摘要
变分模型和优化技术是数字图像处理领域的研究热点之一。本文主要研究灰度值图像的恢复和分割问题，针对当前方法的缺点以及面临的困难，提出了一些有效的模型和算法，所使用的方法主要有：局部或非局部全变分，增广Lagrange方法，低秩矩阵恢复等。本文的研究工作以及创新点主要体现在下面几个方面：
1.分析了当前图像恢复的二阶段方法的局限和缺点，提出了一种图像去模糊的TV-Stokes模型。在提出的模型中，首先利用卷积算子的可交换性，根据Bayesian公式建立了去模糊图像切向量场估计的TV-Stokes模型；然后，定义新的方向匹配的正则项来建立图像恢复的TV模型，该模型克服了先前方法第二阶段中根据估计的切向量场恢复图像的变分模型的缺点。提出的模型可以用增广Lagrange方法有效地求解。
2.经典的TV恢复模型通常使用标量正则化参数，而统一的参数值无法满足同质区域去噪和纹理区域保持细节两方面的要求。本文在非高斯噪声条件下，提出了两种图像恢复的自适应TV模型和算法。首先，定义一个与Gamma噪声有关的随机变量来区分图像的同质和纹理区域，在此基础上建立了Gamma噪声去噪的局部约束的TV模型和调节TV模型矢量参数的迭代算法。其次，针对Poisson噪声下图像去模糊问题，使用Poisson局部差异函数来区分图像的不同区域，进而建立相应的局部约束的TV模型和空间自适应的TV算法。试验结果表明提出模型能够准确识别图像区域，进而自适应调节正则化参数来达到同时消除图像噪声和保持纹理细节的目的。
3.非局部TV模型相比TV恢复模型而言能够更好地保持图像的纹理和细节信息，然而求解方面具有较大计算量。本文基于变量分离和罚函数的思想提出了一种交替极小的快速迭代算法，该方法使用泰勒近似展开和连续性策略来加速算法执行，相比于PBOS算法具有较高的执行效率。先前的NL-TV模型是基于高斯噪声假设的，本文使用最小均方误差估计的方法，进一步建立了Gamma噪声的非局部TV模型和迭代算法，试验表明提出模型相比于TV模型具有更好的恢复效果。
4.研究了图像处理中极小化问题的快速求解，提出了不动点迭代算法和基于子空间加速的增广Lagrange方法。一方面，将极小化问题转化为一个不动点方程，使用Picard迭代来得到相应的不动点，进而求得原问题的解。另一方面，在求解极小化问题的增广Lagrange方法的迭代公式中，使用子空间优化技术求解相应的子问题，从而得到一种基于子空间加速的增广Lagrange方法。数值试验表明提出的算法相比于先前提出的算法具有更高的执行效率。
5.研究了空间域和小波域的图像修复问题，提出了两种新的图像修复模型。在图像域修复方面，基于PCA字典和未知图像信息，通过矩阵分解和PCA变换的方法将图像域修复问题转化为低秩和联合稀疏矩阵恢复问题，从而得到一种基于PCA字典的图像修复模型。在小波域修复方面，基于TV(NL-TV)的小波域修复模型对应算法的计算量较大，本文从图像分解的角度提出了新的小波域修复模型，相应迭代算法具有更加简单的形式，数值试验表明提出算法极大地提高了修复效率。
6.图像分割的EWCVT模型具有结构简单，执行效率高的优点，然而它不能够较好地保持图像的微弱或者奇异边界，而且不适用于非高斯噪声。为了克服EWCVT模型的缺点，本文提出了基于标签函数匹配的正则化CVT模型，相比于EWCVT模型提出方法提高了图像分割精度，具有更好的分割效果。
Thestudyonvariationalmodelsandoptimizationtechniquesisanimportantresearchaspect in the field of digital image processing. This thesis mainly focuses on gray imagerestoration and segmentation problems, and proposes some efficient models and algo-rithms to overcome the shortcomings and difficulties of the previous works. The usedtools mainly include the local/non-local(NL) total variation(TV), augmented Lagrangianapproach, low-rank matrix restoration, etc. The main work and innovation are embodiedas follows.
