Color Image Inpainting via Robust Pure Quaternion Matrix Completion: Error Bound and Weighted Loss

TL;DR

This paper models color image inpainting as a quaternion matrix completion problem, establishing theoretical error bounds and proposing a weighted loss to handle noise, with extensive experiments validating the approach.

Contribution

It introduces a novel quaternion matrix completion framework with theoretical error bounds and a weighted loss for robust color image inpainting.

Findings

Error bounds established for quaternion matrix completion in noisy settings

Weighted loss effectively handles unbalanced and correlated noise

Experimental results confirm theoretical predictions and robustness

Abstract

In this paper, we study color image inpainting as a pure quaternion matrix completion problem. In the literature, the theoretical guarantee for quaternion matrix completion is not well-established. Our main aim is to propose a new minimization problem with an objective combining nuclear norm and a quadratic loss weighted among three channels. To fill the theoretical vacancy, we obtain the error bound in both clean and corrupted regimes, which relies on some new results of quaternion matrices. A general Gaussian noise is considered in robust completion where all observations are corrupted. Motivated by the error bound, we propose to handle unbalanced or correlated noise via a cross-channel weight in the quadratic loss, with the main purpose of rebalancing noise level, or removing noise correlation. Extensive experimental results on synthetic and color image data are presented to confirm and demonstrate our theoretical findings.