信息科学与工程学院知识库

在信息技术迅猛发展的今天，图像对人们的日常社会生活，工业生产以及科学研究产生了极大的影响。而获得清晰和高分辨率图像不仅能为人们的分析和决策提供依据，也可以作为进一步计算机图像理解与识别的基础。但是在成像过程中，各种内在或外在的原因，比如成像热噪声，运动模糊，成像设备受到光的衍射极限导致的低分辨率等等，都可能导致观察图像存在不同类型和不同程度的退化。因此，设计有效的图像恢复方法一直以来都是信号处理领域的研究热点，并具有十分重要的理论价值和现实意义。但是由于图像退化过程往往导致原始清晰图像部分信息的丢失，从而使得图像恢复问题存在病态特性。因此通过构建有效表达图像潜在结构的先验模型，并将其作为正则项来解决图像逆问题所固有的病态性，对于图像恢复问题是至关重要的。

近年来，结构稀疏表达模型备受人们关注，并在很多图像恢复问题中取得了成功。这种策略在使用合适的过表达辞典对信号进行稀疏表达的同时，对于其中基函数的选择进行了有效地约束，获得了相比传统稀疏表达模型更稳定的图像恢复结果，也因此成为目前主流的图像恢复策略之一。本论文以图像的结构稀疏表达为主线，在三种不同类型的图像恢复策略上提出了五个图像恢复算法，并在图像修复，图像去模糊，和图像超分辨率问题上分析和验证了提出算法的有效性，具体内容如下：（1）提出了一个基于最大相关熵准则并使用高斯混合模型作为图像片先验的图像恢复方法MCC_GMM。由于图像片包含的纹理细节越多就越难为其找到足够数量的非局部相似片，因此为当前图像片找到的k近邻片中往往存在与当前片不属于同一分布的“外点”。通过分析目前基于高斯混合模型的图像恢复方法，我们发现其目标函数均可看作是基于最小均方误差准则的有约束优化问题，而最小均方误差极易受到外点的影响，因此导致传统方法对于纹理较为复杂的图像常出现错误的估计结果。为了克服这个问题，我们利用最大相关熵准则的数据自适应性提出了一个在空间域对图像先验信息进行建模的图像恢复方法，提出的方法可以自动确定k近邻片中的外点，并为每一个k近邻片赋予一个自动计算出来的数据自适应权值，借助该权值可以鲁棒地估计高斯分布参数并用来加权放回图像片以此重建清晰图像。最后，我们设计了一个有效求解相应目标函数的迭代优化算法。通过实验可以看到，相比传统基于高斯混合模型的方法，本文提出的MCC_GMM算法可以显著地提升图像修复的性能。虽然提出的方法是在空间域对图像片进行先验建模，但通过理论分析和实际观察可以发现图像片在变换域中的表达系数表现出了高度的结构特性，因此MCC_GMM算法仍然属于基于结构稀疏表达的图像恢复方法。（2）提出了一个基于最大相关熵准则并使用数据自适应稀疏分布对图像片表达系数进行先验建模的结构稀疏模型MCC_DAP。在提出的方法中，我们利用最大相关熵准则的数据自适应性，在变换域设计了一个将图像片的稀疏表达及估计方法与最大相关熵准则相结合的算法框架，并利用自动计算出来的辅助变量作为表达系数的加权系数在变换域对图像片进行了先验建模。提出的方法可以在理论上获得比初始赋予表达系数的拉普拉斯分布更强的稀疏表达能力，而这种特性是以数据自适应的方式获得的，并且相应的目标函数可通过迭代求解两个凸优化子问题得到解决，从而避免了求解由于直接使用更稀疏的先验分布时可能带来的非凸优化问题。在解决图像修复问题的实验中验证了提出方法的有效性。（3）在非局部正则的框架下提出了一个基于图的全局图像恢复算法G3。通过对传统非局部正则方法的分析，我们发现非局部差分算子里的权值函数在建模图像先验中起着很重要的作用。因此，为了增强图像先验建模的灵活性，我们设计了一个参数化的非局部差分算子，并推导出了一个新颖的参数化数据自适应变换矩阵。通过分析，我们发现引入的变换矩阵具有高通滤波器的本质特点，并且该矩阵的特征值可看成是图像对应无向加权图的图频率，而其特征向量可看成是对应图的基函数，因此该矩阵编码了目标图像的潜在结构信息。求解相应目标函数的优化算法简单而且高效，并在去除对称模糊的实验中可以看到相比传统的非局部正则方法本文提出的策略可以更有效地提高恢复图像的质量。（4）在上述G3算法的基础上，提出了一个基于参数化数据自适应变换矩阵的结构稀疏表达方法SPDT。目前大多数结构稀疏模型，需要对大量外部训练数据集进行片聚类，或是需要利用图像的非局部自相似性为当前待处理图像片寻找相似片集合，以希望得到图像结构信息的有效表达。但是对于前者，从外部训练集得到的恢复模型往往很难适应于当前图像；对于后者，文献中已经证实难以为较为复杂的纹理图像片在其一个相对较大窗口内找到足够数量的非局部相似片，因此对于这些图像区域常常无法获得满意的结果。作为一种不同的策略，我们提出的方法直接关注于图像片的潜在结构表达，并使用在G3算法中获得的变换矩阵作为数据自适应滤波器来考察图像片滤波响应的统计特性，以此对图像片的先验信息进行建模。最后，我们提出了一个有效的优化算法来求解相应的目标函数，并通过大量图像去模糊和图像超分辨率的实验展示了提出方法的良好性能。（5）提出了一个基于局部空间自适应图像先验的结构稀疏模型LSAP。传统的稀疏表达方法通常假设互相重叠图像片之间是相互独立的，因此也常孤立地为每个图像片的表达向量建立概率先验模型。虽然目前的同时稀疏编码方法可以克服上述缺点，但通常假设表达系数服从某个固定形状参数的广义高斯分布。在提出的方法中，我们使用同时编码策略约束非局部相似图像片的稀疏表达向量使其在各维上服从同参数的广义双曲先验分布，但不同维度上的形状参数和尺度参数是不同的且是未知的。利用图像不同空间位置的上下文信息，我们获得了具有空间自适应性的图像先验，并通过提出的贝叶斯变分推理可以有效地从非局部相似片集合中联合估计表达系数和模型中的未知参数。最后，我们设计了一个有效的迭代求解算法，并在图像超分辨率的实验中展示了LSAP方法良好的图像重建能力。

With the increasing development of the information technology, digital images have had a significant impact on people's daily social life, industrial production, and scientific research. The clear and high-resolution images can not only be used as the basis for image comprehension and recognition, but also provide the basis for analysis and decision making. However, in the imaging process, a variety of intrinsic or extrinsic factors, such as thermal noise, motion blur, low resolution due to the diffraction limit of light, etc., may result in observed images with different types and varying degrees of degradation. Thus, the design of effective image restoration (IR) methods has always been a hot research topic in the field of signal processing, and has important theoretical and practical implications. However, as part of the information is lost during the image degradation process, IR problem is inherently ill-posed. Therefore, modeling image prior information that can effectively express the underlying structures of the image and using it as a regularizer to overcome the ill-posedness of IR is of great importance to restore degraded images.

