Wavelet shrinkage: unification of basic thresholding functions and thresholds

Abstract

This work addresses the unification of some basic functions and thresholds used in non-parametric estimation of signals by shrinkage in the wavelet domain. The soft and hard thresholding functions are presented as degenerate smooth sigmoid-based shrinkage functions. The shrinkage achieved by this new family of sigmoid-based functions is then shown to be equivalent to a regularization of wavelet coefficients associated with a class of penalty functions. Some sigmoid-based penalty functions are calculated, and their properties are discussed. The unification also concerns the universal and the minimax thresholds used to calibrate standard soft and hard thresholding functions: these thresholds pertain to a wide class of thresholds, called the detection thresholds. These thresholds depend on two parameters describing the sparsity degree for the wavelet representation of a signal. It is also shown that the non-degenerate sigmoid shrinkage adjusted with the new detection thresholds is as performant as the best up-to-date parametric and computationally expensive method. This justifies the relevance of sigmoid shrinkage for noise reduction in large databases or large size images.

Sparse Signal Estimation by Maximally Sparse Convex Optimization

“…Next, sparse deconvolution is performed using three algorithms developed to solve the highly non-convex 0 quasinorm problem, namely the Iterative Support Detection (ISD) algorithm [68], 2 the Accelerated Iterative Hard Thresholding (AIHT) algorithm [7], 3 and the Single Best Replacement (SBR) algorithm [63]. In each case, we used software by the respective authors.…”

“…Each has its disadvantages, and many other thresholding functions that provide a compromise of the soft and hard thresholding functions have been proposed -for example: the firm threshold [32], the non-negative (nn) garrote [26], [31], the SCAD threshold function [24], [73], and the proximity operator of the p quasinorm (0 < p < 1) [44]. Several penalty functions are unified by the two-parameter formulas given in [3], [35], wherein threshold functions are derived as proximity operators [19]. (Table 1.2 of [19] lists the proximity operators of numerous functions.)…”

High order structural image decomposition by using non-linear and non-convex regularizing objectives

“…The PLS problem inherits the convexity [4][5][6][7][8][9][10][11][12][13] or nonconvexity of Q λ [14][15][16][17][18][19][20][21] (the quadratic term involved in this problem is a convex function). Both convex and non-convex forms have shown relevancy in many image processing applications involving • denoising and deconvolution [2,4,5,8,10,14,15,[22][23][24]; • cartoon/texture decomposition and image inpainting [6,7,10,25]; • inverse problems and compressive sampling reconstruction [26][27][28]; etc.…”

“…Note that a PLS problem can be expressed in the image input space [5,8], the Fourier domain [7], [10], the wavelet domain [4,11,13] or via Meyer's oscillatory functions [6,12].…”

Noise reduction using wavelet cycle spinning: analysis of useful periodicities in the z-transform domain

“…Many books and papers have been written on the subject, ranging from theoretical aspects [1][2][3][4] to their applications to different research fields, such as imaging [5][6][7][8][9][10] and acoustics [11][12][13][14], among others.…”