Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage

This paper examines the relationship between wavelet-based image processing algorithms and variational problems. Algorithms are derived as exact 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, minimize over all g in the Besov space B-1(1) (L-1(I)) the functional parallel 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 compression in L-2(I) to derive accurate error bounds for noise removal through wavelet shrinkage applied to images corrupted with i.i.d., mean zero, Gaussian noise. A new signal-to-noise ratio (SNR), which we claim more accurately reflects the visual perception of noise in images, arises in this derivation, We present extensive computations that support the 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 shrinkage parameters yields uniformly better results than Donoho and Johnstone's VisuShrink procedure; an example suggests, however, that Donoho and Johnstone's SureShrink method, which uses a different shrinkage parameter for each dyadic level, achieves lower error than our procedure.