A Subdivision Scheme for Continuous-Scale B-Splines and Affine-Invariant Progressive Smoothing

Multiscale representations and progressive smoothing constitute an important topic in different fields as computer vision, CAGD, and image processing. In this work, a multiscale representation of planar shapes is first described. The approach is based on computing classical B-splines of increasing orders, and therefore is automatically affine invariant. The resulting representation satisfies basic scale-space properties at least in a qualitative form, and is simple to implement. The representation obtained in this way is discrete in scale, since classical B-splines are functions in C^k-2, where k is an integer bigger or equal than two. We present a subdivision scheme for the computation of B-splines of finite support at continuous scales. With this scheme, B-splines representations in C^r are obtained for any real r in [0, ∞), and the multiscale representation is extended to continuous scale. The proposed progressive smoothing receives a discrete set of points as initial shape, while the smoothed curves are represented by continuous (analytical) functions, allowing a straightforward computation of geometric characteristics of the shape.