 Datasets . 77 4.5 Discussion . 120 3.5.2 Visualization . 79 5 Conclusions 122 3.6 Discussion . 85 A Algebraic Topology Basics and Morse Theory 124 4 Scalar Function Comparison 90 A.1 Manifolds .