【作者】 仵冀颖；

【导师】 阮秋琦；

【作者基本信息】 北京交通大学 ， 信号与信息处理， 2009， 博士

【摘要】 图像是人类获取、传递信息的重要媒介,因此图像广泛应用在人们生活中,并且成为一种科学研究和社会生产的重要工具。偏微分方程是一类重要的数学分析模型,不同方程得处理特性由具体的扩散项和扩散方向决定。本文研究偏微分方程作为一类重要的数学工具在数字图像处理中的应用。偏微分方程具有各向异性扩散性能,并且整个扩散过程在局部信息的作用下进行;方程直接使用图像中的几何特征控制扩散项及扩散方向,因此偏微分方程处理图像可在平滑同质区域的同时保持区域边界等几何特征。本文以偏微分方程图像处理模型作为研究对象,运用泛函分析、马尔可夫随机场、小波变换、模式识别等多种理论和方法,研究偏微分方程的有效性及其与其它现代图像处理方法的关系。重点对偏微分方程与基于全局信息图像处理方法的互补性和相关性进行了深入的研究和探讨。本文分别研究偏微分方程及复合模型在图像去噪、图像放大、图像修复以及复杂条件下人脸识别预处理系统中的应用,对偏微分方程各向异性扩散性能、平滑图像及局部特征保持性能进行多层次、多角度的分析和研究。整个研究中的创新性工作主要体现在以下几个方面:1.复合偏微分方程去噪模型。本文根据泛函分析和马尔可夫随机场理论,证明了经典偏微分方程去噪模型与小波阈值收缩模型、高斯-马尔可夫模型的结构一致性,提出了两类偏微分方程与现代图像处理方法结合的复合去噪模型。复合模型在实现去噪的同时保持图像中的边缘、纹理特征,同时减弱了传统偏微分方程模型引起的块状效应,并减弱处理后得到图像中的Gibbs效应;2.提出双层约束的图像放大模型。传统插值操作得到的图像中存在严重的锯齿效应、振铃效应,本文提出在原始未插值图像中真实几何信息约束下处理放大图像,在减弱噪声的同时保持了边缘特征,减弱了振铃、锯齿效应。本文对放大操作的改进包括两部分:提出一种双前向扩散的偏微分方程模型,模型可以在保持边缘特征的同时减弱图像中的锯齿、振铃效应。模型中的扩散项和扩散方向是原始未插值图像中真实几何信息作用的结果;提出一种双层约束的同时基于局部信息和全局信息的图像放大模型。模型使用双向扩散的偏微分方程处理几何特征,在减弱边缘位置的锯齿、振铃效应的同时增强区域边界特征;图像中的纹理和噪声则由基于全局信息的非局部均值算法处理。该模型可直接应用于自然场景图像、纹理图像和噪声图像的放大操作中;3.提出一种适用于缺失较小信息区域及不包含纹理信息图像的偏微分方程修复模型。已知区域的梯度信息通过优化整体变分能量泛函扩散入目标区域,像素灰度信息根据更新的正交几何方向向目标区域内扩散,扩散过程具有形态学不变性,模型在保持原图像几何特征的同时可平滑地修复图像;4.提出一种基于模块匹配的修复模型,可修复缺失较大信息区域及包含复杂纹理特征的自然场景图像。模型以模块为基本单元基于全局信息处理图像,使用切向等照度线扩散的偏微分方程约束模块的修复顺序。偏微分方程具有形态学不变性,且在边缘特征宽度的约束下扩散,因此沿几何特征的模块具有较高的修复优先级,从而模型可以较好地保持图像中的几何特征。本文在CIELAB空间下实现模块相似度匹配,并使用偏微分方程减弱填充模块之间存在的接缝效应。本文最后还对基于模块模型进行扩展,可以实现更多的复杂修复目的;5.提出一种复杂条件下人脸识别系统的预处理模型。模型使用偏微分方程和朗伯表面模型提取图像中具有光照不变性的小尺度特征,同时使用偏微分方程模型和局部直方图均衡化方法生成增强的大尺度特征,然后在特征级融合两类图像特征得到最终的样本图像。模型预处理后的图像不仅对复杂光照条件下的人脸识别具有鲁棒性,同时应用于表情、遮挡等复杂条件下的人脸识别也可提高传统方法的识别率。本文分别从数学角度和图像处理角度分析了偏微分方程不同扩散项和扩散方向的特性,证明了偏微分方程模型与其他现代图像处理方法的区别与联系,并证明了偏微分方程在图像处理中应用的有效性。本文将具有不同扩散性能的偏微分方程模型分别应用于底层、中层和高级图像处理的预处理操作等不同领域,并将其与不同的现代图像处理方法相结合,提出了满足不同应用要求的有效的图像处理模型。更多还原

