2020/12/30 by Guangjing Song, Song, Guangjing, Weiyang Ding +3 · 1 citation
Computer Science · Mathematics · #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #cs.NA #math.NA #math.OC
paper · pdf · doi:10.48550/arxiv.2012.15138
arxiv created 2020/12/30 · arxiv updated 2021/01/01
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.