Beltrami Representation for Diffeomorphisms and Its Applications
We introduce a simple representation of bijective surface maps, which facilitates the process of surface registration and morphometry.In shape analysis, understanding the deformations between different surface data is of great importance.It usually involves finding an optimal 1-1 correspondence between surfaces within a large class of admissible bijective mappings.Such a process is called surface registration.The difficulty lies in the fact that the space of all surface homeomorphisms is a complicated functional space.Hence, the optimization process over the search space of bijective surface maps becomes challenging.To tackle this problem, we describe in this paper a novel representation of surface diffeomorphisms using Beltrami coefficients (BCs), which are complex-valued functions defined on surfaces with supreme norm less than 1.Fixing the correspondences of several points on a pair of surfaces, there is a 1-1 correspondence between the set of surface diffeomorphisms between them and the set of Beltrami coefficients on the source domain.Hence, every surface diffeomorphisms can be represented by a unique Beltrami coefficient.Conversely, given a Beltrami coefficient, we can reconstruct the unique surface map associated to it.Using BCs to represent surface maps is advantageous because it captures most essential features of a surface map.By adjusting BCs,we can adjust the surface homeomorphism accordingly to obtain desired properties of the map.The space of BCs is a simple functional space.Using BCs, we can easily manipulate shape deformations and find an optimal bijective deformations between shapes.In this paper, we will discuss how the Beltrami representation for diffeomorphisms can be computed and how it can be applied to real applications.
Conformal geometry Ricci flow holomorphic differential discrete surface
Lok Ming Lui David Xianfeng Gu Wei Zeng Shing-Tung Yau
DEPARTMENT OF MATHEMATICS, THE CHINESE UNIVERSITY OF HONG KONG RM 207, LAY SHAW BUILDING, SHATIN, HO COMPUTER SCIENCE DEPARTMENT, SUNY AT STONY BROOK COMPUTER SCIENCE BUILDING 2425, STONY BROOK, NY 117 SCHOOL OF COMPUTING AND INFORMATION SCIENCES FLORIDA INTERNATIONAL UNIVERSITY MIAMI, FL 33199, USA MATHEMATICS DEPARTMENT, HARVARD UNIVERSITY 1 OXFORD STREET, CAMBRIDGE, MA 02138, USA
国际会议
The 6th International Congress of Chinese Mathematicians (第6届世界华人数学家大会)
台北
英文
523-552
2013-07-14（万方平台首次上网日期，不代表论文的发表时间）