Multi-scale orientation estimation for unstructured sample points

Orientation of sampled points is a very useful information at various levels of computer vision tasks. Given a set of unstructured points sampled from a curve in an image, we consider the problem estimating orientation, i.e. finding tangent direction (or equivalently normal direction), at each point. The points are discrete samples of curves, and can be perturbated severely by noise. Hoppe et al. (1992) suggests the use of the covariance matrix, constructed from a k-neighborhood of a point x, to estimate the orientation at x. A problem with this approach is the fixed k value, which imposes implicitly a very restrictive condition on the density of sampled points in order to work properly. To solve this problem, we present a multi-scale method for reliable estimation of orientation and curvature for a set of unstructured points. Furthermore, we derive a structural representation for the point set