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<p>In this paper, we propose a novel variational framework for image denoising corrupted by impulsive heavy-tailed Cauchy noise. The proposed model integrates a weighted fractional-order total variation (FOTV) regularization with an adaptive spatial weighting strategy and a nonconvex Cauchy data-fidelity term. The fractional regularizer, formulated in the Caputo sense, introduces a tunable smoothness control through the fractional order. It enables a flexible balance between local edge preservation and global texture smoothing, while the adaptive weight modulates the regularization strength according to local image structures. To efficiently minimize the resulting nonconvex and nonsmooth energy, we derive a split Bregman iterative algorithm coupled with a half-quadratic reformulation of the Cauchy fidelity, leading to tractable weighted least-squares subproblems solved via the conjugate gradient method. Furthermore, a Bayesian optimization framework is incorporated to automatically tune the key model parameters based on the PSNR metric. This data-driven calibration improves convergence robustness and ensures optimal trade-offs between noise suppression and detail preservation without manual intervention. We also establish the existence of minimizers of the model in the weighted fractional bounded-variation space. Numerical experiments on synthetic and benchmark images show that the proposed model achieves superior robustness to outliers and better preservation of fine structures compared to classical TV, FOTV, and nonlocal variational models. Quantitative results in terms of PSNR, SSIM, SNR, and relative error confirm the effectiveness, adaptivity, and stability of the proposed approach.</p>

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A new weighted Caputo fractional-order total variation for Cauchy noise removal with Bayesian optimization

  • 期刊论文

  • 发表日期：

    2026年6月27日

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  • Hamdi Braiek

摘要

In this paper, we propose a novel variational framework for image denoising corrupted by impulsive heavy-tailed Cauchy noise. The proposed model integrates a weighted fractional-order total variation (FOTV) regularization with an adaptive spatial weighting strategy and a nonconvex Cauchy data-fidelity term. The fractional regularizer, formulated in the Caputo sense, introduces a tunable smoothness control through the fractional order. It enables a flexible balance between local edge preservation and global texture smoothing, while the adaptive weight modulates the regularization strength according to local image structures. To efficiently minimize the resulting nonconvex and nonsmooth energy, we derive a split Bregman iterative algorithm coupled with a half-quadratic reformulation of the Cauchy fidelity, leading to tractable weighted least-squares subproblems solved via the conjugate gradient method. Furthermore, a Bayesian optimization framework is incorporated to automatically tune the key model parameters based on the PSNR metric. This data-driven calibration improves convergence robustness and ensures optimal trade-offs between noise suppression and detail preservation without manual intervention. We also establish the existence of minimizers of the model in the weighted fractional bounded-variation space. Numerical experiments on synthetic and benchmark images show that the proposed model achieves superior robustness to outliers and better preservation of fine structures compared to classical TV, FOTV, and nonlocal variational models. Quantitative results in terms of PSNR, SSIM, SNR, and relative error confirm the effectiveness, adaptivity, and stability of the proposed approach.