NONLINEAR WAVELET IMAGE-PROCESSING - VARIATIONAL-PROBLEMS, COMPRESSION, AND NOISE REMOVAL THROUGH WAVELET SHRINKAGE

This paper examines the relationship between wavelet-based image proce ssing algorithms and variational problems. Algorithms are derived as e xact or approximate minimizers of variational problems; in particular, we show that wavelet shrinkage can be considered the exact minimizer of the following problem: Given an image F defined on a square I, mini mize over all g in the Besov space B-1(1) (L-1(I)) the functional para llel to F-g parallel to(L2(I))(2) + lambda parallel to g parallel to B -1(1) ((L1(I))). We use the theory of nonlinear wavelet image compress ion in L-2(I) to derive accurate error bounds for noise removal throug h wavelet shrinkage applied to images corrupted with i.i.d., mean zero , Gaussian noise. A new signal-to-noise ratio (SNR), which we claim mo re accurately reflects the visual perception of noise in images, arise s in this derivation, We present extensive computations that support t he hypothesis that near-optimal shrinkage parameters can be derived if one knows (or can estimate) only two parameters about an image F: the largest alpha for which F epsilon B-q(alpha) (L-q(I)), 1/q = alpha/2 + 1/2, and the norm parallel to F parallel to B-q(alpha)(L-q(I)). Both theoretical and experimental results indicate that our choice of shri nkage parameters yields uniformly better results than Donoho and Johns tone's VisuShrink procedure; an example suggests, however, that Donoho and Johnstone's SureShrink method, which uses a different shrinkage p arameter for each dyadic level, achieves lower error than our procedur e.