In recent years, structured sparse models have attracted much attention and have been successfully applied to many image inverse problems. When obtaining the sparse representation of the signal of interest using some appropriate over-complete dictionary, this strategy imposes constraints on the selection of the basis functions (or atoms) in the dictionary; therefore, it can produce more stable results compared to the conventional sparse representation models. In this thesis, based on the structured sparse representation of natural images, we propose five IR algorithms by utilizing three different types of strategies, and the effectiveness of the suggested methods is verified through three IR applications which are image inpainting, deblurring, and super-resolution. The main contents of this thesis are detailed as follows:(1) A novel maximum correntropy criterion (MCC) based Gaussian mixture image restoration algorithm (MCC_GMM) is proposed. Since the number of similar patches in the neighborhood of an exemplar patch decay exponentially as the complexity of the patch increases, there may exist outliers within the k-Nearest-Neighbor (kNN) patches of a detailed patch. By analyzing the conventional Gaussian mixture model (GMM) based IR methods, we conclude that their objective functions can be expressed as the minimum of mean square error (MMSE) criterion based optimization problem. As the MMSE criterion is sensitive to outliers, the reconstruction error especially for the detailed patches may occur. To solve this problem, we propose a MCC based IR methods with GMM prior for image patches by replacing the MMSE criterion with the recently developed MCC criterion. The proposed method can automatically identify outliers and assign a proper weight for each kNN patch, and thus can robustly estimate the Gaussian parameters which can result in more accurate estimation of the image patches. Although we model image prior information in spatial domain, when viewed in transform domain the proposed algorithm can still be considered as a structured sparse representation-based IR strategy as the sparse representation vector shows high structural characteristics. Finally, an effective iterative optimization algorithm is designed to solve the proposed objective function under the MCC criterion. The experimental results for image inpainting demonstrate the capabilities of our proposed method.(2) A MCC based data-adaptive sparse prior algorithm (MCC_DAP) is proposed. In this work, we integrate the sparse estimation and MCC criterion into one unified framework by utilizing the powerful data-adaptive property of MCC. In the process, an automatically computed auxiliary variable is introduced to weigh differently each representation coefficient of an image patch and can serve as a threshold to shrink the coefficients towards to zero. By taking advantage of this weighing scheme, the proposed method can make the actual prior distribution of the transform coefficients sparser than the initially assigned Laplacian distribution, and more importantly, this sparsity is obtained in a data-adaptive fashion. In contrast to use a prior distribution with a significantly heavier tails than a Laplacian as the existing methods do, which usually results in a more difficult non-convex problem, the proposed objective function can be iteratively solved by alternating between two steps, both of which are convex optimization problems and thus can be computed more efficiently. The experimental results for image inpainting verify the effectiveness of the proposed method.