Relations between nonlinear filters in digital image processing

非线性滤波器在图像恢复中非常受欢迎，因为它们允许去噪而不恶化边缘。虽然许多不同的方法已经并行开发，它们之间的联系几乎没有研究到目前为止。在这个项目中，我们打算研究三个重要的非线性滤波器，即局部离散滤波器，扩散滤波器和小波技术之间的关系。我们将分析这些关系，不仅在理论上，但我们也将研究数值实现，并进行实际评估。为了得到一个统一的描述的三个类，我们调查配方无约束优化问题，离散加权平均法和抛物型偏微分方程（PDE）。我们将比较合成图像以及广泛使用的真实世界的测试图像的不同技术。

Nonlinear filters are very popular for image restoration, since they allow denoising without deterioration of edges. Although many different methodologies have been developed in parallel, their connections have hardly been studied so far. In this project we intend to examine the relations between three important classes of nonlinear filters, namely local discrete filters, diffusion filters and wavelet techniques. We will analyse these relations not only theoretically but we will also study numerical implementations and perform practical evaluations. In order to derive a unifying description of the three classes, we investigate formulations in terms of unconstrained optimization problems, discrete weighted averaging methods and parabolic partial differential equations (PDEs). We will compare the different techniques both for synthetical images as well as for widely used real-world test images.