A Coupled Fourth Order Telegraph Diffusion Framework Using Grayscale Indicators for Image Despeckling

Speckle noise plagues coherent imaging systems like Synthetic Aperture Radar (SAR) and medical ultrasound, degrading image quality. Traditional second-order PDE-based despeckling methods often introduce staircase artifacts and blur fine details. To address this, Manish Kumar and Rajendra K. Ray present a coupled fourth-order telegraph diffusion framework that combines hyperbolic and parabolic PDE components. The model consists of two coupled evolution equations: one enforces fourth-order diffusion for smooth intensity transitions and effective speckle reduction, while the other refines an edge indicator to protect textures and structural features. The diffusion coefficient is adaptively constructed using both image intensity and a grayscale-based indicator function, ensuring structure-aware denoising that avoids blocky artifacts and preserves fine details. The authors prove existence of a weak solution using Schauder's fixed-point theorem and implement a finite-difference scheme with Gauss-Seidel iteration.

Experimental results demonstrate that this approach consistently outperforms both the existing coupled second-order PDE model (HPCPDE) and the fourth-order telegraph diffusion model (TDFM). Quantitative metrics—PSNR, MSSIM, and Speckle Index—show significant improvements on standard grayscale images, real SAR and ultrasound data, and even speckle-corrupted color images. The method's ability to balance noise removal with edge and texture preservation makes it a promising advancement for remote sensing, medical imaging, and other fields relying on coherent imaging. While computationally more intensive than simpler filters, the enhanced quality and theoretical guarantees position this framework as a strong candidate for practical implementation in imaging pipelines.