Comparative Study of Weighted and Coupled Second- and Fourth-Order PDEs for Image Despeckling in Grayscale, Color, SAR, and Ultrasound

2026-04-26

AI Summary

This work addresses the limitations of conventional second-order partial differential equation (PDE)-based despeckling methods, which often produce blocky artifacts, and higher-order models, which may introduce speckle noise or fail to balance noise removal with detail preservation. To overcome these issues, two novel PDE frameworks are proposed: first, a weighted second-/fourth-order hybrid model featuring an adaptive diffusion coefficient constructed from image intensity, gradient, and Laplacian information; and second, a coupled iterative model that solves each order term independently. Both approaches are implemented using explicit finite difference schemes and enhance spatial adaptivity through weighted fusion and coupling strategies. Experimental results demonstrate that the proposed methods consistently outperform TDM and TDFM across grayscale, color, SAR, and ultrasound images, achieving superior performance in terms of PSNR, SSIM, and speckle index, while significantly improving both quantitative metrics and visual quality.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Matrix & Tensor Methods

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

Partial Differential Equation (PDE)-based approaches have gained significant attention in image despeckling due to their strong capability to preserve structural details while suppressing noise. However, conventional second-order PDE models tend to generate blocky artifacts, whereas higher-order models often introduce speckle patterns. To resolve it, this paper proposes and comparatively analyzes two advanced PDE-based frameworks designed for speckle noise suppression while preserving the fine edges. The first model introduces a novel weighted formulation that combines second and fourth-order PDEs through a weighting parameter. The second-order diffusion coefficient employs grayscale and gradient-based indicators, while the fourth-order term is guided solely by a Laplacian-based indicator. The second model constructs a coupled PDE framework, where independent fourth and second-order components are explicitly solved in an iterative manner. In this coupled structure, each diffusion coefficient is defined separately to enhance adaptability in varying image regions. Both models are implemented using the explicit finite difference method. The proposed techniques are extensively evaluated on a variety of datasets, including standard grayscale, color, Synthetic Aperture Radar (SAR), and ultrasound images. Comparative experiments with the existing Telegraph Diffusion Model (TDM) and Fourth-Order Telegraph Diffusion Model (TDFM) demonstrate the superiority of the proposed approaches in reducing speckle noise while effectively preserving fine image structures and edges. Quantitative evaluations using PSNR, SSIM and Speckle Index metrics confirm that the proposed models produce higher image quality and enhanced visual perception. Overall, the presented PDE-based formulations provide a reliable and efficient framework for image despeckling in both natural and medical imaging.

Problem

• image despeckling
• speckle noise
• partial differential equations
• structural preservation
• blocky artifacts

Innovation

• Weighted PDE
• Coupled PDE
• Image Despeckling
• Fourth-order Diffusion
• Speckle Noise Suppression