Two Algorithms for Non-separable Wavelet Transforms and Applications to Image Compression

Previously, the use of non-separable wavelets in image processing has been hindered by the lack of a fast algorithm to perform a non-separable wavelet transform. We present two such algorithms in this paper. The rst algorithm implements a periodic wavelet transform for any valid wavelet lter sequence and dilation matrices satisfying a trace condition. We discuss some of the complicating issues unique to the non-separable case and how to overcome them. The second algorithm links Haar wavelets and complex bases and uses this link to drive the algorithm. For the complex bases case, the asymmetry of the wavelet trees produced leads to a discussion of the complexities in implementing zero-tree and other wavelet compression methods. We describe some preliminary attempts at using this algorithm, with non-separable Haar wavelets, for reducing the blocking artifacts in fractal image compression.