Tensorial total variation-based image and video restoration with optimized projection methods

References

• A.H. Bentbib, A. Bouhamidi, and K. Kreit, A conditional gradient method for primal–dual total variation-based image denoising, Electron. Trans. Numer. Anal. 48 (2018), pp. 310–328.
   Web of Science ®Google Scholar
• A.H. Bentbib, M. El Guide, and K. Jbilou, A generalized matrix Krylov subspace method for TV regularization, J. Comput. Appl. Math. 373 (2019), pp. 112405.
   Web of Science ®Google Scholar
• A.H. Bentbib, M. El Guide, K. Jbilou, and L. Reichel, A global Lanczos method for image restoration, J. Comput. Appl. Math. 300 (2016), pp. 233–244.
   Web of Science ®Google Scholar
• A. Bouhamidi, K. Jbilou, L. Reichel, and H. Sadok, An extrapolated tsvd method for linear discrete ill-posed problems with kronecker structure, Linear. Algebra. Appl. 434 (2011), pp. 1677–1688.
   Web of Science ®Google Scholar
• S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends® Mach. Learn. 3 (2011), pp. 1–122.
   Google Scholar
• S.H. Chan, R. Khoshabeh, K.B. Gibson, P.E. Gill, and T.Q. Nguyen, An augmented lagrangian method for total variation video restoration, IEEE. Trans. Image. Process. 20 (2011), pp. 3097–3111.
   PubMed Web of Science ®Google Scholar
• Z. Chen and L. Lu, A projection method and kronecker product preconditioner for solving sylvester tensor equations, Science China Math. 55 (2012), pp. 1281–1292.
   Web of Science ®Google Scholar
• E.K. Chong and S.H. Zak, An Introduction to Optimization, John Wiley & Sons, Inc, Hoboken (NJ), 2004.
   Google Scholar
• A. Cichocki, R. Zdunek, A.H. Phan, and S.i. Amari, Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation, John Wiley & Sons, Chichester, 2009.
   Google Scholar
• D. Colton, M. Piana, and R. Potthast, A simple method using Morozov's discrepancy principle for solving inverse scattering problems, Inverse. Probl. 13 (1997), pp. 1477–1493.
   Web of Science ®Google Scholar
• K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, Image restoration by sparse 3D transform-domain collaborative filtering, Image Processing: Algorithms and Systems VI, Vol. 6812, International Society for Optics and Photonics, 2008, pp. 681207.
   Google Scholar
• J.P. Dedieu, Points Fixes, Zéros Et La Méthode De Newton. 1.Springer, Berlin, Heidelberg, 2006.
   Google Scholar
• J. Duran, B. Coll, and C. Sbert, Chambolle's projection algorithm for total variation denoising, Image Process. Line 3 (2013), pp. 311–331.
   Google Scholar
• D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Comput. Math. Appl. 2 (1976), pp. 17–40.
   Google Scholar
• P. Getreuer, Rudin–Osher–Fatemi total variation denoising using split Bregman, Image Process. On Line 2 (2012), pp. 74–95.
   Google Scholar
• G. Golub and W. Kahan, Calculating the singular values and pseudo-inverse of a matrix, J. Soc. Indust. Appl. Math., Ser. B: Numer. Anal. 2 (1965), pp. 205–224.
   Google Scholar
• M.E. Guide, A.E. Ichi, F. Beik, and K. Jbilou, Tensor gmres and Golub–Kahan bidiagonalization methods via the Einstein product with applications to image and video processing, preprint (2020), Available at arXiv:2005.07458.
   Google Scholar
• M.E. Guide, A.E. Ichi, K. Jbilou, and R. Sadaka, Tensor Krylov subspace methods via the t-product for color image processing, preprint (2020). Available at arXiv:2006.07133.
   Google Scholar
• X. Guo and Y. Ma, Generalized tensor total variation minimization for visual data recovery, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2015, pp. 3603–3611.
   Google Scholar
• A. Haddad, Stability in a class of variational methods, Appl. Comput. Harmon. Anal. 23 (2007), pp. 57–73.
