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Color image and video restoration using tensor CP decomposition

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Abstract

This paper proposes a new approach to image and video restoration. This approach constructs a degradation model based on a tensor representation, where a color image is represented by a third-order tensor, and a video composed of color images is a fourth-order tensor. Applying tensor CP decomposition to our original problem leads to three subproblems. To solve those subproblems, we apply global LSQR algorithm, and a new algorithm based on Golub Kahan bidiagonalization. Some numerical tests are presented to show the effectiveness of the proposed methods.

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References

  1. Bentbib, A.H., ElGuide, M., Jbilou, K., Onunwor, E., Reichel, L.: Solution methods for linear discrete ill-posed problems for color image restoration. BIT Numer. Math. 58, 555–576 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bentbib, A.H., Khouia, A., Sadok, H.: The LSQR method for solving tensor least-squares problems. Electron. Trans. Numer. Anal. 55, 92–111 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bouhamidi, A., Jbilou, K.: Sylvester Tikhonov-regularization methods in image restoration. J. Comput. Appl. Math. 206, 86–98 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Calvetti, D., Golub, G.H., Reichel, L.: Estimation of the L-curve via Lanczos bidiagonalization. BIT Numer. Math. 39, 603–619 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chung,J., Gazzola,S.: Computational methods for large-scale inverse problems: a survey on hybrid projection methods. arXiv preprint arXiv:2105.07221 (2021)

  6. Chung, J., Nagy, J.G., O’leary, D.P. et al.: A weighted GCV method for Lanczos hybrid regularization. Electron. Trans. Numer. Anal. 28, 149–167 (2008)

  7. Cichocki, A., Zdunek, R., Phan, A.H., Amari, S.: Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-Way Data Analysis and Blind Source Separation. Wiley, New York (2009)

    Book  Google Scholar 

  8. Comon, P., Luciani, X., DeAlmeida, A.L.: Tensor decompositions, alternating least squares and other tales. J. Chemom. A J. Chemometr. Soc. 23, 393–405 (2009)

    Google Scholar 

  9. ElGuide, M., Ichi, A., Jbilou, K., Sadaka, R.: On tensor GMRES and Golub–Kahan methods via the T-product for color image processing. Electron. J. Linear Algebra 37, 524–543 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  10. Engl, H.W.: Regularization Methods for The Stable Solution of Inverse Problems. Univ., Institut für Mathematik (1992)

  11. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems, vol. 375. Springer, New York (1996)

    Book  MATH  Google Scholar 

  12. Fenu, C., Reichel, L., Rodriguez, G., Sadok, H.: GCV for Tikhonov regularization by partial SVD. BIT Numer. Math. 57, 1019–1039 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gazzola, S., Novati, P., Russo, M.R.: Embedded techniques for choosing the parameter in Tikhonov regularization. Numer. Linear Algebra Appl. 21, 796–812 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Golub, G.H., Von Matt, U.: Tikhonov Regularization for Large Scale Problems. Citeseer (1997)

  15. Golub, G.H., Heath, M., Wahba, G.: Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21, 215–223 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hanke, M., Hansen, P.C.: Regularization methods for large-scale problems. Surv. Math. Ind. 3, 253–315 (1993)

    MathSciNet  MATH  Google Scholar 

  17. Hansen, P.C., Nagy, J.G., O’leary, D.P.: Deblurring Images: Matrices, Spectra, and Filtering. SIAM, Philadelphia (2006)

  18. Harshman, R.A. et al.: Foundations of the PARAFAC procedure: models and conditions for an explanatory multimodal factor analysis. In: UCLA Working Papers in Phonetics, vol. 16, pp. 1–84 (1970)

  19. Jbilou, K., Messaoudi, A., Sadok, H.: Global FOM and GMRES algorithms for matrix equations. Appl. Numer. Math. 31, 49–63 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kamm, J., Nagy, J.G.: Kronecker product and SVD approximations in image restoration. Linear Algebra Appl. 284, 177–192 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kiers, H.A.: Towards a standardized notation and terminology in multiway analysis. J. Chemom. A J. Chemom. Soc. 14, 105–122 (2000)

    Google Scholar 

  22. Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51, 455–500 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kruskal, J.B.: Rank, decomposition, and uniqueness for 3-way and n-way arrays. In: Multiway Data Analysis, pp. 7–18. Elsevier (1989)

  24. Liu, S., Trenkler, G., et al.: Hadamard, Khatri-Rao, Kronecker and other matrix products. Int. J. Inf. Syst. Sci. 4, 160–177 (2008)

    MathSciNet  MATH  Google Scholar 

  25. Nagy, J.G., Kilmer, M.E.: Kronecker product approximation for preconditioning in three-dimensional imaging applications. IEEE Trans. Image Process. 15, 604–613 (2006)

    Article  MathSciNet  Google Scholar 

  26. Nagy, J.G., Ng, M.K., Perrone, L.: Kronecker product approximations for image restoration with reflexive boundary conditions. SIAM J. Matrix Anal. Appl. 25, 829–841 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Oseledets, I.V.: Tensor-train decomposition. SIAM J. Sci. Comput. 33, 2295–2317 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Paige, C.C., Saunders, M.A.: Algorithm 583: LSQR: sparse linear equations and least squares problems. ACM Trans. Math. Softw (TOMS) 8, 195–209 (1982)

    Article  Google Scholar 

  29. Paige, C.C., Saunders, M.A.: LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Softw. (TOMS) 8, 43–71 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  30. Peng, Z.-Y.: A matrix LSQR iterative method to solve matrix equation AXB= C. Int. J. Comput. Math. 87, 1820–1830 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Toutounian, F., Karimi, S.: Global least squares method (Gl-LSQR) for solving general linear systems with several right-hand sides. Appl. Math. Comput. 178, 452–460 (2006)

    MathSciNet  MATH  Google Scholar 

  32. VanLoan, C.F.: The ubiquitous Kronecker product. J. Comput. Appl. Math. 123, 85–100 (2000)

    Article  MathSciNet  Google Scholar 

  33. Van Loan, C.F., Pitsianis, N.: Approximation with Kronecker products. In: Linear Algebra for Large Scale and Real-Time Applications, pp. 293–314. Springer (1993)

  34. Wen, Y.-W., Ng, M.K., Huang, Y.-M.: Efficient total variation minimization methods for color image restoration. IEEE Trans. Image Process. 17, 2081–2088 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the two anonymous referees for all the valuable remarks and helpful suggestions. The authors would like also to express their deepest appreciation to Lothar Reichel for his valuable comments.

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Authors and Affiliations

  1. University Cadi Ayyad, LAMAI, Av. Abdelkarim Elkhattabi Gueliz, B.P 549, Marrakech, Morocco

    A. H. Bentbib & A. Khouia

  2. University Littoral Cote d’Opale, LMPA, 50 rue F, Buisson, BP 699, 62228, Calais Cedex, France

    A. Khouia & H. Sadok

Authors
  1. A. H. Bentbib
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  2. A. Khouia
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  3. H. Sadok
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Communicated by Rosemary Anne Renaut.

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Bentbib, A.H., Khouia, A. & Sadok, H. Color image and video restoration using tensor CP decomposition. Bit Numer Math 62, 1257–1278 (2022). https://doi.org/10.1007/s10543-022-00910-6

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  • DOI: https://doi.org/10.1007/s10543-022-00910-6

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