【作者】 王忠谦；

【导师】 朱宁；

【作者基本信息】 苏州大学 ， 应用数学， 2005， 硕士

【摘要】 本文针对近年来在偏微分方程图像处理方法的文献中经常出现的一类能量泛函,系统论述其起源、特点和研究成果并展开深入研究,通过精细的计算,导出相应的Euler-Lagrange方程,又根据图像处理问题具有的离散性特点,利用有关文献的做法,对此方程加以适当简化,得到一个可用于图像处理的高度退化的非线性二阶偏微分方程模型: 其中函数g(x)=1/(1+k|x|~2),G(x)为一个Gauss函数,k>0,d>0,β>0为常数,I(x)为初始图像。 这一问题难以讨论其古典解,因而我们引入偏微分方程的一类广义解——粘性解的概念。注意到过去类似的文献中,有关方程粘性解的定义容易产生歧义,为此,我们重新给出严格的粘性解定义。同时,应用经典的二阶抛物型偏微分方程理论和有关粘性解的知识,补上了以往文献中未加证明的结论,详细证明了上述偏微分方程模型粘性解的存在性、唯一性和稳定性。最后采用迎风有限差分格式对该模型进行含噪图像的降噪和分割实验,从运算结果可以看出,这一模型在保护图像重要局部特征(如边缘、角点等)的同时迭代迅速,并具有明显的降噪效果。更多还原

【Abstract】 The paper studies a type of energy functionals which often occurred in recent articles on PDE image processing. First, we review the source, feature and research outcome systematically, then present an Euler-Lagrange equation by accurate computation. According to the discreteness of image processing problems, we simplify the equation slightly and proposed a highly nonlinear and degenerate second-order PDE of parabolic type.here g(x) =1/(1+k|x|~2) G{x) is a Gauss function, k > 0, d > 0, β > 0 are constants, andI(x) is the origin image.It is difficult to discuss the classical solution of the above equation, so we introduce a sort of PDE’ generalized solution — the viscosity solution. We redefine strictly the notion of so-called viscosity solution. At the same time, some unsolved conclusions of the past articles are done by using the classical theory of second-order parabolic PDE and the knowledge of viscosity solution. And we demonstrate the existence, uniqueness and stability for the viscosity solution of the proposed model in detail. At last, numerical experiments are completed with the upwind finite difference scheme for image denoising and image segmentation. From the result, the model in this paper can preserve local important structures such as edges and corners effectively, and it has good quality of noise removal with high speed.更多还原