Truncated Low-Rank and Total <i>p</i> Variation Constrained Color Image Completion and its Moreau Approximation Algorithm

Abstract

Recently, low-rank (LR) and total variation (TV) constrained tensor completion algorithms have been broadly studied for image restoration. These algorithms, however, ignore the difference of the intrinsic properties along spatial structure, spectral correlation, and unfolded mode. In this paper, we go further by providing a detailed comparison of the LR and TV properties in matrix and tensor cases, and figure out the LRTV constraints for pixel matrices are more evident and accordant than for others. This inspires us to develop a simple yet effective multichannel LRTV model that is capable of genuinely discovering the intrinsic properties with reduced computational cost. Moreover, due to the suboptimality of nuclear norm and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l_{1}$ </tex-math></inline-formula> norm in approximating the essential low rank and low gradient properties, we employ two enhanced constraints, i.e., truncated nuclear norm (TNN) and total <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> variation <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\text{T}}_{p}\text{V}$ </tex-math></inline-formula> , for a better performance. This results in a challenging problem since that both TNN and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\text{T}}_{p}\text{V}$ </tex-math></inline-formula> are nonsmooth and nonconvex. Observing that the Moreau approximation of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\text{T}}_{p}\text{V}$ </tex-math></inline-formula> constraint is a continuous difference-of-convex function, we then develop a first-order method by repeatedly computing two simple proximal operators. Under mild assumption, we further prove that the sequence generated by our method clusters at a stationary point. Extensive experimental results on color image completion show the efficacy and efficiency of our method over state-of-the-art competitors.