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On Variable Splitting and Augmented Lagrangian Method for Total Variation-Related Image Restoration Models

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Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging

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Abstract

Variable splitting and augmented Lagrangian method are widely used in image processing. This chapter briefly reviews its applications for solving the total variation (TV) related image restoration problems. Due to the nonsmoothness of TV, related models and variants are nonsmooth convex or nonconvex minimization problems. Variable splitting and augmented Lagrangian method can benefit from the separable structure and efficient subsolvers, and has convergence guarantee in convex cases. We present this approach for a number of TV minimization models including TV-L2, TV-L1, TV with nonquadratic fidelity term, multichannel TV, high-order TV, and curvature minimization models.

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Acknowledgements

Tai is supported by NSFC/RGC Joint Research Scheme (N_HKBU214/19), Initiation Grant for Faculty Niche Research Areas(RC-FNRA-IG/19-20/SCI/01) and CRF (C1013-21GF).

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Authors and Affiliations

  1. School of Mathematical Sciences, Tianjin Normal University, Tianjin, China

    Zhifang Liu

  2. Center for Applied Mathematics, Tianjin University, Tianjin, China

    Yuping Duan

  3. School of Mathematical Sciences, Nankai University, Tianjin, China

    Chunlin Wu

  4. Hong Kong Center for Cerebro-cardiovascular Health Engineering (COCHE), Shatin, Hong Kong

    Xue-Cheng Tai

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  1. Zhifang Liu
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  2. Yuping Duan
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  3. Chunlin Wu
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  4. Xue-Cheng Tai
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Corresponding author

Correspondence to Chunlin Wu .

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Editors and Affiliations

  1. Department of Mathematical Sciences, The University of Liverpool, Liverpool, UK

    Ke Chen

  2. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK

    Carola-Bibiane Schönlieb

  3. Hong Kong Center for Cerebrocardiovascular Health Engineering (COCHE), Shatin, Hong Kong

    Xue-Cheng Tai

  4. Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD, USA

    Laurent Younes

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Liu, Z., Duan, Y., Wu, C., Tai, XC. (2023). On Variable Splitting and Augmented Lagrangian Method for Total Variation-Related Image Restoration Models. In: Chen, K., Schönlieb, CB., Tai, XC., Younes, L. (eds) Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging. Springer, Cham. https://doi.org/10.1007/978-3-030-98661-2_84

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