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

Abstract

The total variation regularization method was introduced by Rudin, Osher, and Fatemi as an efficient technique for regularizing grayscale images. In this work, we aimed to generalize the total variation method to regularize multidimensional problems such as colour image and video restoration. A degradation model in a tensor format is proposed to recover blurred and noisy colour images and videos. The alternating direction method for multipliers (ADMM) and an optimized form of projection methods have been employed to solve the tensorial total variation minimization problem. The structure of the developed approach allows the selection of the optimal parameter. We use the truncated SVD (TSVD) to reduce the size of the problem and to accelerate the convergence of the algorithm. The convergence analysis of the proposed method is proved using convex optimization. Numerical tests for image and video restoration are given showing the effectiveness of the proposed approaches.

Keywords:

• Tensorial algebra
• total variation
• ADMM
• shrinkage formula
• projection methods
• GMRES-BTF
• CG-BTF
• LSQR-BTF
• discrepancy principle
• QR factorization
• TSVD

Acknowledgements

We would like to thank the anonymous referees for their recommendations and helpful remarks, which improved the quality of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

O. Benchettou

O. Benchettou received her Ms degree in modeling and scientific computing for mathematical engineering from Cadi Ayyad University, Morocco in July 2019. She is currently enrolled in a cotutelle PhD program at Université du Littoral Côte d'Opale, in France, and Cadi Ayyad University in Morocco under the co-supervision of Prof. Bentbib Abdeslem Hafid (Morocco) and Prof. Bouhamidi Abderrahman (France). Her research areas centered around the regularization of multidimensional inverse problems using tensor algebra and their applications to image and video processing.

A. H. Bentbib

A. H. Bentbib is a Professor for Applied Mathematics at Cadi Ayyad University of Marrakech and belongs to the board of the Applied Mathematics and Informatics laboratory (LAMAI). He received the PhD degree in 1993 at University of Lille 1 sciences and technology (French). For the last two decades he has focused on numerical linear and multilinear algebra with applications arising from image and signal processing, compressed sensing, completion, multilinear PageRanking.

A. Bouhamidi

A. Bouhamidi is a Professor of Applied Mathematics at the University Côte d'Opale in Calais, France (ULCO). He received his Ph.D. in 1992 from the University of Nantes, France, in Applied Mathematics. During the last two decades, he has focused on Linear and Multilinear Numerical Algebra with applications in image and signal processing, Approximation Theory, Meshless Methods for PDEs, Control and optimization.