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A Triangulation-Invariant Method for Anisotropic Geodesic Map Computation on Surface Meshes
Oct. 2012 (vol. 18 no. 10)
pp. 1664-1677
Sang Wook Yoo, Korea Advanced Institute of Science and Technology, Daejeon
Joon-Kyung Seong, Soongsil University, Seoul
Min-Hyuk Sung, Korea Institute of Science and Technology, Seoul
Sung Yong Shin, Korea Advanced Institute of Science and Technology, Daejeon
Elaine Cohen, University of Utah, Salt Lake City
This paper addresses the problem of computing the geodesic distance map from a given set of source vertices to all other vertices on a surface mesh using an anisotropic distance metric. Formulating this problem as an equivalent control theoretic problem with Hamilton-Jacobi-Bellman partial differential equations, we present a framework for computing an anisotropic geodesic map using a curvature-based speed function. An ordered upwind method (OUM)-based solver for these equations is available for unstructured planar meshes. We adopt this OUM-based solver for surface meshes and present a triangulation-invariant method for the solver. Our basic idea is to explore proximity among the vertices on a surface while locally following the characteristic direction at each vertex. We also propose two speed functions based on classical curvature tensors and show that the resulting anisotropic geodesic maps reflect surface geometry well through several experiments, including isocontour generation, offset curve computation, medial axis extraction, and ridge/valley curve extraction. Our approach facilitates surface analysis and processing by defining speed functions in an application-dependent manner.

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Index Terms:
Measurement,Equations,Surface treatment,Approximation algorithms,Least squares approximation,shape analysis.,Geodesic,anisotropy,surface mesh,Hamilton-Jacobi-Bellman,curvature minimization,curvature variation minimization
Sang Wook Yoo, Joon-Kyung Seong, Min-Hyuk Sung, Sung Yong Shin, Elaine Cohen, "A Triangulation-Invariant Method for Anisotropic Geodesic Map Computation on Surface Meshes," IEEE Transactions on Visualization and Computer Graphics, vol. 18, no. 10, pp. 1664-1677, Oct. 2012, doi:10.1109/TVCG.2012.29
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