图像处理中几类PDE模型的数值方法
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摘要
随着社会的数字化与信息化,数字图像处理日益成为一门越来越重要的学科,并且在生物、二工业生产、医疗卫生、通信等许多领域都有着极其广泛的应用.总体来说,图像处理包括图像复原、图像增强、图像分割、图像编码、图像压缩、图像增强、图像放大等众多内容.本文主要研究图像处理的两类问题：图像复原问题与图像放大问题.
一般来说,图像处理的方法可以归纳为三类：基于小波的方法,基于概率统计的方法以及基于偏微分方程(PDE)的方法.本文主要着眼于基于PDE的图像处理方法,研究了局部与非局部两类模型：局部模型中,我们分别研究了ROF模型、LLT模型、LOT模型以及TV-Stokes模型；非局部模型中,我们主要考虑了非局部变分模型.研究基于PDE的图像处理方法的优势在于数值计算本身已有许多求解PDE的理论与算法可供使用.事实证明PDE在图像科学的发展中具有举足轻重的地位.
在第三章中,我们提出了求解图像复原问题中四阶模型的修正不动点算法.与标准不动点算法相比,本章所提出的修正不动点算法在迭代过程中避免了直接求解逆矩阵,因此,该算法在加快收敛速度的同时还降低了舍入误差.理论上,我们采用不同的证明方法,分别给出了各向异性与各向同性扩散四阶模型的收敛性分析：对于各向异性扩散四阶模型,我们利用Cauchy-Schwarz不等式及简单的不等式放缩法进行收敛界估计；对于各向同性扩散四阶模型,我们利用矩阵的谱条件数,得到了一个更加简单与精确的收敛界.在数值实验中,通过与时间演化算法,标准不动点算法以及分裂Bregman算法进行比较,不难看出本章所提出的基于四阶模型的修正不动点算法的高效性与优越性.
在第四章中,基于对偶策略,我们提出了两种求解图像放大问题中两步模型的对偶型算法.在4.3节中,我们考虑的是基于修正LOT模型的图像放大方法.经典的LOT模型中,第一步首先直接光滑化图像的法向量.然而,考虑到Chambolle对偶策略中的对偶变量也可以看作图像的法向量,我们提出直接计算对偶变量而不是直接光滑化法向量.在第二步中,我们采用快速的分裂Bregman迭代算法重构所求图像.在4.4节中,我们考虑基于TV-Stokes模型的图像放大方法.在第一步中,我们结合零散度条件对切向量进行光滑化,得到一个Stokes型方程.接下来由第一步求得的光滑化切向量,我们计算得到相应的法向量,类似于LOT模型通过拟合法向量来重构图像.数值求解时,我们采用对偶算法.关于这两类算法,我们都给出了收敛性分析,数值实验也表明了这两类算法的有效性.
在第五章中,我们研究了求解图像放大问题中非局部变分模型(NL-TV)的分裂Bregman迭代算法.不同于局部模型,非局部模型充分利用了图像自身的空间结构信息来对图像进行处理,从而能同时处理光滑区域与纹理区域,更好地保持了图像的细微结构.在算法设计中,一方面,我们使用分裂Brenman算法来求解该模型所对应的欧拉-拉格朗日方程,另一方面,我们在每一步计算中更新权函数,从而更加精确的刻画原始像素之间的相互作用.接下来,我们证明了该算法的收敛性.数值实验表明,与以往多种图像放大方法相比,本章所提出的基于NL-TV模型的图像放大方法得到了更加精确的数值结果.
此博士论文得到了国家自然科学基金(60872129)的资助.
此博士论文用LATEX2ε软件打印.
With the digitalization and informationization of the society, digital image processing is becoming a more and more important subject and has a very wide range of applications in biology, industrial production, health care, communications and so on. In general, there are many problems in image processing including image restoration, image enhancement, image segmentation, image coding, image com-pression, image zooming, et al. In this dissertation, we focus on image restoration and zooming.
Usually, the image processing methods can be summarized into three cate-gories:wavelet-based methods, based on probability and statistics methods, and methods based on partial differential equation (PDE). In this dissertation, we study the methods based on PDE including local models and nonlocal models. To local models, we research the ROF model, LLT model and TV-Stokes model respectively. As for nonlocal models, we research the nonlocal variational model. One advan-tage using PDE in image processing is there are many theories and methods for solving the PDE in numerical computation. available. Experiments demonstrated that PDE plays an important role in the development of image science.
