Steklov Geometry Processing: An Extrinsic Approach to Spectral Shape Analysis by Yu Wang Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Master of Science at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2018 @ Massachusetts Institute of Technology 2018. All rights reserved. Signature redacted A uthor ............. Department of Electrica Engineering and Computer Science May 23, 2018 A /- A Signature redacted Certified by....... V I Justin Solomon Assistant Professor of Electrical Engineering and Computer Science Thesis Supervisor Signature redacted Accepted by ........... k (LI U Leslie A. Kolodziejski Professor of Electrical Engineering and Computer Science Chair, Department Committee for Graduate Students MASSACHUSETTS INSTITUTE OF TECHNOLOGY C JUNj JUN 18 2018 LIBRARIE 2 Steklov Geometry Processing: An Extrinsic Approach to Spectral Shape Analysis by Yu Wang Submitted to the Department of Electrical Engineering and Computer Science on May 23, 2018, in partial fulfillment of the requirements for the degree of Master of Science Abstract We propose using the Dirichlet-to-Neumann operator as an extrinsic alternative to the Laplacian for spectral geometry processing and shape analysis. Intrinsic approaches, usually based on the Laplace-Beltrami operator, cannot capture the spatial embed- ding of a shape up to rigid motion, and many previous extrinsic methods lack theo- retical justification. Instead, we consider the Steklov eigenvalue problem, computing the spectrum of the Dirichlet-to-Neumann operator of a surface bounding a volume. A remarkable property of this operator is that it completely encodes volumetric ge- ometry. We use the boundary element method (BEM) to discretize the operator, ac- celerated by hierarchical numerical schemes and preconditioning; this pipeline allows us to solve eigenvalue and linear problems on large-scale meshes despite the density of the Dirichlet-to-Neumann discretization. We further demonstrate that our oper- ators naturally fit into existing frameworks for geometry processing, making a shift from intrinsic to extrinsic geometry as simple as substituting the Laplace-Beltrami operator with the Dirichlet-to-Neumann operator. Thesis Supervisor: Justin Solomon Title: Assistant Professor of Electrical Engineering and Computer Science 3 4 Acknowledgments I would like to thank my advisor, Justin Solomon, for his tremendous support and guidance during my two years at MIT. His knowledge, insight and passion has always been a source of inspiration for me, and it has been truly a pleasure and enjoyable experience to work with him. The body of this thesis is derived from a project done in collaboration with Mirela Ben-Chen, losif Polterovich and Justin Solomon, all of whom contributed text to the discussion [781. I would like to thank all of my collaborators for their guidance and input to the project, as well as anonymous ACM ToG/SIGGRAPH reviewers for their comment and feedback to improve this work. Many members in MIT Geometric Data Processing (GDP), Medical Vision, Computer Graphics and Computer Vision groups, as well as our colleagues outside MIT provided insightful comments and helped in various aspects. Mikhail Bessmeltsev, Sebastian Claici, Andrew Spielberg and Aric Bartle proofread earlier versions of the paper which evolved into this thesis. Nilima Nigam, Miaomiao Zhang and Maks Ovsjanikov provided many suggestions. I also acknowledge the generous support of the Herbert E. (1933) and Dorothy J. Grier Presidential Fellowship and the Thomas and Stacey Siebel Foundation Scholarship for my graduate study and research at MIT. Finally, thank you to my family for their support and love. 5 6 Contents 1 Introduction 9 1.1 Background ...... ............... 9 1.2 Related work ...... ........ ...... 12 1.2.1 Intrinsic geometry .... .......... 12 1.2.2 Extrinsic geometry ... .......... 12 1.2.3 Numerical PDE ............... 14 2 Method 15 2.1 Mathematical preliminaries ........... .... ... .. .15 2.1.1 Steklov eigenproblem ............ ... .... .15 2.1.2 Boundary representation and operators . .. ... ... .. 17 2.1.3 Calder6n projection & Dirichlet-to-Neumann operator ..... 19 2.2 Discrete Dirichlet-to-Neumann operator .... .. .... ..... 20 2.2.1 Weak form boundary operators ..... .... .... .. 20 2.2.2 Discretized Dirichlet-to-Neumann operator . ....... .. 22 2.3 Matrix-free formulation ... ........ .... .... ...... 23 2.3.1 Expansion ........ ........ .. ..... ..... 23 2.3.2 Symmetrization ........ ....... .......... 24 2.3.3 Preconditioning .. ........ ..... .......... 24 2.3.4 Iterative solvers ..... .......... ... ...... 26 2.3.5 Generalization ..... ........ ... .... ...... 26 2.3.6 Implementation details . ........ .. ... ...... 