Modeling the Space of Point Landmark Constrained Diffeomorphisms

Chengfeng Wen, Yang Guo, Xianfeng Gu ;

Surface registration plays a fundamental role in shape analysis and geometric processing. Generally, there are three criteria in evaluating a surface mapping result: diffeomorphism, small distortion and feature alignment. In order to fulfill these requirements, this work proposes a novel model of the space of point landmark constrained diffeomorphisms. Based on Teichm\""uller theory, this mapping space is generated by the Beltrami coefficients, which are infinitesimally Teichm\""uller equivalent to $0$. These Beltrami coefficients are the solutions to a linear equation group. By using this theoretic model, optimal registrations can be achieved by iterative optimization with linear constraints in the diffeomorphism space, such as harmonic maps and Teichm\""uller maps, which minimize different type of distortions. The theoretic model is rigorous and has practical value. Our experiment results demonstrate the efficiency and efficacy of the proposed method."