Estimating the Tensor of Curvature of a Surface from a Polyhedral Approximation

Estimating principal curvatures and principal directions of a surface from a polyhedral approximation with a large number of small faces, such as those produced by iso-surface construction algorithms, has become a basic step in many computer vision algorithms. Particularly in those targeted at medical applications. The tensor of curvature of the surface instead, and we describe a method to do so. The tensor of curvature of a surface is a function that assigns a quadratic form to each point on the surface, and can be described by a 3 x 3 symmetric matrix for each surface point. The principal curvatures and principal directions of the surface at a point are eigenvalues and eigenvectors of the matrix associated with the point. The third eigenvector is the normal vector, with associated eigenvalue zero. The algorithm introduced in this paper is based on a new integral formula for another family of symmetric 3 x 3matrices, closely related to the tensor of curvature matrices. Although eigenvalues and eigenvectors must be computed to extract principal curvatures and principal directions from these matrices, we show that they can be computed in closed form using well established matrix computations operations. The complexity of the resulting algorithm is linear, both in time and in space, as a function of the number of vertices and faces of the polyhedral surface. We finish the paper with some experiments that evaluate the accuracy of the method in cases where principal curvatures and principal directions can be evaluated analytically.