基于偏微分方程的图像复原技术研究
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摘要
本学位论文综合运用泛函分析理论、凸分析理论、最优化理论、以及偏微分方程(PDE)基本理论等把已有的基于PDE的图像复原模型和算法进行了推广和改进.为快速求得所提出的变分模型的数值解,我们重点研究了两类数值算法:最速下降法和分裂Bregman迭代法.相关的数值试验明显地证明了改进的变分模型和数值算法的优越性.这些创新性的改进,显著地提高了图像恢复的质量,为图像处理和计算机视觉中图像的高层次处理,譬如图像分析和理解、模式识别等研究打下了良好的基础.全文的内容共分为四章.
在第一章中,我们首先介绍了本论文的研究背景和实际意义,并回顾了基于PDE的图像复原算法的发展状况.接着又揭示了本论文所研究的主要内容和主要创新点.最后,我们还简要概括了本文的框架结构.
在第二章中,我们简要回顾了与本文中的变分模型和数值算法紧密相关的一些实分析与泛函分析知识、变换域中一种常用的正交变换:傅立叶变换与反变换、以及数值模拟中常用的数值方法:有限差分法.
在第三章中,我们重点研究了两类改进的变分PDE图像恢复模型:韦伯化的TV-L1模型,和自适应的四阶PDE滤波模型,以及其最速下降法的数值求解问题.其中韦伯化的TV-L1模型,则是在经典的TV-L1模型的基础上,结合物理学中的Weber定律而提出的变分模型.此模型由于充分考虑到了人类的视觉效应,因而由它复原出来的图像在直观上更能满足人类的要求.另一方面,经典的LLT模型虽在去噪的同时很好地克服了平滑区域的“阶梯效应”,但是却不可避免地造成了图像边缘的模糊.为了克服这种弊端,本章中我们所提出的自适应的四阶PDE图像滤波模型就能很好地解决这一矛盾.此模型中的边缘停止函数能根据图像中区域特征的差异性,灵活地调节扩散系数.从而使得新模型在有效去除噪声的同时不但能很好地避免“阶梯效应”的出现,而且还能很好地杜绝边缘模糊现象的发生.本章中相应的数值试验明显地验证了提出的新模型的优越性和鲁棒性.
在第四章中,我们详细研究了快速数值算法:分裂Bregman迭代法,在基于PDE图像复原模型中的应用和推广.首先,我们回顾了与分裂Bregman迭代法紧密相关的Bregman迭代法和线性Bregman迭代法,以及分裂Bregman迭代法的基本理论.接着,我们采用全有界变差代替经典的全变差作为正则项,并提出了三个改进的变分PDE模型:基于全有界变差正则项L2保真项的图像去模糊模型、基于非局部全有界变差正则项L2保真项的图像去噪模型、以及基于全有界变差正则项泊松化图像复原模型.其中相比于全变差正则化的模型,采用全有界变差正则项L2保真项的图像去模糊模型和全有界变差正则项泊松化图像复原模型,则能明显加快分裂Bregman迭代法的收敛速度,具有无可比拟的优越性.至于非局部全有界变差正则化的图像去噪模型,为了体现其有效性,我们将之与全有界变差正则化的图像去噪模型相比,数值试验表明提出的新模型不但能有效地克服平滑区域“阶梯效应”的发生,而且还能显著地加快分裂Bregman迭代法的计算速度.最后,我们又将分裂Bregman迭代法进一步应用于基于H~(-1)保真项的图像分解和复原中.相比于原有的最速下降法,采用分裂Bregman迭代法则能明显地减少数值计算的迭代次数,并大幅度地缩短CPU的计算时间.
In this dissertation, we extend and improve the existing theory of partial differen-tial equation (PDE) based models and algorithms for image restoration, by syntheticallycombing functional analysis theory, theory of convex analysis, optimization theory, andthe basic theory of partial differential equations. To quickly obtain the numerical solutionsof the proposed variational models, we investigate two numerical algorithms in detail:steepest descent scheme, and split Bregman iteration. The corresponding numerical ex-perimentations obviously demonstrate the superiority of the improved variational modelsand numerical methods. All these innovatory improvements enhance the quality of therecovered images evidently, and ground for the higher level image processing in imageprocessing and computer vision well, such as image analysis and understanding, patternrecognition, etc. The content of this thesis comprises four chapters listed as follows.
Firstly, in Chapter 1, we introduce the research background and the practical signif-icance, and review the development of PDE based schemes for image restoration. Then,we present the primary coverage and the innovation. Lastly, the framework of this work isdisplayed in this chapter also.
