A PDE Model for Foggy Image Enhancement Based on Fractional Gradient Field

In order to enhance the texture detail information and brightness, improve the image quality while re- move the fog, a fractional partial differential equation （PDE） model for foggy image enhancement is proposed in this paper. The fractional gradient field of fog-free image is established by combining fractional differential with atmospheric scattering physical model. For highlighting the texture detail information of the image, avoiding the phenomenon that the edge is excessively enhanced or the detail texture is not enhanced com- pletely, an enhancement function of the fractional gradient field is constructed, which can make the frac- tional gradient field enhance non-linearly with the change of the gradient modulus. In order to approximate the gradient field of the fog image to the enhancement gradient field, an energy functional is established in the gradient domain. And then the fractional partial differential equation model for foggy image enhancement is obtained by the variational method. Finally, the finite difference method is used to solve the model nume- rically. The experimental results show that the proposed model can effectively improve the contrast and sharpness of the image while removing the fog. Hence the model is an effective fog image enhancement model.