On the compression of two-dimensional piecewise smooth functions

It is well known that wavelets provide good non-linear approximation of one-dimensional (1-D) piecewise smooth functions. However, it has been shown that the use of a basis with good approximation properties does not necessarily lead to a good compression algorithm. The situation in 2-D is much more complicated since wavelets are not good for modeling piecewise smooth signals (where discontinuities are along smooth curves). The purpose of this work is to analyze the performance of compression algorithms for 2-D piecewise smooth functions directly in a rate distortion context. We consider some simple image models and compute rate distortion bounds achievable using oracle based methods. We then present a practical compression algorithm based on optimal quadtree decomposition that, in some cases, achieve the oracle performance.