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

TL;DR

This paper introduces a hybrid defogging method combining a fourth-order nonlinear PDE with physical haze modeling, improving haze removal while preserving image details.

Contribution

The novel integration of a fourth-order telegraph PDE with physical haze modeling enhances defogging effectiveness and numerical stability compared to existing techniques.

Findings

The method achieves comparable visual quality to state-of-the-art techniques.

It effectively preserves structural details during haze removal.

Experimental results show improved stability and convergence.

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.