$L_{2,1}$-Norm Regularized Quaternion Matrix Completion Using Sparse Representation and Quaternion QR Decomposition

TL;DR

This paper introduces a novel quaternion matrix completion method using quaternion QR decomposition and $L_{2,1}$-norm regularization, improving computational efficiency and performance in color image completion tasks.

Contribution

It proposes a new quaternion matrix completion algorithm based on quaternion QR decomposition and $L_{2,1}$-norm, with enhancements for reweighted minimization and sparse regularization.

Findings

IRQLNM-QQR outperforms the basic QLNM-QQR method.

QLNM-QQR-SR surpasses several state-of-the-art methods.

The approach reduces computational complexity by avoiding QSVD calculations.

Abstract

Color image completion is a challenging problem in computer vision, but recent research has shown that quaternion representations of color images perform well in many areas. These representations consider the entire color image and effectively utilize coupling information between the three color channels. Consequently, low-rank quaternion matrix completion (LRQMC) algorithms have gained significant attention. We propose a method based on quaternion Qatar Riyal decomposition (QQR) and quaternion $L_{2,1}$-norm called QLNM-QQR. This new approach reduces computational complexity by avoiding the need to calculate the QSVD of large quaternion matrices. We also present two improvements to the QLNM-QQR method: an enhanced version called IRQLNM-QQR that uses iteratively reweighted quaternion $L_{2,1}$-norm minimization and a method called QLNM-QQR-SR that integrates sparse regularization. Our experiments on natural color images and color medical images show that IRQLNM-QQR outperforms QLNM-QQR and that the proposed QLNM-QQR-SR method is superior to several state-of-the-art methods.