In Chapter 2, some necessary real analysis and functional analysis knowledge, onefrequently-used orthogonal transform in transform domain: Fourier transformation andFourier inverse transformation, and the classical numerical method of numerical simula-tion: finite difference method are retrospected here in brief.
Then, Chapter 3 focuses on two improved variational PDE models: Weberized TV-L1 model, and adaptive fourth-order PDE filter for image restoration, and adopting thesteepest descent scheme for obtaining their numerical solutions. As for the Weberized TV-L1 model, which is a variational model based on the noted TV-L1 model, and combinedwith Weber law closely. Adequately considered the in?uence of human vision psychol-ogy, the proposed model can accurately recover the degraded images, and satisfy humanrequirements well. On the other hand, the classical LLT model effectively overcomes the“staircasing effects”in ?at regions while removing noise, however it results in edge blurryfrequently. In order to conquer this drawback, our proposed adaptive fourth-order PDE fil-ter can do it excellently. According to the distinctiveness of the provincial characteristicsof the image, the edge-stopping function adaptively regulates the diffusivity coefficient.Thus our novel model not only surmounts the“staircasing effects”well while removing noise, but availably avoids the blurring effects. Synchronously, the related numerical re-sults distinctly show its superiority and robustness over the traditional method.
Finally, in Chapter 4, the application and generalization of the fast numerical strat-egy: split Bregman iteration, in PDE based models for image restoration are investigateddetailedly. First of all, we reiterate the fundamental theory of the related computationalmethods, i.e. Bregman iteration, linearized Bregman iteration, and split Bregman itera-tion. Secondly, replacing total variation by total bounded variation for regularization, wepropose three improved variational PDE models: total bounded variation regularization L2fidelity based scheme for image deblurring, nonlocal total bounded variation regularizationL2 fidelity based model for image denoising, and total bounded variation regularizationtechnique based Poissonian images recovery. Compared with the total variation based ver-sions, the total bounded variation regularization L2 fidelity based image deblurring model,and total bounded variation regularization technique based Poissonian images recoverymodel markedly accelerate the convergence speed of split Bregman iteration, and have theunexampled superiority. As for the nonlocal total bounded variation regularization L2 fi-delity based model for image denoising, we also demonstrate the efficiency by comparingwith its corresponding total variation based one. Numerical simulations illustrate that ourproposed scheme not only overcomes the“staircasing effects”well, but effectively speedsup the calculating speed of split Bregman iteration. Lastly, we employ the fast split Breg-man iteration for solving the H?1 fidelity based image decomposition and restoration.Contrasted with the primary steepest descent scheme, experimental results demonstratethat the proposed algorithm can reduce the iterations drastically, and obviously shorten theCPU time for computing.
在第一章中,我们首先介绍了本论文的研究背景和实际意义,并回顾了基于PDE的图像复原算法的发展状况.接着又揭示了本论文所研究的主要内容和主要创新点.最后,我们还简要概括了本文的框架结构.
在第二章中,我们简要回顾了与本文中的变分模型和数值算法紧密相关的一些实分析与泛函分析知识、变换域中一种常用的正交变换:傅立叶变换与反变换、以及数值模拟中常用的数值方法:有限差分法.
在第三章中,我们重点研究了两类改进的变分PDE图像恢复模型:韦伯化的TV-L1模型,和自适应的四阶PDE滤波模型,以及其最速下降法的数值求解问题.其中韦伯化的TV-L1模型,则是在经典的TV-L1模型的基础上,结合物理学中的Weber定律而提出的变分模型.此模型由于充分考虑到了人类的视觉效应,因而由它复原出来的图像在直观上更能满足人类的要求.另一方面,经典的LLT模型虽在去噪的同时很好地克服了平滑区域的“阶梯效应”,但是却不可避免地造成了图像边缘的模糊.为了克服这种弊端,本章中我们所提出的自适应的四阶PDE图像滤波模型就能很好地解决这一矛盾.此模型中的边缘停止函数能根据图像中区域特征的差异性,灵活地调节扩散系数.从而使得新模型在有效去除噪声的同时不但能很好地避免“阶梯效应”的出现,而且还能很好地杜绝边缘模糊现象的发生.本章中相应的数值试验明显地验证了提出的新模型的优越性和鲁棒性.