27 2.4 Eigenvalue normalization ..... ......... .......... 29 7 2.5 Validation . ... ... ... ... ... .. ... ... ... ... 3 2 2.5.1 Robustness to topological change . .. ... ... ... ... 3 2 2.5.2 Robustness to surface sampling . ... .... ..... .. 3 2 2.5.3 Robustness to vertex noise .. ... .... .... ..... 3 4 2.5.4 Stability test for volume isometries . ..... .... ..... 3 6 2.5.5 Conditioning . ... ... ... .. ..... .... ..... 3 8 2.5.6 Convergence to analytical eigenvalues . .... ..... .... 3 8 2.5.7 Tim ing . .. .. ... .. .. ... .... ..... .... 3 8 2.5.8 Comparison with FEM .. ... .. .. .... ..... ... 3 9 3 Results 45 3.1 Experiments and applications .. .. .. .. .... ..... ... 4 5 3.1.1 Steklov spectrum .. .. ... .. .. .... ..... ... 4 8 3.1.2 Kernel-based descriptors . .. .. .. ... ... ... ... 4 9 3.1.3 Spectral distance . ... ... .. .. ... ... ... ... -3 3.1.4 Volume-aware segmentation . .. .. .... .... .... 5 3.1.5 Shape differences and variability . 3.1.6 Shape exploration and retrieval . ... ... ... ... .. 6 1 3.1.7 Comparison with the Dirac operator .. ... ... ... .. 6 K5 4 Conclusion 71 8 Chapter 1 Introduction There has been the dichotomy in geometry processing and shape analysis. Many existing approaches view a shape as a thin shell given in the form of a surface, but often the shape really should be understood as a solid whose boundary is represented by the surface. With the popularity and proliferation of intrinsic Laplacian geometry processing, this subtle difference is often overlooked and ignored. In this thesis, we propose to resolve this dichotomy with a simple systematic solution using the DtN operator and corresponding Steklov spectrum. 1.1 Background Geometry processing and shape analysis tools for computer graphics typically draw from two complementary theories of geometry. To distinguish these, consider a closed surface embedded in three dimensions. From the viewpoint of extrinsic geometry, we might examine the surface as the outer boundary of a volume. This approach relies on distances and other measurements taken from the space surrounding the surface to understand its shape. Contrastingly, many techniques in differential geometry de- couple extrinsic shape from intrinsic geometry, which is concerned with quantities like geodesic distances that can be measured without leaving the outer surface. A crowning achievement of classical differential geometry shows that certain quantities like Gaussian curvature can be measured intrinsically; this basic observation inspired 9 theoretical exploration of purely intrinsic techniques. Computational geometry pro- cessing embraced the intrinsic perspective early on, leading to numerous applications of intrinsic computations. Intrinsic geometry, however, is an ineffective de- scription of shape for many applications. First, the spatial embedding information is lost. The variabil- ity in embedding can be essential. As an extreme Figure 1-1 example, consider searching a database of origami models: From an intrinsic perspective, all origami is equivalent to the same piece of flat paper. Second, the intrinsic perspective is a counterintuitive way to describe the shapes of many real-world objects, e.g. identifying the inward and outward bumps on the cubes in Figure 1-1 . A naive approach to extrinsic geometry could be to use the (x, y, z) coordinates of the embedding. This incorporates information about the embedding but is not invariant to rigid motion. Alternatively, rotation-invariant shape descriptors (e.g., built from spherical harmonic power spectra) usually involve extrinsic information about a surface as a shell rather than as the boundary of a volume. We follow the standard practice of operator-basedapproaches in computer graphics and shape analysis. Operator-based approaches usually consider a geometric varia- tional problem that is well-defined in the continuous case; minimizing of the vari- ational objective, which is often a quadratic bilinear form corresponding to some smoothness energy with physical interpretation, yields a linear partial differential equation (PDE) defined using a linear operator with certain boundary conditions. Then techniques from numerical analysis are used to discretize the variational prob- lem (or equivalently, the resulting PDE), yielding a matrix which discretizes the quadratic energy (and equivalently, the corresponding linear operator). The reason that operator-based methods have been popular is that they nat- urally are consistent and invariant to geometric representation: Same results will be obtained, regardless of the underlying shape representation used: For example, the geometric domain may be given in the form of a triangle mesh, quadrilateral 10 mesh, polygon mesh, triangle/polygon