Frequency and spatially adaptive wavelet packets

We consider a method for image compression based on frequency and spatially adaptive wavelet packets. We present a new fast directed acyclic graph (DAG) structured decomposition, with both spatial segmentation and orthogonal frequency branching from each node. Whereas traditional wavelet packet decomposition adapts to a global frequency distribution, this technique finds the best joint spatial segmentation and local frequency basis. The algorithm is derived from the fast double tree algorithm proposed by Herley, et. al. (see IEE Transactions on Signal Processing, December 1993), for 1-D signals, with an extension to 2-D and modification to include spatial segmentation of frequency nodes. By collecting redundant nodes in this full adaptive tree, we have derived a directed acyclic graph (DAG) structure which contains the same number of nodes as the double tree, but includes new connections between nodes. We present the adaptive wavelet packet DAG algorithm and examine image compression performance on test images.