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Symmetrised fractional variation with L1 fidelity for signal denoising via Grünwald-Letnikov scheme

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  • Lanza, Alessandro
  • Leaci, Antonio
  • Morigi, Serena
  • Tomarelli, Franco

Abstract

We define, study and implement the model SFV-L1: a variational approach to signal analysis exploiting the Riemann-Liouville (RL) fractional calculus. This model incorporates an L1 fidelity term alongside fractional derivatives of the right and left RL operators to act as regularizers. This approach aims to achieve an orientation-independent protocol. The model is studied in the continuous setting and discretized in one dimension by means of a second-order consistent scheme based on approximating the RL fractional derivatives by a truncated Grünwald-Letnikov (GL) scheme. The discrete optimization problem is solved using an iterative approach based on the alternating direction method of multipliers, with guaranteed convergence.

Suggested Citation

  • Lanza, Alessandro & Leaci, Antonio & Morigi, Serena & Tomarelli, Franco, 2025. "Symmetrised fractional variation with L1 fidelity for signal denoising via Grünwald-Letnikov scheme," Applied Mathematics and Computation, Elsevier, vol. 500(C).
    • Handle: RePEc:eee:apmaco:v:500:y:2025:i:c:s0096300325001560
    DOI: 10.1016/j.amc.2025.129429
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    References listed on IDEAS

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    1. Yi-Fei Pu & Ji-Liu Zhou & Patrick Siarry & Ni Zhang & Yi-Guang Liu, 2013. "Fractional Partial Differential Equation: Fractional Total Variation and Fractional Steepest Descent Approach‐Based Multiscale Denoising Model for Texture Image," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    2. Chen, Dali & Chen, YangQuan & Xue, Dingyu, 2015. "Fractional-order total variation image denoising based on proximity algorithm," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 537-545.
    3. Dirk A. Lorenz & Quoc Tran-Dinh, 2019. "Non-stationary Douglas–Rachford and alternating direction method of multipliers: adaptive step-sizes and convergence," Computational Optimization and Applications, Springer, vol. 74(1), pages 67-92, September.
    4. Yi-Fei Pu & Ji-Liu Zhou & Patrick Siarry & Ni Zhang & Yi-Guang Liu, 2013. "Fractional Partial Differential Equation: Fractional Total Variation and Fractional Steepest Descent Approach-Based Multiscale Denoising Model for Texture Image," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-19, November.
    5. Fausto Ferrari, 2018. "Weyl and Marchaud Derivatives: A Forgotten History," Mathematics, MDPI, vol. 6(1), pages 1-25, January.
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