Upper bounds of wavelets on 2-D lipschitz class and zerotrees

The wavelet transform method has become one of the most powerful tools in image compression applications. It has a number of advantages with respect to other spectral transforms. With lossy compression, after a suitable wavelet transform is chosen two basic procedures — zonal and threshold coding — are usually applied to the spectral image [1]. In zonal coding only a fixed small "zone" of transformed image is encoded and the optimal zonal coding method ensures the best selection of spectral zones at which a minimum mean-square error of reconstruction is achieved. In order to determine optimal zonal coding method for the chosen transform one has to obtain the estimates of its spectra on a given class of input images. We suggest in this paper a unified approach to obtain exact upper bounds of a given wavelet transform on the class of discrete images with bounded first order finite differences. The presented estimates allow to apriori identify the spectral "zones" that are likely to be of significance in image reconstruction for a given wavelet transform.