In Chapter3, we propose a modified fixed point iterative algorithm for solv-ing the fourth-order PDE model in image restoration. Compared with the stan-dard fixed point algorithm, the proposed algorithm needn't to compute inverse matrices so that it can speed up the convergence and reduce the roundoff er-ror. Theoretically, we give the convergence analysis for the isotropic diffusion and anisotropic diffusion models respectively by means of different strategies. Using the Cauchy-Schwarz inequality and some properties of inequality, we give the con-vergence analysis for the anisotropic diffusion model. For the isotropic diffusion model, a sophisticated strategy on spectral properties of matrices is analyzed in convergence theorems so that a simpler and better bound is deduced. Furthermore, we give some experimental results to illustrate the effectiveness and advantages of the proposed algorithms by comparing with the standard fixed point algorithm, the time marching algorithm and the split Bregman algorithm.
In Chapter4, we propose two two-step methods for image zooming using duality strategies. In the first method, instead of smoothing the normal vector directly as did in the first step of the classical LOT model, we reconstruct the unit normal vector by means of Chambolle's dual formulation. Then, we adopt the split Bregman iteration to obtain the zoomed image in the second step. In the second method, we propose an image zooming method based on the TV-Stokes model using the dual formulation. By imposing the divergence free condition on the tangential vector field, we got a nonlinear TV-Stokes image zooming model. Once the regularized tangent vector is obtained in the first step, the corresponding regularized normal vector can be computed. So, in the second step, we solve the same problem as in the LOT model. Furthermore, we give the convergence analysis of the proposed algorithms. Numerical experiments show the efficiency of the proposed methods.
In Chapter5, we study the nonlocal total variation (NL-TV) regularization technique for image zooming, which exploits the spatial interactions in images. Due to using the nonlocal regularization in image zooming, our model preserves the smooth region, fine structure and texture well. To solve the nonlinear Euler-Lagrange equation associated with the NL-TV regularization framework, we pro-pose a split Bregman NL-TV image zooming method. Furthermore, we update the weight function during the image zooming which contains a better similarity information between two pixels. Then, we give the convergence analysis of the pro-posed algorithm. Experimental results illustrate the effectiveness and reliability of our method by comparing it with some previous methods.
This dissertation is supported by the National Natural Science Foundation of China (60872129).
This dissertation is typeset by software LATEX2ε.
一般来说,图像处理的方法可以归纳为三类：基于小波的方法,基于概率统计的方法以及基于偏微分方程(PDE)的方法.本文主要着眼于基于PDE的图像处理方法,研究了局部与非局部两类模型：局部模型中,我们分别研究了ROF模型、LLT模型、LOT模型以及TV-Stokes模型；非局部模型中,我们主要考虑了非局部变分模型.研究基于PDE的图像处理方法的优势在于数值计算本身已有许多求解PDE的理论与算法可供使用.事实证明PDE在图像科学的发展中具有举足轻重的地位.
在第三章中,我们提出了求解图像复原问题中四阶模型的修正不动点算法.与标准不动点算法相比,本章所提出的修正不动点算法在迭代过程中避免了直接求解逆矩阵,因此,该算法在加快收敛速度的同时还降低了舍入误差.理论上,我们采用不同的证明方法,分别给出了各向异性与各向同性扩散四阶模型的收敛性分析：对于各向异性扩散四阶模型,我们利用Cauchy-Schwarz不等式及简单的不等式放缩法进行收敛界估计；对于各向同性扩散四阶模型,我们利用矩阵的谱条件数,得到了一个更加简单与精确的收敛界.在数值实验中,通过与时间演化算法,标准不动点算法以及分裂Bregman算法进行比较,不难看出本章所提出的基于四阶模型的修正不动点算法的高效性与优越性.
在第四章中,基于对偶策略,我们提出了两种求解图像放大问题中两步模型的对偶型算法.在4.3节中,我们考虑的是基于修正LOT模型的图像放大方法.经典的LOT模型中,第一步首先直接光滑化图像的法向量.然而,考虑到Chambolle对偶策略中的对偶变量也可以看作图像的法向量,我们提出直接计算对偶变量而不是直接光滑化法向量.在第二步中,我们采用快速的分裂Bregman迭代算法重构所求图像.在4.4节中,我们考虑基于TV-Stokes模型的图像放大方法.在第一步中,我们结合零散度条件对切向量进行光滑化,得到一个Stokes型方程.接下来由第一步求得的光滑化切向量,我们计算得到相应的法向量,类似于LOT模型通过拟合法向量来重构图像.数值求解时,我们采用对偶算法.关于这两类算法,我们都给出了收敛性分析,数值实验也表明了这两类算法的有效性.
