The heat kernel describes the amount of heat that diffuses from one point of a manifold to another over a given time t. The behavior of this kernel contains surprisingly rich geometric information about the manifold itself. The central principle of heat methods is to use short-time diffusion as a starting point for geometric computation. Since diffusion processes are linear, local, and extremely well-studied in scientific computing, this principle translates into efficient and scalable numerical algorithms that easily generalize to a broad range of geometric discretizations (polygon meshes, point clouds, etc.). This talk will first review the original heat method for geodesic distance computation, and an overview of subsequent extensions that improve its accuracy and efficiency. It will also introduce a recent generalization to vector-valued data that allows the heat method to be applied to a much broader range of problems, including: extrapolation of level set velocities, surface parameterization via the logarithmic map, robust computation of Karcher means and geometric medians, multiply-connected centroidal Voronoi diagrams, and intrinsic landmarks for shape correspondence.
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