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New Fourth-Order Grayscale Indicator-Based Telegraph Diffusion Model for Image Despeckling

Rajendra K. Ray, Manish Kumar

TL;DR

This work addresses despeckling of multiplicative speckle noise in SAR and ultrasound images, where the noise follows a gamma distribution and denoising must balance noise suppression with detail preservation. It introduces a novel fourth-order nonlinear PDE that fuses diffusion and wave dynamics via a telegraph-type term, with a diffusion coefficient $C(I_\xi,|\Delta I_\xi|)$ modulated by intensity and Laplacian information and stabilized by Gaussian smoothing $I_\xi = J_\xi \ast I$. The model is discretized with an explicit finite-difference scheme and applied to grayscale and color images, demonstrating superior PSNR and MSSIM and lower Speckle Index compared to state-of-the-art second-order methods such as Shan, TDE, and related approaches. The results indicate reduced staircase artifacts and better texture/edge preservation in despeckling, particularly for SAR data, with scope for faster solvers to enable broader practical adoption. Overall, the fourth-order PDE framework marks a significant improvement in robust image restoration under challenging multiplicative noise conditions.

Abstract

Second-order PDE models have been widely used for suppressing multiplicative noise, but they often introduce blocky artifacts in the early stages of denoising. To resolve this, we propose a fourth-order nonlinear PDE model that integrates diffusion and wave properties. The diffusion process, guided by both the Laplacian and intensity values, reduces noise better than gradient-based methods, while the wave part keeps fine details and textures. The effectiveness of the proposed model is evaluated against two second-order anisotropic diffusion approaches using the Peak Signal-to-Noise Ratio (PSNR) and Mean Structural Similarity Index (MSSIM) for images with available ground truth. For SAR images, where a noise-free reference is unavailable, the Speckle Index (SI) is used to measure noise reduction. Additionally, we extend the proposed model to study color images by applying the denoising process independently to each channel, preserving both structure and color consistency. The same quantitative metrics PSNR and MSSIM are used for performance evaluation, ensuring a fair comparison across grayscale and color images. In all the cases, our computed results produce better results compared to existing models in this genre.

New Fourth-Order Grayscale Indicator-Based Telegraph Diffusion Model for Image Despeckling

TL;DR

This work addresses despeckling of multiplicative speckle noise in SAR and ultrasound images, where the noise follows a gamma distribution and denoising must balance noise suppression with detail preservation. It introduces a novel fourth-order nonlinear PDE that fuses diffusion and wave dynamics via a telegraph-type term, with a diffusion coefficient modulated by intensity and Laplacian information and stabilized by Gaussian smoothing . The model is discretized with an explicit finite-difference scheme and applied to grayscale and color images, demonstrating superior PSNR and MSSIM and lower Speckle Index compared to state-of-the-art second-order methods such as Shan, TDE, and related approaches. The results indicate reduced staircase artifacts and better texture/edge preservation in despeckling, particularly for SAR data, with scope for faster solvers to enable broader practical adoption. Overall, the fourth-order PDE framework marks a significant improvement in robust image restoration under challenging multiplicative noise conditions.

Abstract

Second-order PDE models have been widely used for suppressing multiplicative noise, but they often introduce blocky artifacts in the early stages of denoising. To resolve this, we propose a fourth-order nonlinear PDE model that integrates diffusion and wave properties. The diffusion process, guided by both the Laplacian and intensity values, reduces noise better than gradient-based methods, while the wave part keeps fine details and textures. The effectiveness of the proposed model is evaluated against two second-order anisotropic diffusion approaches using the Peak Signal-to-Noise Ratio (PSNR) and Mean Structural Similarity Index (MSSIM) for images with available ground truth. For SAR images, where a noise-free reference is unavailable, the Speckle Index (SI) is used to measure noise reduction. Additionally, we extend the proposed model to study color images by applying the denoising process independently to each channel, preserving both structure and color consistency. The same quantitative metrics PSNR and MSSIM are used for performance evaluation, ensuring a fair comparison across grayscale and color images. In all the cases, our computed results produce better results compared to existing models in this genre.

Paper Structure

This paper contains 5 sections, 45 equations, 16 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Diffusion Models
  3. Proposed Model
  4. Numerical Implementation
  5. Conclusion

Figures (16)

  • Figure 1: Grayscale images: (a) Boat, (b) Texture.
  • Figure 2: Grayscale images: (a) test_011, (b) test_015, (c) test_019, (d) test_022, (e) test_037.
  • Figure 3: color Images: (a) Baboon, (b)Pepper.
  • Figure 4: The first column contains noisy boat images with noise level Look = 1, 3, 5, 10. Subsequent columns: Restored boat images using different models (SHAN Model, TDM Model, Proposed Model).
  • Figure 5: The first column contains noisy texture images with noise levels Look = 1, 3, 5, 10. Subsequent columns: Restored texture images using different models (SHAN, TDM, and Proposed Model).
  • ...and 11 more figures