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Total Fractional-Order Variation-Based Constraint Image Deblurring Problem

Author

Listed:
  • Shahid Saleem

    (Abdus Salam School of Mathematical Sciences (AS-SMS), Government College University, Lahore 54000, Pakistan)

  • Shahbaz Ahmad

    (Abdus Salam School of Mathematical Sciences (AS-SMS), Government College University, Lahore 54000, Pakistan)

  • Junseok Kim

    (Department of Mathematics, Korea University, Seoul 02841, Republic of Korea)

Abstract

When deblurring an image, ensuring that the restored intensities are strictly non-negative is crucial. However, current numerical techniques often fail to consistently produce favorable results, leading to negative intensities that contribute to significant dark regions in the restored images. To address this, our study proposes a mathematical model for non-blind image deblurring based on total fractional-order variational principles. Our proposed model not only guarantees strictly positive intensity values but also imposes limits on the intensities within a specified range. By removing negative intensities or constraining them within the prescribed range, we can significantly enhance the quality of deblurred images. The key concept in this paper involves converting the constrained total fractional-order variational-based image deblurring problem into an unconstrained one through the introduction of the augmented Lagrangian method. To facilitate this conversion and improve convergence, we describe new numerical algorithms and introduce a novel circulant preconditioned matrix. This matrix effectively overcomes the slow convergence typically encountered when using the conjugate gradient method within the augmented Lagrangian framework. Our proposed approach is validated through computational tests, demonstrating its effectiveness and viability in practical applications.

Suggested Citation

  • Shahid Saleem & Shahbaz Ahmad & Junseok Kim, 2023. "Total Fractional-Order Variation-Based Constraint Image Deblurring Problem," Mathematics, MDPI, vol. 11(13), pages 1-26, June.
  • Handle: RePEc:gam:jmathe:v:11:y:2023:i:13:p:2869-:d:1179994
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    References listed on IDEAS

    as
    1. Dali Chen & YangQuan Chen & Dingyu Xue, 2013. "Fractional-Order Total Variation Image Restoration Based on Primal-Dual Algorithm," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-10, November.
    2. Xi-Le Zhao & Ting-Zhu Huang & Xian-Ming Gu & Liang-Jian Deng, 2017. "Vector Extrapolation Based Landweber Method for Discrete Ill-Posed Problems," Mathematical Problems in Engineering, Hindawi, vol. 2017, pages 1-8, November.
    3. Guo, Lin & Zhao, Xi-Le & Gu, Xian-Ming & Zhao, Yong-Liang & Zheng, Yu-Bang & Huang, Ting-Zhu, 2021. "Three-dimensional fractional total variation regularized tensor optimized model for image deblurring," Applied Mathematics and Computation, Elsevier, vol. 404(C).
    4. Shuren Qi & Yushu Zhang & Chao Wang & Rushi Lan, 2023. "Representing Blurred Image without Deblurring," Mathematics, MDPI, vol. 11(10), pages 1-11, May.
    5. Dayi Yang & Xiaojun Wu & Hefeng Yin, 2022. "Blind Image Deblurring via a Novel Sparse Channel Prior," Mathematics, MDPI, vol. 10(8), pages 1-17, April.
    6. S. M. A. Sharif & Rizwan Ali Naqvi & Zahid Mehmood & Jamil Hussain & Ahsan Ali & Seung-Won Lee, 2022. "MedDeblur: Medical Image Deblurring with Residual Dense Spatial-Asymmetric Attention," Mathematics, MDPI, vol. 11(1), pages 1-16, December.
    7. Edmundo Capelas de Oliveira & José António Tenreiro Machado, 2014. "A Review of Definitions for Fractional Derivatives and Integral," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-6, June.
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