GAMES Webinar 2019 – 105期 |Hanxiao Shen（New York University），Yu Wang（Massachusetts Institute of Technology）
【GAMES Webinar 2019-105期】
报告嘉宾：Hanxiao Shen，New York University
Tutte embedding is one of the most common building blocks in geometry processing algorithms due to its simplicity and provable guarantees. Although provably correct in infinite precision arithmetic, it fails in challenging cases when implemented using floating point arithmetic, largely due to the induced exponential area changes. We propose Progressive Embedding, with similar theoretical guarantees to Tutte embedding, but more resilient to the rounding error of floating point arithmetic. Inspired by progressive meshes, we collapse edges on an invalid embedding to a valid, simplified mesh, then insert points back while maintaining validity. We demonstrate the robustness of our method by computing embeddings for a large collection of disk topology meshes. By combining our robust embedding with a variant of the matchmaker algorithm, we propose a general algorithm for the problem of mapping multiply connected domains with arbitrary hard constraints to the plane, with applications in texture mapping and remeshing.
Hanxiao Shen is a Ph.D. student in the Geometric Computing Lab at New York University. He earned his B.E. in Computer Science at Zhejiang University, and his M.S. in Computer Science in Courant Institute, NYU. His research interests are geometry processing and physically-based simulation.
报告嘉宾：Yu Wang，Massachusetts Institute of Technology
报告题目：Steklov Spectral Geometry for Extrinsic Shape Analysis
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 embedding of a shape up to rigid motion, and many previous extrinsic methods lack theoretical 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 extrinsic and volumetric geometry. We use the boundary element method (BEM) to discretize the operator, accelerated 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 operators 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.
Yu Wang is a Ph.D. student in the Department of Electrical Engineering and Computer Science at MIT, conducting research in the Geometric Data Processing group. He is interested in applying mathematical, statistical, and computational approaches to the processing, analysis, and understanding of geometric data, with applications to computer graphics and vision, graph analysis, and machine learning. Previously he obtained his B.Eng. degree from Tsinghua University. He also has industry experience through internships in research labs at Microsoft and Adobe.
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