An innovative non-variational framework for denoising impulsive Cauchy noise in medical imaging

Hssaine Oummi; Abdelmajid EL Hakoume; Amine Laghrib; Mourad Nachaoui

doi:10.3934/naco.2025021

Article Contents

2026, Volume 17: 23-54. Doi: 10.3934/naco.2025021

This volume Previous Article Moment-SOS hierarchy and exit location of stochastic processes Next Article Open-loop Nash equilibrium for networked control systems with asymmetric information

An innovative non-variational framework for denoising impulsive Cauchy noise in medical imaging

1.
Ibn Zohr University, Faculty of Applied Sciences Ait Melloul, Morocco
2.
Mathematics and Computer Science and Decision Making Laboratory, ENSA Tétouan, Abdelmalek Essaadi University, Tétouan, Morocco
3.
EMI, Université Sultan Moulay Slimane, Khouribga, Morocco

^*Corresponding author: Amine Laghrib
^*Corresponding author: Amine Laghrib

Received: October 2024

Revised: June 2025

Early access: August 14, 2025

Published online: August 14, 2025

Abstract / Introduction Full Text(HTML) Figure(16) Related Papers Cited by

Abstract

We introduce an innovative non-variational framework designed to address impulsive Cauchy noise in images. This approach utilizes a coupled system that integrates image decomposition techniques with the $p(x, t)$-Laplacian operator, successfully preserving texture and edge details. Our initial analysis focuses on the theoretical foundations of the proposed model, where we employ the Galerkin method to confirm its well-posedness.In addition, we apply our method to COVID MRI chest images, demonstrating its effectiveness in reducing noise while maintaining critical anatomical details. Experimental results show that our approach consistently outperforms existing denoising techniques, highlighting its robustness and effectiveness in medical imaging contexts, particularly in enhancing the quality of MRI scans affected by impulsive noise.

Keywords:

Mathematics Subject Classification: Primary: 34K29; Secondary: 49J20.

Citation:

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Figure 1. Restored MRI Brain image through the introduced PDE with involved components when the Cauchy noise is applied with arbitrary ratio. Note that last figure represents the computed $ p $ at time $ T $

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Figure 2. Restored MRI Head image through the introduced PDE with involved components when the Cauchy noise is applied with arbitrary ratio. Note that last figure represents the computed $ p $ at time $ T $

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Figure 3. Comparison of different denoising methods for removing Cauchy noise (with parameter $ \gamma = 10 $) from the noisy Eyes image

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Figure 4. The relative error $ RelErr $ colorbars between the clean images and restored Eyes images in Fig. 3. The colorbars shows a more efficient restoration if the color is more accentuated

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Figure 5. Comparison of different denoising methods for removing Cauchy noise (with parameter $ \gamma = 20 $) from the noisy Cube image

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Figure 6. The relative error $ RelErr $ colorbars between the clean images and restored Cube images in Fig. 5. The colorbar shows a more efficient restoration if the color is more accentuated

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Figure 7. Comparison of different denoising methods for removing Cauchy noise (with parameter $ \gamma = 30 $) from the noisy Bear image

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Figure 8. The relative error $ RelErr $ colorbars between the clean images and restored Bear images in Fig. 7. The colorbar shows a more efficient restoration if the color is more shaded

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Figure 9. The variation of the PSNR values with respect to iteration number for the recovered clean images of the three above tests: Eyes, Square and Bears

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Figure 10. The variation of the $ \log_{10}(RelErr) $ with respect to iteration number for the recovered clean images of the three above tests: Eyes, Square and Bears

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Figure 11. Obtained results on the MRI Chest 1 image

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Figure 12. Obtained coloured version of the MRI Chest 1 image

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Figure 13. Obtained results on the MRI Chest 2 image

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Figure 14. Obtained coloured version of the MRI Chest 2 image

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Figure 15. The restored real Brain MRI image using our algorithm and compared to three denoising approaches. Note that we show the respective 3D surfaces to better see the tiny recovered details of each restored image

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Figure 16. The restored real Head MRI image using our algorithm and compared to three denoising approaches. Note that we show the respective 3D surfaces to better see the tiny recovered details of each restored image

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