【Abstract】 Image is an important method to obtaining and conveying information.It is widely used in human life,and is critical for scientific research 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.In this paper,the usage of PDE in image processing is deeply analyzed and compared.The PDE anisotropiccally 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.In this thesis,we focus on the research on the PDE models used in image processing.We give the profound research on functional theory,Markov random filed theory,Wavelet transform analysis,pattern recognition etc..We analyze the validity of PDE models and prove the relationship between PDE and other modern image processing methods.Our objective is to explore the interdependency and complementarity between PDE and the image processing methods based on the global information.We explore the effective PDE models and the composite models used in image denoising,image magnification,image inpainting,and the pre-processing step for face recognition under complex conditions.The anisotropic diffusion property, smoothing and locality preserving property of PDE are analyzed and proved from various aspects and feature levels.The main creative work in this thesis is summarized as follows:1.The composite image denoising models.In this paper,the relationship between PDE and the Wavelet shrinkage denoising method,and the relationship between PDE and the Gaussian-Markov model are proved based on the Functional theory and the Markov random field theory,respectively.Then two composite denoising models are proposed.The composite models could denoise while preserving the edge and textured information in image.Also it decreases the blocky effects caused by PDE. There is no Gibbs phenomenon in the denoised results;2.The two-layer constrained image magnification models.There are severe zigzag effects and ringing effects in the interpolated image processed by the conventional magnification models.In the two-layer constrained models,the processing procedures are executed under the constraint of the geometric information in the original image.So the processed image is smoothed and preserves the geometric property;there is less zigzag and ringing effect.The two-layer constrained model has two forms:A PDE model which forward diffuses along two orthogonal directions.It preserves the linear structure and removes the zigzag and ringing effects.The diffusion items and directions in this PDE model are the constrained result of the geometric properties in the original image;A magnification model which is based on both local and global image information.This model uses a bi-directional diffused PDE to preserve the geometric property.Then the zigzag and ringing effects are weakened while preserving the linear structure.The textured and noisy information in the image is processed by the non-local means algorithm.This model could be directly used to magnify the natural image,texture image and noisy image;3.A PDE inpainting model which is used to restore the image with a small target region and containing no textured information.The gradient information is diffused into the target region by minimizing the total variation energy function.The pixel’s gray value is diffused along two orthogonal directions based on the updated geometric vectors.The diffusion PDEs are all morphological invariant.The novel model could smoothly restore the image while preserving the geometric property;4.An exemplar-based model which is used to restore the natural image with the large target region and containing the textured information.The exemplar is the basic processing unit and its processing order is determined by a cross isophote diffused PDE.This PDE is morphological invariant,and it is diffused under the constraint of the edge width.The exemplar along the linear structure has a higher processing priority.So the model could preserve the geometric property of image better.The similarity comparison of exemplar is implemented in the CIELAB space and PDE is used to remove the seams between exemplars.The model is further extended for more complex restoring tasks;5.A pre-processing model for face recognition under complex conditions.The model uses PDE and the Lambertian surface model to extract the illumination invariant small-scaled image features.Also it uses PDE and the region-based histogram equalization method to extract the enhanced large-scaled image features.Last,two scaled image features are fused at the feature-level.The face sample processed by this model is robust to the lighting conditions.This model achieves good performances in the face databases with the large variations of expression,occlusion, etc.In this paper,the diffusion items and diffusion directions of PDE are analyzed based on the mathematical theory and image processing theory,respectively.The difference and connection between PDE and other modern image processing methods are proved.Different PDEs are used in low-level,middle-level and pre-processing for high-level image processing fields.They are combined with other modern image processing methods and achieve good performances in different image processing tasks.更多还原