Convex 1-D Total Variation Denoising with Non-convex Regularization

Abstract

Total variation (TV) denoising is an effective noise suppression method when the derivative of the underlying signal is known to be sparse. TV denoising is defined in terms of a convex optimization problem involving a quadratic data fidelity term and a convex regularization term. A non-convex regularizer can promote sparsity more strongly, but generally leads to a non-convex optimization problem with non-optimal local minima. This letter proposes the use of a non-convex regularizer constrained so that the total objective function to be minimized maintains its convexity. Conditions for a non-convex regularizer are given that ensure the total TV denoising objective function is convex. An efficient algorithm is given for the resulting problem.

Total Variation Denoising Via the Moreau Envelope

“…Figure 2 shows that the proposed penalty yields the lowest average RMSE for all σ 0.4. However, at low noise levels, separable convexitypreserving penalties [48] perform better than the proposed non-separable convexity-preserving penalty.…”

“…This denoised signal consistently underestimates the amplitudes of jump discontinuities, especially those occurring near other jump discontinuities of opposite sign. Figure 1(c) shows the result using a separable non-convex penalty [48]. This method can use any nonconvex scalar penalty satisfying a prescribed set of properties.…”

Total Variation Regularization Term-Based Low-Rank and Sparse Matrix Representation Model for Infrared Moving Target Tracking

“…Noise is easy to be separated into S for the reason that the rank of target matrix T is limited and the noise which may distribute in any part of the image also possesses sparse property. In order to distinguish the noise from S, we introduce a total variation model that is used in image denoising [41] and object detection [42] to impose a more accurate regularization on S:…”

“…where, G(S) is a differentiable and convex function with Lipschitz continuous gradient and H(x) is a non-smooth but convex function [41].…”

Reconstruction of noisy signals by minimization of non-convex functionals

“…Besides the trend-setting works of Geman and McClure [4] and Chipot et al [6], the practical effectiveness of non-convex functionals have also been shown in other studies. In signal/image denoising, the researches of Aubert et al [1], Nikolova [17] and Selesnick et al [18] illustrate by concrete examples the better performance of some non-convex functionals as compared to convex ones. Vese [19] shows the ability of several non-convex potentials to reconstruct signals, being particularly accurate at restoring edges and linear regions.…”