Tree Partitioning Compression of Piecewise Polynomial Functions

In this paper compression of piecewise polynomial functions is considered. The importance of this class of functions is due to the fact that most of the signals observed in the nature are piecewise smooth and piecewise polynomials provide an efficient model for representing piecewise smooth signals. We compute the e-entropy of the class of piecewise polynomials and show that the rate-distortion behavior of the traditional compression algorithms is far from optimum. Then we will discuss the binary partitioning algorithm, and will demonstrate that with a simple modification of this algorithm, it is possible to get close to the optimal performance. Moreover, this modification will retain all the advantages of the binary tree algorithms such as simplicity, automatic parameter selection and extension to higher dimensions.