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LOW RANK PURE QUATERNION APPROXIMATION FOR PURE QUATERNION MATRICES.
- Source :
-
SIAM Journal on Matrix Analysis & Applications . 2021, Vol. 42 Issue 1, p58-82. 25p. - Publication Year :
- 2021
-
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 pro jection on a low-rank quaternion matrix manifold and a projection on a quaternion matrix with zero real component, and develop an alternating pro jections algorithm to find such optimal low-rank pure quaternion matrix approximation. The convergence of the pro jection 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 pro jection algorithm can find optimal low-rank pure quaternion approximation for pure quaternion matrices or color images. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 08954798
- Volume :
- 42
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- SIAM Journal on Matrix Analysis & Applications
- Publication Type :
- Academic Journal
- Accession number :
- 149750774
- Full Text :
- https://doi.org/10.1137/19M1307329