CONFORMAL WASSERSTEIN DISTANCE : II

This paper is a companion paper to [19]. We provide numerical procedures and algorithms for computing the alignment of and distance between two disk type surfaces. We provide a convergence analysis of the discrete approximation to the arising mass-transportation problems. We furthermore generalize the framework to support sphere-type surfaces, and prove a result connecting this distance to local geodesic distortion. Lastly, we provide numerical experiments on several surfaces’ datasets and compare to state of the art method.

[1]  Alexander M. Bronstein,et al.  Efficient Computation of Isometry-Invariant Distances Between Surfaces , 2006, SIAM J. Sci. Comput..

[2]  R. Bass,et al.  Review: P. Billingsley, Convergence of probability measures , 1971 .

[3]  J L Lancaster,et al.  Automated Talairach Atlas labels for functional brain mapping , 2000, Human brain mapping.

[4]  Guillermo Sapiro,et al.  A Theoretical and Computational Framework for Isometry Invariant Recognition of Point Cloud Data , 2005, Found. Comput. Math..

[5]  V. Deineko,et al.  The Quadratic Assignment Problem: Theory and Algorithms , 1998 .

[6]  Daniela Giorgi,et al.  SHape REtrieval Contest 2007: Watertight Models Track , 2007 .

[7]  I. Daubechies,et al.  Conformal Wasserstein distances: Comparing surfaces in polynomial time , 2011, 1103.4408.

[8]  Lei Zhu,et al.  Optimal Mass Transport for Registration and Warping , 2004, International Journal of Computer Vision.

[9]  Mikael Fortelius,et al.  High-level similarity of dentitions in carnivorans and rodents , 2007, Nature.

[10]  A. Dale,et al.  High‐resolution intersubject averaging and a coordinate system for the cortical surface , 1999, Human brain mapping.

[11]  K. Polthier,et al.  On the convergence of metric and geometric properties of polyhedral surfaces , 2007 .

[12]  Alexander M. Bronstein,et al.  Numerical Geometry of Non-Rigid Shapes , 2009, Monographs in Computer Science.

[13]  Ulrich Pinkall,et al.  Computing Discrete Minimal Surfaces and Their Conjugates , 1993, Exp. Math..

[14]  M. Gromov.  Metric Structures for Riemannian and Non-Riemannian Spaces , 1999 .

[15]  Facundo Mémoli,et al.  Eurographics Symposium on Point-based Graphics (2007) on the Use of Gromov-hausdorff Distances for Shape Comparison , 2022 .

[16]  S. Yau,et al.  Global conformal surface parameterization , 2003 .

[17]  T. Funkhouser,et al.  Möbius voting for surface correspondence , 2009, SIGGRAPH 2009.

[18]  Ron Kimmel,et al.  Generalized multidimensional scaling: A framework for isometry-invariant partial surface matching , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[19]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[20]  G. Dziuk.  Finite Elements for the Beltrami operator on arbitrary surfaces , 1988 .