(3) A generalized graph-based global IR algorithm (G3) is proposed in the context of nonlocal regularization. Based on the analysis of the existing nonlocal regularization methods, we find that the weights in the definition of the nonlocal difference operator play an important role in modeling image prior information. Thus, in this work, we introduce a new parametric nonlocal difference operator, which not only adds flexibility to the prior model of the clean image, but also leads to a novel parametric data-adaptive transformation matrix. Through analysis, we find that the proposed transformation matrix has a high-pass filtering nature, and its eigenvalues and eigenvectors can be considered as the graph frequencies and basis functions of the underlying undirected weighted graph which corresponds to the image under test; therefore, this transformation matrix encodes the underlying structure of the image content. Finally, a simple but effective algorithm is designed to solve the corresponding objective function, and the experimental results in the symmetric blur can verify the effectiveness of this parametric scheme.(4) Based on the parametric data-adaptive transformation matrix introduced in G3 algorithm, we propose a novel structured sparse model for IR (called SPDT). Basically, the existing structured sparse representation based methods can be categorized into two classes: performing patch clustering on lots of external data to model image patch prior, or exploiting the nonlocal similarities in natural images to find similar grouped patches within the image under consideration. For the former case, the learned model from the external training data may not be able to adapt to the current image. And for the latter one, as the number of the similar patches for each exemplar patch within a spatially constrained window decay exponentially as the complexity of the patch increases, these methods may not produce satisfactory results for the detailed patches due to the lack of a sufficient number of similar patches. Therefore, as an alternative strategy, we directly focus on the underlying structure representation of an image patch, and model image prior information by taking advantage of the sparse nature of responses of the data-adaptive filters (i.e., the proposed parametric data-adaptive transformation matrix). Finally, an effective optimization algorithm is designed to solve the corresponding sparse inverse problem. Extensive experimental comparisons with state-of-the-art image deblurring and super-resolution algorithms validate the effectiveness of our proposed method.(5) A local spatial adaptation image prior based structured sparse model (LSAP) for IR is proposed. The traditional sparse representation methods assume that the overlapping patches are independent from each other and thus do not take into consideration the dependency of the representation coefficients of the patches in an image. To address this issue, the existing simultaneous sparse coding based methods usually encourage the representation coefficients of the nonlocal similar patches to admit the same generalized Gaussian distribution with some hand-selected shape parameter. Since different image texture region may possess different statistical characteristics, it is inappropriate to impose a single image prior distribution everywhere in the image. Therefore, in this work, by taking advantage of the expressiveness of generalized hyperbolic (GH) distribution, we assume that the representation coefficients of nonlocal similar patches on the same dimension follow a GH distribution with the shared shape and scale parameters but varying at different image spatial location. Utilizing the proposed Bayesian variational inference algorithm, we can jointly estimate the sparse representation coefficients and the unknown parameters in the prior distribution from the limited local spatial information. Finally, an effective iterative algorithm is designed to solve the corresponding optimization problem, and the numerical results demonstrate the performance of the proposed method.