1. This thesis analyzes the limitations and shortcomings of the previous two-stepmethods in the field of image restoration, and proposes a TV-Stokes model for image de-convolution. In the new model, we first utilize the commuting property of the convolutionoperators and the Bayesian formula to establish the TV-Stokes model for the tangentialfield estimation of the deblurred image. Then a new defined regularization term which isonly orientation-matching is included in the TV model for image restoration, and it over-comes the defects of the variational models which utilize the estimated tangential vectorto reconstruct the image in the second step of previous two-step methods. The proposedmodel can be solved efficiently by the augmented Lagrangian approach.
2. The classical TV restoration models always use scalar regularization parameters,but they are unable to meet the demand of removing the noise in the homogeneous regionsand preserving the details in the texture areas meanwhile with the uniform parameter val-ues. This thesis proposes two adapted TV models and corresponding algorithms. Firstly,some random variable with respect to Gamma noise is defined for the distinguishment ofthe homogeneous regions and texture areas. Then the local constrained TV model andthe corresponding iterative algorithm of adjusting the vector parameter of the TV modelare established for the Gamma noise removal. Next, the problem of poissonian image de-blurring is considered. The (Poisson) local discrepancy function is used to distinguish thehomogeneous and texture regions, and then the local constrained TV model and spatiallyadapted TV algorithm are proposed. Experimental results demonstrate that the proposedmodel can recognize the image regions exactly, and then adjust the regularization param-eters adaptively to remove the noise and retain the image details simultaneously.
3. Compared with the TV models, the nonlocal TV models are able to better preservethe texture and details of images. However, the computational amount of the nonlocalmodels is very large. This thesis proposes a fast alternative minimization algorithm basedon the the idea of variable splitting and penalty techniques in optimization. The proposedalgorithm uses Taylor series approximation and a continuation scheme to accelerate itsimplementation,andisprovedtobemoreefficientthanthePBOSalgorithm. ThepreviousNL-TV models are based on the assumption of Gaussian noise. This thesis further usesthe minimize mean-square error (MMSE) estimator to establish the nonlocal TV modeland corresponding iterative algorithm for Gamma noise removal. Experiments show thatthe proposed NL-TV model outperforms the corresponding TV model.
4. This thesis researches the fast computation of the minimization problem in thefield of image processing, and proposes a fixed point algorithm and a subspace optimiza-tion accelerating augmented Lagrangian approach. On the one hand, we transform theminimization problem into a fixed point equation, and use the Picard iteration to obtainthe corresponding fixed point. Then the solution of the original problem can be obtainedby the fixed point; on the other hand, a subspace optimization technique is adopted tosolve the sub-minimization problem in each iteration of the augmented Lagrangian ap-proach, and then a subspace optimization accelerating augmented Lagrangian approach isproposed for solving the TV deblurring model. Numerical experiments demonstrate thatthe proposed algorithms are very efficient.
5. The inpainting problems in the space and wavelet domains are investigated, andtwo inpainting models are proposed in this thesis. In the field of image inpainting, weconsider the PCA dictionaries and the unknown image information, and transform the in-painting problem into low-rank and joint-sparse matrix recovery based on matrix decom-position and PCA transform. Then a new inpainting model based on the PCA dictionariesis obtained. In the field of wavelet domain inpainting, the computational amount of thealgorithms corresponding to TV(NL-TV) wavelet inpainting models is very large. Thisthesisproposesanewwaveletinpaintingmethodbasedonimagedecomposition. Thecor-responding iterative algorithm has a simpler structure, and it can improve the efficiencyof wavelet inpainting dramatically.
6. TheEWCVTmodelforimagesegmentationisverysimpleandefficient, however,it is unable to preserve irregular edges such as some thin and slight edges, and is not suit- able for the non-Gaussian noise. In order to overcome the shortcomings of the EWCVTmodel,thisthesisproposesanovellabel-matchingregularizationCVT(centroidalVoronoitessellation) model. Compared with the EWCVT model, the proposed methods can im-prove the image segmentation accuracy, and therefore obtain the superior segmentationperformance.
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