Non-convex Total Variation Regularization for Convex Denoising of Signals

• Published in: Journal of mathematical imaging and vision Vol. 62; no. 6-7; pp. 825 - 841
• Main Authors: Selesnick, Ivan, Lanza, Alessandro, Morigi, Serena, Sgallari, Fiorella
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2020; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Computer Science; Convexity; Cost function; Image Processing and Computer Vision; Mathematical Methods in Physics; Noise reduction; Parameters; Random noise; Regularization; Signal,Image and Speech Processing; Forward-backward splitting algorithm; Signal denoising; Total variation regularization; Convex non-convex regularization
• ISSN: 0924-9907; 1573-7683
• DOI: 10.1007/s10851-019-00937-5

Total variation (TV) signal denoising is a popular nonlinear filtering method to estimate piecewise constant signals corrupted by additive white Gaussian noise. Following a ‘convex non-convex’ strategy, recent papers have introduced non-convex regularizers for signal denoising that preserve the convexity of the cost function to be minimized. In this paper, we propose a non-convex TV regularizer, defined using concepts from convex analysis, that unifies, generalizes, and improves upon these regularizers. In particular, we use the generalized Moreau envelope which, unlike the usual Moreau envelope, incorporates a matrix parameter. We describe a novel approach to set the matrix parameter which is essential for realizing the improvement we demonstrate. Additionally, we describe a new set of algorithms for non-convex TV denoising that elucidate the relationship among them and which build upon fast exact algorithms for classical TV denoising.