基于偏微分方程和变分的方法在图像去噪、分割、压缩、修复、重建和目标跟踪等方面发挥了越来越不可替代的作用，对应的偏微分方程主要是二阶和高阶方程。高阶非线性方程图像模型一方面具有二阶方程模型所没有的图像处理效果，另一方面通常具有强非线性、退化、可能不适定等特点。本项目拟主要研究几类重要的高阶非线性退化图像方程解的适定性，解的性质如正性、支撑集的演化、零点集的估计和正则性等；探讨高阶非线性方程特别是四阶方程的基础算法，提出具有一定普适性的有效的自适应快速算法和逼近格式，建立相应的收敛性、精度和稳定性等算法理论；建立目标跟踪、非刚性配准和融合的高阶PDE图像模型，并研究解的性质和基础算法。开展这些问题的研究不光对图像处理具有重要的意义，而且无论是在非线性高阶方程的研究还是在其它应用领域如材料科学、地球物理等都是非常必要的。

The methods based on partial differential equations and calculus of variations have been play more and more important roles in image processing such as denoising, segmentation, compression, inpainting, reconstruction and tracking. The corresponding PDEs are of second order or of higher order. Higher order nonlinear PDE image models have not only powerful processing effects which second order PEDs have no, but also strong nonlinear, degenerate and sometimes ill-posed. The aim of this project is mainly to investigate well-posedness and properties of solutions to several classes of important higher order nonlinear degenerate image equations, including positivity, evolution of support sets, estimates of nodal sets and regularity of solutions etc. We intend to study basic algorithms of higher order nonlinear equations, especially the fourth order equations. We will try to propose some useful and applicable efficient adaptive fast algorithms and approximate schemes and establish algorithm theory including convergence, error estimates and stability. We are also going to establish higher order PDE image models for target tracking and nonrigid registration and fusion, and investigate their solution properties and basic algorithm. These researches are not only very imortant to image processing but also very necessary to the areas of higher order nonlinear PDEs as well as to many other areas such as material science and geophysics.