This paper focuses on the problem of image restoration under Cauchy noise. The variational method, which constructs the data fidelity term involving the Cauchy distribution by MAP estimator, has been proven to be a successful approach. In this paper, a nonlinear diffusion equation is proposed to deal with it. The main ingredients of the proposed equation are a gray level based diffusivity that estimates the amplitude of the noise and a classical gradient based diffusivity that controls the anisotropic diffusion according to the image’s local structure. The proposed equation has the nondivergence form, and its properties, including the existence, uniqueness, and stability of solutions, are established by the notion of viscosity solution. Experimental results show the superiority of the proposed equation over variational methods in restoring small details of images.

本文重点讨论柯西噪声下的图像恢复问题。用MAP估计器构造涉及Cauchy分布的数据保真度项的变分方法已被证明是一种成功的方法。本文提出了一种非线性扩散方程来处理它。提出的方程式的主要成分是估计噪声幅度的基于灰度的扩散率和根据图像的局部结构控制各向异性扩散的经典的基于梯度的扩散率。该方程具有非散度形式，其性质包括溶液的存在性，唯一性和稳定性，是通过粘性溶液的概念建立的。