Well-posedness of a Variable-Exponent Telegraph Equation Applied to Image Despeckling

TL;DR

This paper introduces a variable-exponent telegraph diffusion model for image despeckling, proving its mathematical well-posedness and demonstrating superior performance over existing methods in noise reduction and edge preservation.

Contribution

It develops a novel variable-exponent telegraph diffusion model for despeckling, with rigorous proof of existence and uniqueness of solutions, and validates its effectiveness through numerical experiments.

Findings

Outperforms nonlocal speckle removal in noise elimination

Effectively preserves edges while despeckling

Proven well-posedness of the model mathematically

Abstract

In this paper, we present a telegraph diffusion model with variable exponents for image despeckling. Moving beyond the traditional assumption of a constant exponent in the telegraph diffusion framework, we explore three distinct variable exponents for edge detection. All of these depend on the gray level of the image or its gradient. We rigorously prove the existence and uniqueness of weak solutions of our model in a functional setting and perform numerical experiments to assess how well it can despeckle noisy gray-level images. We consider both a range of natural images contaminated by varying degrees of artificial speckle noise and synthetic aperture radar (SAR) images. We finally compare our method with the nonlocal speckle removal technique and find that our model outperforms the latter at speckle elimination and edge preservation.