Optimal decomposition for quad-trees with leaf dependencies

We propose a fast and e cient algorithm which nds the optimal quad-tree (QT) decomposition with leaf dependencies in the rate distortion sense. The underlying problem is the encoding of an image by a variable block size scheme, where the block size is encoded using a QT, each block is encoded by one of the admissible quantizers and the quantizers are transmitted using a rst order di erential pulse code modulation (DPCM) scheme along the scanning path. First we de ne an optimal scanning path for a QT such that successive blocks are always neighboring blocks. Then we propose a procedure which infers such an optimal path from the QTdecomposition and introduce a special optimal path which is based on a Hilbert curve. Then we consider the case where the image is losslessly encoded using a QT structure and propose a dynamic programming (DP) based multi-level approach to nd the optimal QT-decomposition and the optimal quantizer selection. We then apply the Lagrangian multiplier method to solve the lossy case, and show that the unconstrained problem of the Lagrangian multiplier method can be solved using the algorithm introduced for the lossless case. Finally we present a mean value QT-decomposition example, where the mean values are DPCM encoded.