一类非线性方程解的存在和唯一性及在图像中的应用

THE EXISTENCE AND UNIQUENESS OF THE SOLUTION FOR A NONLINEAR DIFFERENTIAL EQUATION AND THE APPLICATION TO IMAGE THEORY

本文我们给出一个修正的非线性扩散方程模型 ,与 Cotte Lions和 Morel的模型相比该模型有许多实质上的优点 .主要的想法是把原来去噪声部分 :卷积 Gauss过程替代为解一个有界区域上的线性抛物方程问题 ,因此避开了对初始数值如何全平面延拓的问题 .我们从数学上的证明该问题解的存在性和适定性 ,同时给出对矩形域情况的解的级数形式 .最后我们给基于本模型的数值计算差分模型 ,并且给出几个具体图像在该模型下处理结果 .

A modified version of the Lions, Morel and Coll theory for image selective smoothing and edge detection is proposed.Comparing with their model, the most important advantage of this modification is that the convolution with Gaussians is the filtering process is replaced by solving an initial boundary value problem for the heat equation,which simplifies the mumerical scheme to some extent. We prove the existence and uniqueness of the solution of the modified model and give some exact expression of the filter of the model.