Estimating normal vectors and curvatures by centroid weights

The tensors of curvature play an important role in differential geometry. In surface theory (1990), it is determined by the derivative of unit normal vectors of tangent spaces of the underlying surface. However every geometric object in computation is a discrete model. We can only approximate them. In estimating the curvature on polyhedral surfaces, how to approximate normal vectors is a crucial step. Chen and Schmitt (1992) and Taubin (1995) described two simple methods to estimate the principal curvatures. The weights they choose is related to the triangle areas. But this choice not the best. Max (1999) presented a new kind of weight to estimate the normal vector.In this paper, we will present a new set of weights from duality and gravity. We choose the centroid weights to approximate normal vectors and estimate the principle curvatures. This will lead to a better estimation than the area-weights. Our results are also comparable with Max's weights.

[1]  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.

[2]  Anil K. Jain,et al.  On reliable curvature estimation , 1989, CVPR.

[3]  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.

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

[5]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

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

[7]  H. Piaggio.  Differential Geometry of Curves and Surfaces , 1952, Nature.