Scale-Space Derived From B-Splines

It is well-known that the linear scale-space theory in computer vision is mainly based on the Gaussian kernel. The purpose of the paper is to propose a scale-space theory based on B-spline kernels. Our aim is twofold. On one hand, we present a general framework and show how B-splines provide a flexible tool to design various scale-space representations: continuous scalespace, dyadic scale-space frame, and compact scale-space representation. In particular, we focus on the design of continuous scale-space and dyadic scale-space frame representation. A general algorithm is presented for fast implementation of continuous scale-space at rational scales. In the dyadic case, efficient frame algorithms are derived using B-spline techniques to analyze the geometry of an image. Moreover, the image can be synthesized from its multiscale local partial derivatives. Also, the relationship between several scale-space approaches is explored. In particular, the evolution of wavelet theory from traditional scale-space filtering can be well understood in terms of B-splines. On the other hand, the behavior of edge models, the properties of completeness, causality, and other properties in such a scale-space representation are examined in the framework of B-splines. It is shown that, besides the good properties inherited from the Gaussian kernel, the B-spline derived scale-space exhibits many advantages for modeling visual mechanism with regard to the efficiency, compactness, orientation feature, and parallel structure.