Nonconvex Robust Quaternion Matrix Completion for Imaging Processing

TL;DR

This paper introduces a nonconvex robust quaternion matrix completion model for improved image recovery, utilizing MCP and quaternion Lp norms, with an ADMM algorithm and a nonlocal self-similarity approach, outperforming existing methods.

Contribution

It proposes a novel nonconvex quaternion matrix completion model with MCP and Lp norms, along with an ADMM algorithm and a nonlocal self-similarity method for large-scale data.

Findings

Outperforms existing methods in image and video recovery tasks.

Demonstrates improved low-rank and sparse representation with nonconvex regularization.

Provides convergence analysis for the proposed ADMM algorithm.

Abstract

One of the tasks in color image processing and computer vision is to recover clean data from partial observations corrupted by noise. To this end, robust quaternion matrix completion (QMC) has recently attracted more attention and shown its effectiveness, whose convex relaxation is to minimize the quaternion nuclear norm plus the quaternion $L_1$-norm. However, there is still room to improve due to the convexity of the convex surrogates. This paper proposes a new nonconvex robust QMC model, in which the nonconvex MCP function and the quaternion $L_p$-norm are used to enhance the low-rankness and sparseness of the low-rank term and sparse term, respectively. An alternating direction method of multipliers (ADMM) algorithm is developed to solve the proposed model and its convergence is given. Moreover, a novel nonlocal-self-similarity-based nonconvex robust quaternion completion method is proposed to handle large-scale data. Numerical results on color images and videos indicate the advantages of the proposed method over some existing ones.