Robust quaternion matrix completion with applications to image inpainting

Zhigang Jia, Kwok Po NG*, Guang Jing Song

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

76 Citations (Scopus)


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 ℓ1-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.

Original languageEnglish
Article numbere2245
JournalNumerical Linear Algebra with Applications
Issue number4
Publication statusPublished - Aug 2019

Scopus Subject Areas

  • Algebra and Number Theory
  • Applied Mathematics

User-Defined Keywords

  • color images
  • convex optimization
  • low rank
  • matrix recovery
  • quaternion


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