Inverse Problems and Imaging (IPI)

Subdivision connectivity remeshing via Teichmüller extremal map

Pages: 825 - 855, Volume 11, Issue 5, October 2017      doi:10.3934/ipi.2017039

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Chi Po Choi - Department of Statistics, University of California, Davis, United States (email)
Xianfeng Gu - Department of Computer Science, State University of New York at Stony Brook, United States (email)
Lok Ming Lui - Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China (email)

Abstract: Curvilinear surfaces in 3D Euclidean spaces are commonly represented by triangular meshes. The structure of the triangulation is important, since it affects the accuracy and efficiency of the numerical computation on the mesh. Remeshing refers to the process of transforming an unstructured mesh to one with desirable structures, such as the subdivision connectivity. This is commonly achieved by parameterizing the surface onto a simple parameter domain, on which a structured mesh is built. The 2D structured mesh is then projected onto the surface via the parameterization. Two major tasks are involved. Firstly, an effective algorithm for parameterizing, usually conformally, surface meshes is necessary. However, for a highly irregular mesh with skinny triangles, computing a folding-free conformal parameterization is difficult. The second task is to build a structured mesh on the parameter domain that is adaptive to the area distortion of the parameterization while maintaining good shapes of triangles. This paper presents an algorithm to remesh a highly irregular mesh to a structured one with subdivision connectivity and good triangle quality. We propose an effective algorithm to obtain a conformal parameterization of a highly irregular mesh, using quasi-conformal Teichmüller theories. Conformality distortion of an initial parameterization is adjusted by a quasi-conformal map, resulting in a folding-free conformal parameterization. Next, we propose an algorithm to obtain a regular mesh with subdivision connectivity and good triangle quality on the conformal parameter domain, which is adaptive to the area distortion, through the landmark-matching Teichmüller map. A remeshed surface can then be obtained through the parameterization. Experiments have been carried out to remesh surface meshes representing real 3D geometric objects using the proposed algorithm. Results show the efficacy of the algorithm to optimize the regularity of an irregular triangulation.

Keywords:  Surface remeshing, quasi-conformal, Beltrami coefficient, Beltrami's equation, Teichmüller map.
Mathematics Subject Classification:  Primary: 94A08, 49Q10, 65E05; Secondary: 65S05, 49M30.

Received: March 2016;      Revised: June 2017;      Available Online: July 2017.