  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.
