Two Mathematical Tools for Shape Matching by Bruno Lévy

Prof. Bruno Lévy is a senior researcher with Inria Nancy-Grand Est, and a member of the LORIA lab. He defended his Ph.D. thesis in 1999 and did a post-doc in Stanford. He was hired in 2000 by Inria. He is currently the head of the ALICE research team, that he created in 2004. He received the Inria young researcher award in 2011. His main interest is numerical geometry, that is to say mathematical algorithms for acquiring, transforming and optimizing the representation of 3D shapes. He developed several algorithms for geometry processing and mesh generation, such as Least Squares Conformal Maps, used to generate (u,v) texture mapping coordinates in several softwares, Manifold Harmonics, a Fourier-like spectral mesh analysis algorithm, and more recently, Voronoi Parallel Linear Enumeration, an algorithm for re-meshing surfaces and volumes.

Abstract

In this talk, I will focus on the problem of finding correspondences between two different shapes and present two fundamentally different approaches to tackle this issue:
  • The first one takes a geometric point of view, and tries to find a map that transforms one shape into the other one while minimizing a criterion. As such, the notion of optimal transport map is well formalized and can be computed by an elegant algorithm that bridges the gap between computational geometry and numerical analysis.
  • The second approach takes an analytic point of view, and considers each shape to be analyzed as a function space. This function space can be parameterized by a function basis, defined as the eigenfunctions of a linear operator. As such, the eigenfunctions of the Laplace operator have interesting properties, and they match more usual function bases when computed on canonical shapes (discrete cosine transform on the square and spherical harmonics on the sphere). They can be used for a wide variety of tasks, such as geometric filtering, detection of characteristic points, and pose-invariant shape matching.
The talk will be illustrated by live demos. Most of the source-code is available in the Graphite and in the (upcoming) Geogram open-source softwares.