Low Rank Quaternion Matrix Recovery via Logarithmic Approximation

TL;DR

This paper introduces a novel quaternion matrix logarithmic norm for color image completion, effectively preserving image structure and improving rank approximation, leading to superior image restoration results.

Contribution

It proposes a new quaternion matrix logarithmic norm for rank approximation, combining truncated and factorization strategies within an alternating minimization framework for color image completion.

Findings

Outperforms traditional methods in color image completion tasks.

Preserves image structure better by using quaternion representation.

Achieves more accurate rank approximation with logarithmic surrogate.

Abstract

In color image processing, image completion aims to restore missing entries from the incomplete observation image. Recently, great progress has been made in achieving completion by approximately solving the rank minimization problem. In this paper, we utilize a novel quaternion matrix logarithmic norm to approximate rank under the quaternion matrix framework. From one side, unlike the traditional matrix completion method that handles RGB channels separately, the quaternion-based method is able to avoid destroying the structure of images via putting the color image in a pure quaternion matrix. From the other side, the logarithmic norm induces a more accurate rank surrogate. Based on the logarithmic norm, we take advantage of not only truncated technique but also factorization strategy to achieve image restoration. Both strategies are optimized based on the alternating minimization framework. The experimental results demonstrate that the use of logarithmic surrogates in the quaternion domain is more superior in solving the problem of color images completion.