# Color image recovery using low-rank quaternion matrix completion algorithm

arXiv: https://arxiv.org/abs/1909.06567 · HTML: https://mlsanity.com/paper/1909.06567/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Sep 2019.

## Built on by (most cited first, 14 of 14)

- [Color Image Inpainting via Robust Pure Quaternion Matrix Completion: Error Bound and Weighted Loss](https://mlsanity.com/paper/2202.02063/) (2022)
- [Quaternion higher-order singular value decomposition and its applications in color image processing](https://mlsanity.com/paper/2101.00364/) (2021)
- [Quaternion-based dynamic mode decomposition for background modeling in color videos](https://mlsanity.com/paper/2112.13982/) (2021)
- [On the Partial Convexification for Low-Rank Spectral Optimization: Rank Bounds and Algorithms](https://mlsanity.com/paper/2305.07638/) (2023)
- [Approximation of Images via Generalized Higher Order Singular Value Decomposition over Finite-dimensional Commutative Semisimple Algebra](https://mlsanity.com/paper/2202.00450/) (2022)
- [Quaternion Generative Adversarial Neural Networks and Applications to Color Image Inpainting](https://mlsanity.com/paper/2406.11567/) (2024)
- [Non-Negative Reduced Biquaternion Matrix Factorization with Applications in Color Face Recognition](https://mlsanity.com/paper/2408.05582/) (2024)
- [Quaternion Nuclear Norms Over Frobenius Norms Minimization for Robust Matrix Completion](https://mlsanity.com/paper/2504.21468/) (2025)
- [Quaternion tensor left ring decomposition and application for color image inpainting](https://mlsanity.com/paper/2307.10620/) (2023)
- [A Theoretically Guaranteed Quaternion Weighted Schatten p-norm Minimization Method for Color Image Restoration](https://mlsanity.com/paper/2307.12656/) (2023)
- [L2,1-Norm Regularized Quaternion Matrix Completion Using Sparse Representation and Quaternion QR Decomposition](https://mlsanity.com/paper/2309.03764/) (2023)
- [Quaternionic Reweighted Amplitude Flow for Phase Retrieval in Image Reconstruction](https://mlsanity.com/paper/2501.02180/) (2025)
- [Depth-Aided Color Image Inpainting in Quaternion Domain](https://mlsanity.com/paper/2503.16818/) (2025)
- [RGNMR: A Gauss-Newton method for robust matrix completion with theoretical guarantees](https://mlsanity.com/paper/2505.12919/) (2025)

## Abstract (arXiv)

As a new color image representation tool, quaternion has achieved excellent results in color image processing problems. In this paper, we propose a novel low-rank quaternion matrix completion algorithm to recover missing data of color image. Motivated by two kinds of low-rank approximation approaches (low-rank decomposition and nuclear norm minimization) in traditional matrix-based methods, we combine the two approaches in our quaternion matrix-based model. Furthermore, the nuclear norm of the quaternion matrix is replaced by the sum of Frobenius norm of its two low-rank factor quaternion matrices. Based on the relationship between quaternion matrix and its equivalent complex matrix, the problem eventually is converted from quaternion number field to complex number field. An alternating minimization method is applied to solve the model. Simulation results on real world color image recovery show the superior performance and efficiency of the proposed algorithm over some state-of-the-art tensor-based ones.