Rate-distortion optimal tree algorithms for piecewise polynomials

In this paper, we are interested in a coding scheme, which achieves oracle like asymptotic rate-distortion (R-D) behavior with polynomial complexity for 1-D as well as 2-D signals. In particular, for the 1-D case we present a coding scheme which utilizes binary tree segmentation with optimal bit allocation among different segments. Investigation of the algorithm reveals the inherent weakness in the initial coding scheme, leading to a suboptimal performance. The optimal binary tree scheme in the 1-D case can be easily extended to the 2-D case as an optimal quadtree scheme with the similar computational complexity. The proposed optimal quadtree scheme also achieves the oracle like R-D performance for some simple classes of images whereas for the 2-D case there is no known algorithm achieving the right R-D behavior at reasonable computational cost.