Total Variation Denoising With Non-Convex Regularizers

Abstract

Total variation (TV) denoising has attracted considerable attention in 1-D and 2-D signal processing. For image denoising, the convex cost function can be viewed as the regularized linear least squares problem ( $\ell _{1}$ regularizer for anisotropic case and $\ell _{2}$ regularizer for isotropic case). However, these convex regularizers often underestimate the high-amplitude components of the true image. In this paper, non-convex regularizers for 2-D TV denoising models are proposed. These regularizers are based on the Moreau envelope and minimax-concave penalty, which can maintain the convexity of the cost functions. Then, efficient algorithms based on forward–backward splitting are proposed to solve the new cost functions. The numerical results show the effectiveness of the proposed non-convex regularizers for both synthetic and real-world image.

Keywords

• Total variation denoising
• l₁ norm%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">l</italic>₁ norm
• l₂ norm%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">l</italic>₂ norm
• non-convex regularizer
• forward-backward splitting