Two-channel nonseparable wavelets statistically matched to 2-D images

This paper presents a new approach for the estimation of 2-channel nonseparable wavelets matched to images in the statistical sense. To estimate a matched wavelet system, first, we estimate the analysis wavelet filter of a 2-channel nonseparable filterbank using the minimum mean square error (MMSE) criterion. The MMSE criterion requires statistical characterization of the given image. Because wavelet basis expansion behaves as Karhunen-Loève type expansion for fractional Brownian processes, we assume that the given image belongs to a 1st order or a 2nd order isotropic fractional Brownian field (IFBF). Next, we present a method for the design of a 2-channel two-dimensional finite-impulse response (FIR) biorthogonal perfect reconstruction filterbank (PRFB) leading to the estimation of a compactly supported statistically matched wavelet. The important contribution of the paper lies in the fact that all filters are estimated from the given image itself. Several design examples are presented using the proposed theory. Because matched wavelets will have better energy compaction, performance of estimated wavelets is evaluated by computing the transform coding gain. It is seen that nonseparable matched wavelets give better coding gain as compared to nonseparable non-matched orthogonal and biorthogonal wavelets.