Low Rank Pure Quaternion Approximation for Pure Quaternion Matrices

TL;DR

This paper introduces an optimal low-rank pure quaternion matrix approximation method for color images, using projections on quaternion matrix manifolds to improve image representation.

Contribution

It develops an alternating projections algorithm to find the best low-rank pure quaternion approximation, addressing limitations of previous methods.

Findings

The algorithm effectively finds optimal low-rank pure quaternion approximations.

Numerical experiments demonstrate improved image representation quality.

The convergence of the method is theoretically established.

Abstract

Quaternion matrices are employed successfully in many color image processing applications. In particular, a pure quaternion matrix can be used to represent red, green and blue channels of color images. A low-rank approximation for a pure quaternion matrix can be obtained by using the quaternion singular value decomposition. However, this approximation is not optimal in the sense that the resulting low-rank approximation matrix may not be pure quaternion, i.e., the low-rank matrix contains real component which is not useful for the representation of a color image. The main contribution of this paper is to find an optimal rank-$r$ pure quaternion matrix approximation for a pure quaternion matrix (a color image). Our idea is to use a projection on a low-rank quaternion matrix manifold and a projection on a quaternion matrix with zero real component, and develop an alternating projections algorithm to find such optimal low-rank pure quaternion matrix approximation. The convergence of the projection algorithm can be established by showing that the low-rank quaternion matrix manifold and the zero real component quaternion matrix manifold has a non-trivial intersection point. Numerical examples on synthetic pure quaternion matrices and color images are presented to illustrate the projection algorithm can find optimal low-rank pure quaternion approximation for pure quaternion matrices or color images.