Optimal wavelet filter banks for regularized restoration of noisy images

Regularized image restoration methods efficiently handle the ill-posed problem of image restoration. Nevertheless, the issue of selecting the regularization parameter as well as the smoothing filter still constitutes an open research topic. A model of regularized image restoration is introduced and analyzed in this paper. The proposed model assumes that wavelet filter banks replace the smoothing filter of conventional regularized restoration. Filter factorizations for the optimal design of wavelet filter banks using the generalized-cross-validation (GCV) criterion are presented, and novel expressions of the influence matrix, which is used to calculate the GCV error, are derived. The error of the GCV method is expressed in terms of the modulation matrix of the filter bank and the modulation vector of the degradation filter. The expressions are given in general form for optimal wavelet filter bank design upon arbitrary sampling lattices. The numerical examples of image restoration using the proposed method that are presented indicate significant signal-to-noise ratio improvement, ΔSNR, compared to image restoration methods that employ the Laplacian as the smoothing filter.