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

Antonin Chambolle, Ronald A. DeVore, Nam-Yong Lee, Bradley J. Lucier

Abstract: 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 |F-g| L2 (I) 2 +λ|g|(B 1 1 (L 1(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 α for which F∈B q α (L q (I)),1/q=α/2+1/2, and the norm |F|B q α (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 (1994, 1995, 1996) SureShrink method, which uses a different shrinkage parameter for each dyadic level, achieves a lower error than our procedure.