Estimating curvatures and their derivatives on triangle meshes

The computation of curvature and other differential properties of surfaces is essential for many techniques in analysis and rendering. We present a finite-differences approach for estimating curvatures on irregular triangle meshes that may be thought of as an extension of a common method for estimating per-vertex normals. The technique is efficient in space and time, and results in significantly fewer outlier estimates while more broadly offering accuracy comparable to existing methods. It generalizes naturally to computing derivatives of curvature and higher-order surface differentials.

[1]  Carlo H. Séquin,et al.  Functional optimization for fair surface design , 1992, SIGGRAPH.

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

[3]  Robert B. Fisher,et al.  Experiments in Curvature-Based Segmentation of Range Data , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[4]  Victoria Interrante,et al.  Enhancing transparent skin surfaces with ridge and valley lines , 1995, Proceedings Visualization '95.

[5]  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.

[6]  Pascal Fua,et al.  Using crest lines to guide surface reconstruction from stereo , 1996, Proceedings of 13th International Conference on Pattern Recognition.

[7]  Roberto Cipolla,et al.  The visual motion of curves and surfaces , 1998, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[8]  Mark Meyer,et al.  Implicit fairing of irregular meshes using diffusion and curvature flow , 1999, SIGGRAPH.

[9]  Nelson L. Max,et al.  Weights for Computing Vertex Normals from Facet Normals , 1999, J. Graphics, GPU, & Game Tools.

[10]  Aaron Hertzmann,et al.  Illustrating smooth surfaces , 2000, SIGGRAPH.

[11]  Joonki Paik,et al.  Robust crease detection and curvature estimation of piecewise smooth surfaces from triangle mesh approximations using normal voting , 2001, Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001.

[12]  Adam Finkelstein,et al.  Real-time hatching , 2001, SIGGRAPH.

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

[14]  Ilan Shimshoni,et al.  Estimating the principal curvatures and the Darboux frame from real 3D range data , 2002, Proceedings. First International Symposium on 3D Data Processing Visualization and Transmission.

[15]  Jens Gravesen,et al.  Constructing Invariant Fairness Measures for Surfaces , 2002, Adv. Comput. Math..

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

[17]  Marc Pouget,et al.  Estimating differential quantities using polynomial fitting of osculating jets , 2003, Comput. Aided Geom. Des..

[18]  David Cohen-Steiner,et al.  Restricted delaunay triangulations and normal cycle , 2003, SCG '03.

[19]  Adam Finkelstein,et al.  Suggestive contours for conveying shape , 2003, ACM Trans. Graph..

[20]  Pierre Alliez,et al.  Anisotropic polygonal remeshing , 2003, ACM Trans. Graph..

[21]  Victoria Interrante,et al.  A novel cubic-order algorithm for approximating principal direction vectors , 2004, TOGS.

[22]  Adam Finkelstein,et al.  Interactive rendering of suggestive contours with temporal coherence , 2004, NPAR '04.