Discrete Curvature Estimation Methods for Triangulated Surfaces

We review some recent approaches to estimate discrete Gaussian and mean curvatures for triangulated surfaces, and discuss their characteristics. We focus our attention on concentrated curvature which is generally used to estimate Gaussian curvature. We present a result that shows that concentrated curvature can also be used to estimate mean curvature and hence principal curvatures. This makes concentrated curvature one of the fundamental notions in discrete computational geometry.

[1]  D. Levin,et al.  Optimizing 3D triangulations using discrete curvature analysis , 2001 .

[2]  Leila De Floriani,et al.  Discrete Distortion for Surface Meshes , 2009, ICIAP.

[3]  M. Troyanov.  Les surfaces euclidiennes à singularités coniques. (Euclidean surfaces with cone singularities). , 1986 .

[4]  Michela Spagnuolo,et al.  Shape Analysis and Structuring (Mathematics and Visualization) , 2007 .

[5]  Ross T. Whitaker,et al.  Partitioning 3D Surface Meshes Using Watershed Segmentation , 1999, IEEE Trans. Vis. Comput. Graph..

[6]  G. Sandini,et al.  Computer Vision — ECCV'92 , 1992, Lecture Notes in Computer Science.

[7]  Lyuba Alboul,et al.  Discrete curvatures and Gauss maps for polyhedral surfaces , 2005, EuroCG.

[8]  Ernest M. Stokely,et al.  Surface Parametrization and Curvature Measurement of Arbitrary 3-D Objects: Five Practical Methods , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

[9]  Francis Schmitt,et al.  Intrinsic Surface Properties from Surface Triangulation , 1992, ECCV.

[10]  Dereck S. Meek,et al.  On surface normal and Gaussian curvature approximations given data sampled from a smooth surface , 2000, Comput. Aided Geom. Des..

[11]  Jia-Guang Sun,et al.  Convergence Analysis of Discrete Differential Geometry Operators over Surfaces , 2005, IMA Conference on the Mathematics of Surfaces.

[12]  Leila De Floriani,et al.  Discrete Distortion in Triangulated 3‐Manifolds , 2008, Comput. Graph. Forum.

[13]  Jean-Marie Morvan,et al.  On the angular defect of triangulations and the pointwise approximation of curvatures , 2003, Comput. Aided Geom. Des..

[14]  Gabriel Taubin,et al.  Estimating the tensor of curvature of a surface from a polyhedral approximation , 1995, Proceedings of IEEE International Conference on Computer Vision.

[15]  Mario Vento,et al.  Image Analysis and Processing - ICIAP 2009, 15th International Conference Vietri sul Mare, Italy, September 8-11, 2009, Proceedings , 2009, ICIAP.

[16]  Mark Meyer,et al.  Discrete Differential-Geometry Operators for Triangulated 2-Manifolds , 2002, VisMath.

[17]  Kouki Watanabe,et al.  Detection of Salient Curvature Features on Polygonal Surfaces , 2001, Comput. Graph. Forum.

[18]  Guoliang Xu,et al.  Convergence analysis of a discretization scheme for Gaussian curvature over triangular surfaces , 2006, Comput. Aided Geom. Des..

[19]  Gershon Elber,et al.  A comparison of Gaussian and mean curvatures estimation methods on triangular meshes , 2003, 2003 IEEE International Conference on Robotics and Automation (Cat. No.03CH37422).

[20]  Emanuele Danovaro,et al.  Surface Segmentation through Concentrated Curvature , 2007, 14th International Conference on Image Analysis and Processing (ICIAP 2007).

[21]  Cindy Grimm,et al.  Estimating Curvature on Triangular Meshes , 2006, Int. J. Shape Model..

[22]  Mongi A. Abidi,et al.  Perception-based 3D triangle mesh segmentation using fast marching watersheds , 2003, 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings..

[23]  Ergun Akleman,et al.  Insight for Practical Subdivision Modeling with Discrete Gauss-Bonnet Theorem , 2006, GMP.

[24]  Manfredo P. do Carmo,et al.  Differential geometry of curves and surfaces , 1976 .

[25]  Andrew M. Wallace,et al.  Computation of Local Differential Parameters on Irregular Meshes , 2000, IMA Conference on the Mathematics of Surfaces.

[26]  Leila De Floriani,et al.  A Geometric Approach to Curvature Estimation on Triangulated 3D Shapes , 2010, GRAPP.