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A PDE-Based Image Dehazing Method via Atmospheric Scattering Theory

This paper presents a novel partial differential equation (PDE) framework for single-image dehazing. By integrating the atmospheric scattering model with nonlocal regularization and dark channel prior, we propose the improved PDE: \[ -\text{div}\left(D(\nabla u)\nabla u\right) + \lambda(t) G(u) = \Phi(I,t,A) \] where $D(\nabla u) = (|\nabla u| + \epsilon)^{-1}$ is the edge-preserving diffusion coefficient, $G(u)$ is the Gaussian convolution operator, and $\lambda(t)$ is the adaptive regularization parameter based on transmission map $t$. We prove the existence and uniqueness of weak solutions in $H_0^1(\Omega)$ using Lax-Milgram theorem, and implement an efficient fixed-point iteration scheme accelerated by PyTorch GPU computation. The experimental results demonstrate that this method is a promising deghazing solution that can be generalized to the deep model paradigm.

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