Heat Kernel Laplace-Beltrami Operator on Digital Surfaces

Many problems in image analysis, digital processing and shape optimization can be expressed as variational problems involving the discretization of the Laplace-Beltrami operator. Such discretizations have have been widely studied for meshes or polyhedral surfaces. On digital surfaces, direct applications of classical operators are usually not satisfactory (lack of multigrid convergence, lack of precision.. .). In this paper, we first evaluate previous alternatives and propose a new digital Laplace-Beltrami operator showing interesting properties. This new operator adapts Belkin et al. [1] to digital surfaces embedded in 3D. The core of the method relies on an accurate estimation of measures associated to digital surface elements. We experimentally evaluate the interest of this operator for digital geometry processing tasks.

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

[2]  Anil N. Hirani,et al.  Discrete exterior calculus , 2005, math/0508341.

[3]  T. Regge.  General relativity without coordinates , 1961 .

[4]  Bruno Lévy,et al.  Spectral Mesh Processing , 2009, SIGGRAPH '10.

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

[6]  Azriel Rosenfeld,et al.  Digital geometry - geometric methods for digital picture analysis , 2004 .

[7]  Eitan Grinspun,et al.  Discrete laplace operators: no free lunch , 2007, Symposium on Geometry Processing.

[8]  Mikhail Belkin,et al.  Discrete laplace operator on meshed surfaces , 2008, SCG '08.

[9]  Jacques-Olivier Lachaud,et al.  Properties of Gauss Digitized Shapes and Digital Surface Integration , 2016, Journal of Mathematical Imaging and Vision.

[10]  Jacques-Olivier Lachaud,et al.  Multigrid convergent principal curvature estimators in digital geometry , 2014, Comput. Vis. Image Underst..

[11]  Mikhail Belkin,et al.  Towards a theoretical foundation for Laplacian-based manifold methods , 2005, J. Comput. Syst. Sci..

[12]  Rémy Malgouyres,et al.  Mesh Parameterization with Generalized Discrete Conformal Maps , 2012, Journal of Mathematical Imaging and Vision.

[13]  Hao Zhang.  Discrete Combinatorial Laplacian Operators for Digital Geometry Processing , 2004 .

[14]  Konrad Polthier,et al.  Identifying Vector Field Singularities Using a Discrete Hodge Decomposition , 2002, VisMath.

[15]  Stanislav Molchanov,et al.  DIFFUSION PROCESSES AND RIEMANNIAN GEOMETRY , 1975 .

[16]  S. Varadhan.  On the behavior of the fundamental solution of the heat equation with variable coefficients , 2010 .

[17]  Jacques-Olivier Lachaud,et al.  Parameter-Free and Multigrid Convergent Digital Curvature Estimators , 2014, DGCI.

[18]  Neil A. Dodgson,et al.  Advances in Multiresolution for Geometric Modelling , 2005 .

[19]  Keenan Crane,et al.  Geodesics in heat: A new approach to computing distance based on heat flow , 2012, TOGS.

[20]  Alexander I. Bobenko,et al.  A Discrete Laplace–Beltrami Operator for Simplicial Surfaces , 2005, Discret. Comput. Geom..

[21]  Jacques-Olivier Lachaud,et al.  Continuous Analogs of Digital Boundaries: A Topological Approach to Iso-Surfaces , 2000, Graph. Model..

[22]  S. Rosenberg.  The Laplacian on a Riemannian Manifold: The Construction of the Heat Kernel , 1997 .

[23]  Christian Mercat,et al.  Discrete Complex Structure on Surfel Surfaces , 2008, DGCI.

[24]  Tristan Roussillon,et al.  Multigrid Convergence of Discrete Geometric Estimators , 2012 .