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

arXiv cs.CV / 5/5/2026

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Key Points

The paper addresses single-image defogging as an ill-posed inverse problem caused by unknown scene depth, atmospheric scattering, and the lack of ground-truth haze-free images.
It proposes a hybrid approach that combines a physical haze formation model with a fourth-order nonlinear telegraph-type PDE for image restoration.
Atmospheric parameters are estimated using the Dark Channel Prior to produce a guidance image, while the final reconstruction is carried out through PDE-based evolution using an edge-adaptive diffusion coefficient and a fidelity term tied to the transmission map.
The authors argue that fourth-order diffusion better suppresses haze while preserving structural details, and the hyperbolic telegraph formulation improves numerical stability and convergence.
Experiments compare against Dark Channel Prior variants and variational single-image defogging methods, using both reference and no-reference metrics, and report visually comparable results with preserved structure on real foggy images.

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.