Survey of image denoising techniques <s> a. Deterministic <s> The wavelet multiscale edge representation of signals developed by S.G. Mallat and S. Zhong (1991) provides a new tool for signal and image processing. A tree structure was built from wavelet-transform (WT) maxima, and metrics for analyzing the WT maxima tree were developed. These metrics fall into two classes corresponding to two perceptual criteria (scaling and spatial stabilities) for discriminating features from background noise. Identified noisy branches are trimmed off the WT maxima tree. This technique of noise reduction preserves edges well while suppressing noise. The results of processing simulated and medical images with this technique are presented. > <s> BIB001 </s> Survey of image denoising techniques <s> a. Deterministic <s> We study what we call "dyadic CART"-a method of nonparametric regression which constructs a recursive partition by optimizing a complexity penalized sum of squares, where the optimization is over all recursive partitions arising from midpoint splits. We show that the method is adaptive to unknown degrees of anisotropic smoothness. Specifically, consider the anisotropic smoothness classes of Nikol'skii, consisting of bivariate functions f(xl, x2) whose finite difference of distance h in direction i is bounded in LP norm by Ch8i, i = 1, 2. We show that dyadic CART, with an appropriate complexity penalty parameter A - u2 Const log(n), is within logarithmic terms of minimax over every anisotropic smoothness class 0 < C < x, 0 < 81, 82 < 1. The proof shows that dyadic CART is identical to a certain adaptive best-ortho-basis algorithm based on the library of all anisotropic Haar bases. Then it applies empirical basis selection ideas of Donoho and Johnstone. The basis empirically selected by dyadic CART is shown to be nearly as good as a basis ideally adapted to the underlying f. The risk of estimation in an ideally adapted anisotropic Haar basis is shown to be comparable to the minimax risk over anisotropic smoothness classes. Underlying the success of this argument is harmonic analysis of anisotropic smoothness classes. We show that, for each anisotropic smoothness class, there is an anisotropic Haar basis which is a best orthogonal basis for representing that smoothness class; the basis is optimal not just within the library of anisotropic Haar bases, but among all orthogonal bases of L2[0, 1]2. <s> BIB002 </s> Survey of image denoising techniques <s> a. Deterministic <s> The more a priori knowledge we encode into a signal processing algorithm, the better performance we can expect. In this paper, we overview several approaches to capturing the structure of singularities (edges, ridges, etc.) in wavelet-based signal processing schemes. Leveraging results from approximation theory, we discuss nonlinear approximations on trees and point out that an optimal tree approximant exists and is easily computed. The optimal tree approximation inspires a new hierarchical interpretation of the wavelet decomposition and a tree-based wavelet denoising algorithm that suppresses spurious noise bumps. <s> BIB003
The Deterministic method of modeling involves creating tree structure of wavelet coefficients with every level in the tree representing each scale of transformation and nodes representing the wavelet coefficients. This approach is adopted in BIB003 . The optimal tree approximation displays a hierarchical interpretation of wavelet decomposition. Wavelet coefficients of singularities have large wavelet coefficients that persist along the branches of tree. Thus if a wavelet coefficient has strong presence at particular node then in case of it being signal, its presence should be more pronounced at its parent nodes. If it is noisy coefficient, for instance spurious blip, then such consistent presence will be missing. Lu et al. BIB001 , tracked wavelet local maxima in scalespace, by using a tree structure. Other denoising method based on wavelet coefficient trees is proposed by Donoho BIB002 .