## 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 a 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 nontrivial 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.

Original language | English |
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Pages (from-to) | 58-82 |

Number of pages | 25 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 42 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2021 |

## Scopus Subject Areas

- Analysis

## User-Defined Keywords

- Color images
- Low-rank approximation
- Manifolds
- Pure quaternion matrices