Beijing Institute of Mathematical Sciences and Applications

Zheng Wei Liu , Shing-Tung Yau , Hui Zhao

Edward Chien

Friday, April 29, 2022 9:30 AM - 10:30 AM

Zoom 388 528 9728 (BIMSA)

Tutte’s harmonic parameterization method is a discrete realization of the Rado-Kneser-Choquet theorem and is a basic tool for many graphics and geometry processing pipelines. Given a mesh of disk topology, with boundary mapped to the boundary of a convex polygon, a simple linear solve results in a guaranteed bijective parameterization. In two recent graphics works, we generalize this parametrization method for locally injective “seamless” parameterizations of meshes of arbitrary topology (cut to a disk). Analogous boundary conditions are found for this scenario, and a discrete index argument on harmonic forms is used to prove the result. I will sketch this argument, note the relevance of seamless parameterizations to quadrilateral meshing, and discuss our numerical implementation of the result. Time permitting, possible extensions and related works on conformal mapping will be discussed. [1] A. Bright, E. Chien, O. Weber, Harmonic global parameterization with rational holonomy, ACM Transactions on Graphics Vol. 36, No. 4 (SIGGRAPH 2017) [2] E. F. Hefetz, E. Chien, O. Weber, A subspace method for fast locally-injective harmonic mapping, Computer Graphics Forum Vol. 38, No. 2 (Eurographics 2019)

Edward Chien is an Assistant Professor of Computer Science at Boston University, having joined recently in 2020. His research applies tools and insights from differential geometry and topology to solve problems in graphics, computational engineering, and machine learning. His work has been published in venues such as SIGGRAPH, SGP, NeurIPS, and ICML. Prior to BU, he was a Postdoc in the Geometric Data Processing group at MIT, led by Justin Solomon, and in Ofir Weber’s lab at Bar-Ilan University. He earned his PhD in Mathematics from Rutgers in 2015 working with Feng Luo.