【作者】 吴静；

【导师】 朱宁；

【作者基本信息】 苏州大学 ， 应用数学， 2005， 硕士

【摘要】 本文通过对近期文献中一类常见的用于图像处理的能量泛函的深入分析、研究,经过精确的计算,并应用文献[1]中对于离散问题的一种特殊处理法,对问题作相应的简化,又根据文献[2]提出的可用耦合方程简化求偏微分方程数值解的思想,提出如下一类图像处理的偏微分方程模型——具有时滞的正则化技术和空间正则化技术相结合的耦合系统模型: 其中,I（x）是待处理的图像,u是处理后的图像,β为一参数,τ为时间尺度参数,G_σ=（1/（4πκ））^n/2exp（-|x|²/（4σ²））（σ>0）为高斯光滑核,g（s）=1/（1+Ks²）（K>0）,c为一常数。 该问题是一类高度退化的非线性偏微分方程定解问题,我们不能指望其有古典解,为此,我们引入偏微分方程的粘性解。注意到以往文献中类似的退化方程粘性解定义的不合理之处,我们作了合适的修正,给出了严格的粘性解定义。此外,我们应用经典偏微分方程理论知识及有关的偏微分方程粘性解的理论,详细地导出方程解的一系列估计,也补上了以往相关文献中未证明的有关细节及不足,严格地证明了耦合偏微分方程模型粘性解的存在性、唯一性和稳定性。最后我们给出该模型在图像处理中的实际处理效果。从实验验证可以看到,该模型可将图像函数过去的梯度信息传递到当前时刻,在保护特殊边缘（角点、铰接点）方面具有明显的优点。因此无论从理论上还是从实际应用上看,该模型都是一类有意义的、合理的且具有很好处理效果的偏微分方程图像处理模型。更多还原

【Abstract】 This thesis makes an in-depth study and accurate calculation on a type of energy functional used for image processing which often appeared in recent papers on image processing based on partial differential equations. Based on a special method for discrete problem which is proposed in [1] and the idea of [2] of using a coupled PDE to simplify the numerical implementation, this thesis proposed a new PDE model which incorporates time-delay regularization and curvature-based diffusion. Our model takes the following form:where I(x) is the image to be processed, u is its smoothed version, /3 is a weightingparameter, r is a time-scale factor, exp is a Gaussion smoothingkernel with a pre-specified σ, g(s) with parameter K > 0, c is a constant.The model is highly nonlinear and degenerate, therefore we can’t hope it has a solution which is differential in the classical sense. So we introduce the definition of so-called viscosity solutions. We make a suitable revision of the definition which has been used in the past papers on the similar problems and re-define the viscosity solutions strictly. Meanwhile, applying the theory of classical partial differential equation and viscosity solutions, a series of estimations of viscosity solutions are derived, some details and deficiencies not proved in relevant papers are supplemented and the existence, uniqueness and stability of the viscosity solution of the coupled PDE model are proved. Finally we give the actual results of our model in image processing. The experimental results show that our model can incorporate the past information of the gradient of u into the diffusion process and has a superiority in preserving sharp edgesand fine structures. So our model is a meaningful, reasonable and effective PDE image processing model not only in theory but also in practice.更多还原