各向异性高阶几何变分与PDE图像去噪模型

Image restoration is one of the basic and fundamental problems in image processing. Three characters are required: removing noise, preserving edges and recovering smoothing transition regions. The classical TV model can produce piecewise constant image and is therefore suffered from staircase effect while preserving edges. The previous high-order models can not preserving edges. Therefore, we propose a new construction strategy by using the edge information with the first order derivatives and the smoothing information with the second order ones. By considering the gray intensity image as the surface in three dimensional space, the following work will be carried out: (1) We will construct total bending model based on the second fundamental form and the bendness of the surface. This model can restore piecewise smoothing image up to the first order. (2) We propose a problem to restore a special second order smoothing image--developable image. (3)A high-order model driven by Gaussian curvature will be presented, which can restore the developable image. . Our research will provide two high order regularization terms for image processing. It can improve the smoothing degree of the image restoration and are helpful to form a whole framework of geometric image processing. The corresponding Euler-Lagrange equations are a combination of the fourth-order terms and three-order term, which provides a new architecture for the high-order models.

图像恢复是图像处理中的一个基础而重要的问题。去噪、保持边界、恢复光滑过渡区域是图像恢复的基本要求。经典的全变分模型可以保持边界但只能恢复出分片常数图像，引起阶梯效应；现有的高阶模型在边界保持方面表现不佳。为此，本项目利用一阶导数表示的边界信息和二阶导数表示的光滑信息，提供一类新的高阶模型构造策略，并从图像处理的几何框架入手，将图像视为三维空间中的曲面，开展如下研究工作：(1)将建立基于第二基本型和曲面弯曲程度的全弯曲模型，恢复一次光滑即分片线性图像；(2)提出一类特殊二次光滑图像--可展图像--的恢复问题；(3)将建立各向异性高斯曲率高阶变分模型，恢复可展图像。. 本项目的研究将给出两种图像处理的高阶正则项，提高图像恢复的光滑程度，完善现有的几何处理框架，同时得到四阶和三阶混合的高阶PDE模型也将丰富和完善现有的高阶模型体系。

本项目基于微分几何中的曲面理论，构造了分片光滑图像恢复的乘性变分框架，并分别构造了用于分片线性光滑和分片二次光滑图像恢复的高阶变分和PDE模型。同时利用引导滤波、松弛中值滤波等技术与PDE方法相结合，恢复分片光滑图像。并在此基础上，开展了对色调映射等相关问题的研究和处理。