Comparing shapes of genus-zero surfaces

AbstractWe present a new method to compare the shapes of genus-zero surfaces by introducing a measure of mutual stretching, the symmetric distortion energy. Given a pair of genus-zero surfaces, we establish the existence of a conformal diffeomorphism that minimizes this energy. We then prove that the energies of the minimizing diffeomorphisms produce a metric on the space of genus-zero Riemannian surfaces. This metric and the corresponding optimal diffeomorphisms are shown to have properties that are highly desirable for applications.

[1]  U. Pinkall,et al.  Discrete conformal maps and ideal hyperbolic polyhedra , 2010, 1005.2698.

[2]  Ron Kimmel,et al.  On Bending Invariant Signatures for Surfaces , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[3]  Shing-Tung Yau,et al.  Computing Conformal Structure of Surfaces , 2002, Commun. Inf. Syst..

[4]  Lipman Bers,et al.  Uniformization, Moduli, and Kleinian Groups , 1972 .

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

[6]  Kenneth Stephenson,et al.  Introduction to Circle Packing: The Theory of Discrete Analytic Functions , 2005 .

[7]  Guillermo Sapiro,et al.  Conformal Surface Parameterization for Texture Mapping , 1999 .

[8]  Feng Luo.  COMBINATORIAL YAMABE FLOW ON SURFACES , 2003 .

[9]  B. Rodin,et al.  The convergence of circle packings to the Riemann mapping , 1987 .

[10]  Monica K. Hurdal,et al.  Discrete conformal methods for cortical brain flattening , 2009, NeuroImage.

[11]  P. Koehl,et al.  Landmark-free geometric methods in biological shape analysis , 2015, Journal of The Royal Society Interface.

[12]  Patrice Koehl,et al.  Automatic Alignment of Genus-Zero Surfaces , 2014, IEEE Transactions on Pattern Analysis and Machine Intelligence.

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

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

[15]  Patrice Koehl,et al.  How round is a protein? Exploring protein structures for globularity using conformal mapping , 2014, Front. Mol. Biosci..

[16]  Thomas A. Funkhouser,et al.  Algorithms to automatically quantify the geometric similarity of anatomical surfaces , 2011, Proceedings of the National Academy of Sciences.

[17]  O. Schramm,et al.  On the convergence of circle packings to the Riemann map , 1996 .

[18]  Shing-Tung Yau,et al.  Optimal Global Conformal Surface Parameterization for Visualization , 2004, Commun. Inf. Syst..