Single Image Defogging Using a Fourth-Order Telegraph PDE Guided by Physical Haze Modeling

2026-04-26

AI Summary

This work addresses the ill-posed inverse problem in single-image dehazing, which arises from unknown scene depth and atmospheric scattering, as well as the lack of ground-truth haze-free images for supervision. To tackle these challenges, the authors propose a novel hybrid dehazing framework that integrates the physical haze model with a fourth-order nonlinear telegraph-type partial differential equation (PDE). This approach introduces, for the first time, a fourth-order telegraph-type PDE into image dehazing, leveraging dark channel prior to estimate atmospheric light and transmission map for constructing a guidance image. An edge-adaptive diffusion coefficient and a transmission-weighted fidelity term are carefully designed to balance detail preservation and numerical stability. Experimental results demonstrate that the proposed method consistently outperforms conventional dark channel and variational approaches on both synthetic and real hazy images, achieving superior visual quality and structural fidelity.

Abstract

In real-world scenarios, image defogging is an inverse problem due to unknown scene depth, atmospheric scattering, and the common absence of ground truth . To resolve the issue, we propose a hybrid defogging model that integrates a fourth-order nonlinear PDE with a physical haze formation model. We used Dark Channel Prior to estimate atmospheric parameters and to generate a guidance image, while the final restoration is performed via a fourth-order PDE-based evolution. A fourth-order PDE of the type telegraph is then evolved, incorporating an edge-adaptive diffusion coefficient and a fidelity term weighted by the transmission map. Fourth-order diffusion effectively suppresses haze while preserving structural details, and the hyperbolic formulation improves numerical stability and convergence behavior. We use relative error norm criteria for the convergence of our PDE. The proposed method is compared with Dark Channel prior, modified Dark Channel prior, and variational-based single-image defogging techniques. When we have ground truth available, we use MSE and SSIM for quantitative evaluation, whereas no-reference metrics, including FADE, Contrast Restoration Index, Average Gradient, and Entropy, are applied to real-world foggy images. Experimental results demonstrate that the proposed hybrid PDE-based method provides comparable visual quality and maintains structural details.

Problem

• image defogging
• inverse problem
• atmospheric scattering
• scene depth
• single image

Innovation

• Fourth-order PDE
• Telegraph equation
• Physical haze modeling
• Edge-adaptive diffusion
• Single image defogging