Applied Math And Analysis Seminar
Thursday, November 2, 2017, 2:00pm, Gross 330
Nadav Dym (Weizmann Institute of Science, Israel)
Provably good convex methods for mapping problems
Abstract:
Computing mappings or correspondences between surfaces is an important tool for many applications in computer graphics, medical imaging, morphology and related fields. Mappings of least angle distortion (conformal) and distance distortion (isometric) are of particular interest. The problem of finding conformal/isometric mappings between surfaces is typically formulated as a difficult non-convex optimization problem. Convex relaxations approximate the non-convex problem by a convex problem which can then be solved globally. In this talk we will discuss convex relaxations for both the problem of computing conformal mappings, and the problem of computing isometries. For each problem we will show that the solution of the relaxed problem is in fact the solution of the original problem! We will pay special attention to the more challenging case of symmetric surfaces. For non-isometric surfaces we search for the most-isometric mapping between the surfaces. Several relaxations have been proposed for such problems, where more accurate relaxations are typically also more time consuming. We propose two algorithms which strike a good balance between accuracy and efficiency: The DS++ algorithm and the Sinkhorn-JA algorithm. We utilize these algorithms to achieve state of the art results for shape matching and image arrangement.

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