   Web of Science ®Google Scholar
• P.C. Hansen, The truncated svd as a method for regularization, BIT Numer. Math. 27 (1987), pp. 534–553.
   Web of Science ®Google Scholar
• B. He, H. Yang, and S. Wang, Alternating direction method with self-adaptive penalty parameters for monotone variational inequalities, J. Optim. Theory Appl. 106 (2000), pp. 337–356.
   Web of Science ®Google Scholar
• B. Huang and C. Ma, Global least squares methods based on tensor form to solve a class of generalized Sylvester tensor equations, Appl. Math. Comput. 369 (2020), pp. 124892.
   Web of Science ®Google Scholar
• S. Karimi and M. Dehghan, Global least squares method based on tensor form to solve linear systems in Kronecker format, Trans. Inst. Meas. Control 40 (2018), pp. 2378–2386.
   Web of Science ®Google Scholar
• T.G. Kolda and B.W. Bader, Tensor decompositions and applications, SIAM Rev. 51 (2009), pp. 455–500.
   Web of Science ®Google Scholar
• N. Lee and A. Cichocki, Fundamental tensor operations for large-scale data analysis in tensor train formats, preprint (2014). Available at arXiv:1405.7786.
   Google Scholar
• C. Li, An efficient algorithm for total variation regularization with applications to the single pixel camera and compressive sensing, Ph.D. diss., Rice University, 2010.
   Google Scholar
• M.J. Powell, A method for nonlinear constraints in minimization problems, Optimization, Academic Press1969), pp. 283–298.
   Google Scholar
• Z. Qin, D. Goldfarb, and S. Ma, An alternating direction method for total variation denoising, Optim. Methods Softw. 30 (2015), pp. 594–615.
   Web of Science ®Google Scholar
• D. Ren, H. Zhang, D. Zhang, and W. Zuo, Fast total-variation based image restoration based on derivative alternated direction optimization methods, Neurocomputing 170 (2015), pp. 201–212.
   Web of Science ®Google Scholar
• L.I. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D Nonlinear Phenomena 60 (1992), pp. 259–268.
   Web of Science ®Google Scholar
• X.C. Tai and C. Wu, Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model, International conference on scale space and variational methods in computer vision, Springer, 2009, pp. 502–513.
   Google Scholar
• Y.W. Wen, M.K. Ng, and Y.M. Huang, Efficient total variation minimization methods for color image restoration, IEEE. Trans. Image. Process. 17 (2008), pp. 2081–2088.
   PubMed Web of Science ®Google Scholar
• B. Wohlberg, Admm penalty parameter selection by residual balancing, preprint (2017). Available at arXiv:1704.06209.
   Google Scholar
• C. Wu, J. Zhang, and X.C. Tai, Augmented Lagrangian method for total variation restoration with non-quadratic fidelity, Inverse Probl. Imaging 5 (2011), pp. 237–261.
   Web of Science ®Google Scholar
• P. Xu, Truncated svd methods for discrete linear ill-posed problems, Geophys. J. Int. 135 (1998), pp. 505–514.
   Web of Science ®Google Scholar
• S. Yang, J. Wang, W. Fan, X. Zhang, P. Wonka, and J. Ye, An efficient ADMM algorithm for multidimensional anisotropic total variation regularization problems, Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data mining, 2013, pp. 641–649.
   Google Scholar
• T.J. Ypma, Historical development of the Newton–Raphson method, SIAM Rev. 37 (1995), pp. 531–551.
   Web of Science ®Google Scholar
• J. Zhang, D. Zhao, R. Xiong, S. Ma, and W. Gao, Image restoration using joint statistical modeling in a space-transform domain, IEEE. Trans. Circuits. Syst. Video. Technol. 24 (2014), pp. 915–928.
   Web of Science ®Google Scholar
• Y. Zheng, B. Jeon, J. Zhang, and Y. Chen, Adaptively determining regularisation parameters in non-local total variation regularisation for image denoising, Electron. Lett. 51 (2015), pp. 144–145.
   Web of Science ®Google Scholar
• M. Zhu and T. Chan, An efficient primal–dual hybrid gradient algorithm for total variation image restoration, Tech. Rep. UCLA CAM Report 34, 2008.
   Google Scholar