数字图像处理中的偏微分方程方法

Partial Differential Equation(PDE) Method on Digital Image Processing

图像是获取信息的重要媒介,图像处理技术广泛用于人类生活及社会生产中。偏微分方程是一种数学分析方法,并且它的特性由扩散方向和扩散项决定,这一点对图像处理大为有利。本文综述了近年来受到广泛重视的数字图像处理中的偏微分方程方法。给出了偏微分方程在图像去噪、图像放大、图像分割、图像修复中主要算法的优势及不足,同时给出了我们的改进算法,并展示了实验结果。

Image is an important media to obtaining and conveying information.It is widely used in human life and social production.Partial Differential Equation（PDE） is an important mathematical analysis method,and its property is determined by the diffusion directions and diffusion items in the equation.Its properties are benefit to image processing.The usage of PDE in image processing is analyzed and compared in this paper.The PDE anisotropic diffuses in image domain and the diffusion procedure is constrained by the local geometric information.The diffusion items and directions could be computed by the geometric properties in image directly.So PDE smoothes image while preserving the edge information. We focus on the research on the PDE models used in image processing in this paper.We give the profound research on basic functional theory,Markov random filed theory,Wavelet transform analysis etc.We analyze the validity of PDE models and other modern image processing methods.Our objective is to improve on the performance of PDE on the image processing,and some improvement PDE methods about image processing are given.We summarize the effective PDE models and the composite models used in image denoising,image magnification,image segmentation and image inpainting.At the same time,the examples of the image processing by the improvement PDE methods are shown in the paper.