Premium
Robust quaternion matrix completion with applications to image inpainting
Author(s) -
Jia Zhigang,
Ng Michael K.,
Song GuangJing
Publication year - 2019
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2245
Subject(s) - quaternion , matrix completion , mathematics , inpainting , matrix norm , semidefinite programming , matrix (chemical analysis) , epipolar geometry , norm (philosophy) , algorithm , mathematical optimization , computer vision , image (mathematics) , computer science , physics , geometry , materials science , composite material , quantum mechanics , gaussian , eigenvalues and eigenvectors , political science , law
Summary In this paper, we study robust quaternion matrix completion and provide a rigorous analysis for provable estimation of quaternion matrix from a random subset of their corrupted entries. In order to generalize the results from real matrix completion to quaternion matrix completion, we derive some new formulas to handle noncommutativity of quaternions. We solve a convex optimization problem, which minimizes a nuclear norm of quaternion matrix that is a convex surrogate for the quaternion matrix rank, and the ℓ 1 ‐norm of sparse quaternion matrix entries. We show that, under incoherence conditions, a quaternion matrix can be recovered exactly with overwhelming probability, provided that its rank is sufficiently small and that the corrupted entries are sparsely located. The quaternion framework can be used to represent red, green, and blue channels of color images. The results of missing/noisy color image pixels as a robust quaternion matrix completion problem are given to show that the performance of the proposed approach is better than that of the testing methods, including image inpainting methods, the tensor‐based completion method, and the quaternion completion method using semidefinite programming.