在第四章中,我们详细研究了快速数值算法:分裂Bregman迭代法,在基于PDE图像复原模型中的应用和推广.首先,我们回顾了与分裂Bregman迭代法紧密相关的Bregman迭代法和线性Bregman迭代法,以及分裂Bregman迭代法的基本理论.接着,我们采用全有界变差代替经典的全变差作为正则项,并提出了三个改进的变分PDE模型:基于全有界变差正则项L2保真项的图像去模糊模型、基于非局部全有界变差正则项L2保真项的图像去噪模型、以及基于全有界变差正则项泊松化图像复原模型.其中相比于全变差正则化的模型,采用全有界变差正则项L2保真项的图像去模糊模型和全有界变差正则项泊松化图像复原模型,则能明显加快分裂Bregman迭代法的收敛速度,具有无可比拟的优越性.至于非局部全有界变差正则化的图像去噪模型,为了体现其有效性,我们将之与全有界变差正则化的图像去噪模型相比,数值试验表明提出的新模型不但能有效地克服平滑区域“阶梯效应”的发生,而且还能显著地加快分裂Bregman迭代法的计算速度.最后,我们又将分裂Bregman迭代法进一步应用于基于H~(-1)保真项的图像分解和复原中.相比于原有的最速下降法,采用分裂Bregman迭代法则能明显地减少数值计算的迭代次数,并大幅度地缩短CPU的计算时间.
In this dissertation, we extend and improve the existing theory of partial differen-tial equation (PDE) based models and algorithms for image restoration, by syntheticallycombing functional analysis theory, theory of convex analysis, optimization theory, andthe basic theory of partial differential equations. To quickly obtain the numerical solutionsof the proposed variational models, we investigate two numerical algorithms in detail:steepest descent scheme, and split Bregman iteration. The corresponding numerical ex-perimentations obviously demonstrate the superiority of the improved variational modelsand numerical methods. All these innovatory improvements enhance the quality of therecovered images evidently, and ground for the higher level image processing in imageprocessing and computer vision well, such as image analysis and understanding, patternrecognition, etc. The content of this thesis comprises four chapters listed as follows.
Firstly, in Chapter 1, we introduce the research background and the practical signif-icance, and review the development of PDE based schemes for image restoration. Then,we present the primary coverage and the innovation. Lastly, the framework of this work isdisplayed in this chapter also.
In Chapter 2, some necessary real analysis and functional analysis knowledge, onefrequently-used orthogonal transform in transform domain: Fourier transformation andFourier inverse transformation, and the classical numerical method of numerical simula-tion: finite difference method are retrospected here in brief.
Then, Chapter 3 focuses on two improved variational PDE models: Weberized TV-L1 model, and adaptive fourth-order PDE filter for image restoration, and adopting thesteepest descent scheme for obtaining their numerical solutions. As for the Weberized TV-L1 model, which is a variational model based on the noted TV-L1 model, and combinedwith Weber law closely. Adequately considered the in?uence of human vision psychol-ogy, the proposed model can accurately recover the degraded images, and satisfy humanrequirements well. On the other hand, the classical LLT model effectively overcomes the“staircasing effects”in ?at regions while removing noise, however it results in edge blurryfrequently. In order to conquer this drawback, our proposed adaptive fourth-order PDE fil-ter can do it excellently. According to the distinctiveness of the provincial characteristicsof the image, the edge-stopping function adaptively regulates the diffusivity coefficient.Thus our novel model not only surmounts the“staircasing effects”well while removing noise, but availably avoids the blurring effects. Synchronously, the related numerical re-sults distinctly show its superiority and robustness over the traditional method.
Finally, in Chapter 4, the application and generalization of the fast numerical strat-egy: split Bregman iteration, in PDE based models for image restoration are investigateddetailedly. First of all, we reiterate the fundamental theory of the related computationalmethods, i.e. Bregman iteration, linearized Bregman iteration, and split Bregman itera-tion. Secondly, replacing total variation by total bounded variation for regularization, wepropose three improved variational PDE models: total bounded variation regularization L2fidelity based scheme for image deblurring, nonlocal total bounded variation regularizationL2 fidelity based model for image denoising, and total bounded variation regularizationtechnique based Poissonian images recovery. Compared with the total variation based ver-sions, the total bounded variation regularization L2 fidelity based image deblurring model,and total bounded variation regularization technique based Poissonian images recoverymodel markedly accelerate the convergence speed of split Bregman iteration, and have theunexampled superiority. As for the nonlocal total bounded variation regularization L2 fi-delity based model for image denoising, we also demonstrate the efficiency by comparingwith its corresponding total variation based one. Numerical simulations illustrate that ourproposed scheme not only overcomes the“staircasing effects”well, but effectively speedsup the calculating speed of split Bregman iteration. Lastly, we employ the fast split Breg-man iteration for solving the H?1 fidelity based image decomposition and restoration.Contrasted with the primary steepest descent scheme, experimental results demonstratethat the proposed algorithm can reduce the iterations drastically, and obviously shorten theCPU time for computing.
引文
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