在第五章中,我们研究了求解图像放大问题中非局部变分模型(NL-TV)的分裂Bregman迭代算法.不同于局部模型,非局部模型充分利用了图像自身的空间结构信息来对图像进行处理,从而能同时处理光滑区域与纹理区域,更好地保持了图像的细微结构.在算法设计中,一方面,我们使用分裂Brenman算法来求解该模型所对应的欧拉-拉格朗日方程,另一方面,我们在每一步计算中更新权函数,从而更加精确的刻画原始像素之间的相互作用.接下来,我们证明了该算法的收敛性.数值实验表明,与以往多种图像放大方法相比,本章所提出的基于NL-TV模型的图像放大方法得到了更加精确的数值结果.
此博士论文得到了国家自然科学基金(60872129)的资助.
此博士论文用LATEX2ε软件打印.
With the digitalization and informationization of the society, digital image processing is becoming a more and more important subject and has a very wide range of applications in biology, industrial production, health care, communications and so on. In general, there are many problems in image processing including image restoration, image enhancement, image segmentation, image coding, image com-pression, image zooming, et al. In this dissertation, we focus on image restoration and zooming.
Usually, the image processing methods can be summarized into three cate-gories:wavelet-based methods, based on probability and statistics methods, and methods based on partial differential equation (PDE). In this dissertation, we study the methods based on PDE including local models and nonlocal models. To local models, we research the ROF model, LLT model and TV-Stokes model respectively. As for nonlocal models, we research the nonlocal variational model. One advan-tage using PDE in image processing is there are many theories and methods for solving the PDE in numerical computation. available. Experiments demonstrated that PDE plays an important role in the development of image science.
In Chapter3, we propose a modified fixed point iterative algorithm for solv-ing the fourth-order PDE model in image restoration. Compared with the stan-dard fixed point algorithm, the proposed algorithm needn't to compute inverse matrices so that it can speed up the convergence and reduce the roundoff er-ror. Theoretically, we give the convergence analysis for the isotropic diffusion and anisotropic diffusion models respectively by means of different strategies. Using the Cauchy-Schwarz inequality and some properties of inequality, we give the con-vergence analysis for the anisotropic diffusion model. For the isotropic diffusion model, a sophisticated strategy on spectral properties of matrices is analyzed in convergence theorems so that a simpler and better bound is deduced. Furthermore, we give some experimental results to illustrate the effectiveness and advantages of the proposed algorithms by comparing with the standard fixed point algorithm, the time marching algorithm and the split Bregman algorithm.
In Chapter4, we propose two two-step methods for image zooming using duality strategies. In the first method, instead of smoothing the normal vector directly as did in the first step of the classical LOT model, we reconstruct the unit normal vector by means of Chambolle's dual formulation. Then, we adopt the split Bregman iteration to obtain the zoomed image in the second step. In the second method, we propose an image zooming method based on the TV-Stokes model using the dual formulation. By imposing the divergence free condition on the tangential vector field, we got a nonlinear TV-Stokes image zooming model. Once the regularized tangent vector is obtained in the first step, the corresponding regularized normal vector can be computed. So, in the second step, we solve the same problem as in the LOT model. Furthermore, we give the convergence analysis of the proposed algorithms. Numerical experiments show the efficiency of the proposed methods.
In Chapter5, we study the nonlocal total variation (NL-TV) regularization technique for image zooming, which exploits the spatial interactions in images. Due to using the nonlocal regularization in image zooming, our model preserves the smooth region, fine structure and texture well. To solve the nonlinear Euler-Lagrange equation associated with the NL-TV regularization framework, we pro-pose a split Bregman NL-TV image zooming method. Furthermore, we update the weight function during the image zooming which contains a better similarity information between two pixels. Then, we give the convergence analysis of the pro-posed algorithm. Experimental results illustrate the effectiveness and reliability of our method by comparing it with some previous methods.
This dissertation is supported by the National Natural Science Foundation of China (60872129).
This dissertation is typeset by software LATEX2ε.
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