Colloquium Computer Science - Prof. Kai Hormann, Università della Svizzera italiana

Title: Maximum Likelihood Coordinates

Abstract:

Any point inside a d-dimensional simplex can be expressed in a unique way as a convex combination of the simplex's vertices, and the coefficients of this combination are called the barycentric coordinates of the point. The idea of barycentric coordinates extends to general polytopes with n vertices, but they are no longer unique if n>d+1. Several constructions of such generalized barycentric coordinates have been proposed, in particular for polygons and polyhedra, but most approaches cannot guarantee the non-negativity of the coordinates, which is important for applications like image warping and mesh deformation. We present a novel construction of non-negative and smooth generalized barycentric coordinates for arbitrary simple polygons, which extends to higher dimensions and can include isolated interior points. Our approach is inspired by maximum entropy coordinates, as it also uses a statistical model to define coordinates for convex polygons, but our generalization to non-convex shapes is different and based instead on the project-and-smooth idea of iterative coordinates. We show that our coordinates and their gradients can be evaluated efficiently and provide several examples that illustrate their advantages over previous constructions.

Short bio:

Kai Hormann is a full professor in the Faculty of Informatics at the Università della Svizzera italiana (USI) in Lugano. He received a Diploma in Mathematics in 1997 and a Ph.D. in Computer Science in 2002, both from the University of Erlangen-Nürnberg. Before moving to Lugano in 2009, he worked as a postdoctoral research fellow at Caltech in Pasadena and the CNR in Pisa, and as an assistant professor at Clausthal University of Technology. He was a visiting BMS substitute professor at Freie Universität Berlin during the winter term 2007/2008 and a visiting professor at NTU Singapore in 2018.

His research interests are focussed on the mathematical foundations of geometry processing algorithms as well as their applications in computer graphics and related fields. In particular, he is working on generalized barycentric coordinates, subdivision of curves and surfaces, barycentric rational interpolation, and dynamic geometry

processing.