Constrained low-rank approximation of quaternion matrices and beyond

Pure quaternion matrices have been widely used in color image processing. However, existing methods often overlook a fundamental fact: the pixels of an image in $b$-bit format can only take integer values from the set $\{0,\dots ,2^b-1\}$. In this paper, we consider this important constraint and propose a constrained model that simultaneously incorporates the pure, integer and box constraints for low-rank quaternion matrix approximation. Our model can precisely obtain the optimal fixed-rank approximation while preserving these essential properties of quaternion matrices. Furthermore, we introduce a universal framework for constrained low-rank quaternion matrix completion tailored to color image inpainting, supported by rigorous theoretical convergence analysis. Experimental results demonstrate the superiority of our algorithms over state-of-the-art methods.