Color Image Recovery Using Generalized Matrix Completion over HigherOrder Finite Dimensional Algebra
Abstract
:1. Introduction
1.1. Background and Related Works
1.2. Contributions and Organization of This Paper
 This research presents a tmatrix model that can extend traditional matrix methods to a higher order. The higherorder algorithm, termed “HigherOrder TNN”, is designed to exploit intricate structures in highdimensional data and refines classical lowerorder algorithms for missing entry recovery of RGB images. Compared to its predecessors, HigherOrder TNN offers significantly improved recovery performance.
 Using the tmatrix model over a finitedimensional algebra, several image analysis algorithms are extended to a higher order using a novel pixel neighborhood strategy.
 Many constructions in matrix and vector analysis are extended to the tmatrix model. Examples include rank, norm, and inner product. In addition, tmatrix versions of Lagrange multipliers are defined.
2. TMatrices
2.1. TScalars
2.2. TScalars as FiniteDimensional Linear Operators
2.3. TMatrices
2.4. Singular Value Decomposition of a TMatrix
Algorithm 1 Tensorial Singular Value Decomposition via Spectral Slices 

Algorithm 2 Tensorial Singular Value Thresholding via Spectral Slices 

3. LowRank Matrix Completion and Its Generalizations
3.1. LowRank Matrix Completion
3.2. Generalization of Matrix Completion over HigherOrder TScalars
Algorithm 3 ADMM for solving Equation (7) 

Algorithm 4 HigherOrder TNN: ADMM for recovering an image with missing values 

3.3. Computational Complexity
4. Rank Considerations
4.1. Tubal Rank and Average Rank
4.2. HigherOrder Rank and Its Trace Variant
5. Experiments
5.1. Experiments on Simulated Random Data
5.2. Experiments on BSD Color Images
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. A Mathematical Justification
Appendix A.1. Matrix Representation for TScalars and HigherOrder Measures
Appendix A.2. A Representation Model for TMatrices and HigherOrder Measures
Appendix A.3. Lagrange Multiplier with TMatrix Variables
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Liao, L.; Guo, Z.; Gao, Q.; Wang, Y.; Yu, F.; Zhao, Q.; Maybank, S.J.; Liu, Z.; Li, C.; Li, L. Color Image Recovery Using Generalized Matrix Completion over HigherOrder Finite Dimensional Algebra. Axioms 2023, 12, 954. https://doi.org/10.3390/axioms12100954
Liao L, Guo Z, Gao Q, Wang Y, Yu F, Zhao Q, Maybank SJ, Liu Z, Li C, Li L. Color Image Recovery Using Generalized Matrix Completion over HigherOrder Finite Dimensional Algebra. Axioms. 2023; 12(10):954. https://doi.org/10.3390/axioms12100954
Chicago/Turabian StyleLiao, Liang, Zhuang Guo, Qi Gao, Yan Wang, Fajun Yu, Qifeng Zhao, Stephen John Maybank, Zhoufeng Liu, Chunlei Li, and Lun Li. 2023. "Color Image Recovery Using Generalized Matrix Completion over HigherOrder Finite Dimensional Algebra" Axioms 12, no. 10: 954. https://doi.org/10.3390/axioms12100954