∈-entropy of piecewise polynomial functions and tree partitioning compression

Most of the signals in nature are piecewise smooth. One of the simple and yet efficient models for representing smooth signals is the class of piecewise polynomials. In this paper compression of this class of functions is considered. Some bounds are derived for the epsiv-entropy of this class of functions. These bounds show us the best performance the optimum compression scheme can have. By comparing it with the performance of traditional binary trees, it is demonstrated that the rate- distortion behavior of binary tree is far from optimum. We will then show that a simple modification of binary trees results in much better performance binary tree algorithms. This modification will retain all the advantages of binary trees.