Analysis of bounded variation penalty methods for ill-posed problems

Abstract

This paper presents an abstract analysis of bounded variation (BV) methods for ill-posed operator equations Au = z. Let T(u) def = kAu ? zk 2 + J(u); where the penalty, or \regularization", parameter > 0 and the functional J(u) is the BV norm or seminorm of u, also known as the total variation of u. Under mild restrictions on the operator A and the functional J(u), it is shown that the functional T(u) has a unique minimizer which is stable with respect to certain perturbations in the data z, the operator A, the parameter , and the functional J(u). In addition, convergence results are obtained which apply when these perturbations vanish and the regularization parameter is chosen appropriately.

Extended HALS algorithm for nonnegative Tucker decomposition and its applications for multiway analysis and classification

“…With the first order difference operator L with J n points, the cost function (12) becomes similar to the total-variation (TV) regularization [45,46], and leads to an update rule robust to noise but rather high computational cost. To considerably reduce the complexity of the algorithm we present a second-order smoothing operator L in the following form:…”

Tomographic retrieval of cloud liquid water fields from a single scanning microwave radiometer aboard a moving platform – Part 2: Observation system simulation experiments

“…Non-linear regularization techniques such as the L 1 norm TV regularization are also becoming more and more popular. The main advantage of the TV regularization is that it does not penalize discontinuities in the solution, while simultaneously not penalizing smoothness in the solution; thus under certain conditions it can preserve the discontinuous structures in the solution (Acar and Vogel, 1994;Chambolle and Lions, 1997). Let us use a simple example presented in Pederson (2005) to illustrate the point.…”

Split Bregman iteration for hybrid regularization based image restoration

“…For this reason, the classical ROF method has been studied by scholars at home and abroad, and emerged some fast numerical algorithms, such as fixed-point iteration method [3][4][5][6][7], dual method [8][9], augmented Lagrangian method [10][11][12][13], Bregman iteration [14][15][16], linearized Bregman iteration [17][18][19][20][21], and split Bregman iteration [16,[22][23][24][25], alternating minimization and alternating direction algorithm [26,27], etc.…”