图像稀疏表示模型及其在图像处理反问题中的应用
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摘要
图像的过完备稀疏表示作为一种新兴的图像模型,能够用尽可能简洁的方式表示图像,即大部分原子系数为零,只有很少的非零大系数,非零系数揭示了图像的内在结构与本质属性,并且冗余系统能够对噪声与误差更为稳健,从而有利于后续的图像处理。同时稀疏表示模型能够有效匹配哺乳动物本原视觉皮层中神经元的稀疏编码机制。稀疏表示理论已经引起了国内外广大学者的普遍关注,是当前的研究热点与难点。本文主要围绕图像稀疏表示理论中过完备字典设计、稀疏分解(逼近)算法以及过完备稀疏表示模型在图像处理反问题中的应用三个方面进行了系统和深入的研究,取得的主要研究成果及创新点包括：
(1)根据图像的几何结构特性,从人类视觉系统感知特性出发,选取二维Gabor函数作为字典的生成函数,建立了匹配各层面图像结构的Gabor感知多成分字典,包含平滑、边缘与纹理三种结构类型的子成分字典,同时依据视觉皮层中神经元感受野的结构特性与图像几何结构特征来分配生成函数中自由参数的采样密度,大幅度缩减了原子个数,进而提出了一种高效的基于匹配追踪的图像稀疏分解算法。实验结果表明Gabor感知多成分字典具有对图像中平滑、边缘与纹理结构的自适应性,与Anisotropic refinement-Gauss混合字典相比能够以较少的原子实现对图像更为高效的稀疏逼近。
(2)提出了一种图像结构自适应的子空间匹配追踪稀疏分解快速算法,首先对待分解图像进行结构自适应的四叉树区域剖分,并将剖分后的每一子块分类为平滑、边缘或纹理三种结构类型之一,进一步将每一子块只在与其结构类型一致的单一子成分字典中进行低维子空间的匹配追踪搜索,从而降低了图像维数与字典搜索复杂度,大幅度提高了稀疏分解效率。
(3)基于图像在过完备字典下的稀疏表示,在Bayesian-MAP框架下,建立了针对泊松噪声的稀疏性正则化图像去噪与恢复凸变分模型,采用负log的泊松似然函数作为数据保真项,模型中非光滑的正则性约束图像表示系数的稀疏性,并附加非负性约束保证恢复图像的非负性。进一步基于分裂Bregman方法,提出了数值求解该模型的多步迭代快速算法,通过引入辅助变量与Bregman距离可将原问题转化为两个简单子问题的迭代求解,降低了计算复杂性。实验结果验证了本文模型与数值算法的有效性。
(4)利用图像的稀疏性先验知识,建立了稀疏性正则化的图像恢复模型,目标泛函建模为实希尔伯特空间中两个下半连续凸泛函之和,并无光滑性要求。根据恢复模型中稀疏性正则项的不同,分为分解(Decompsition)与综合(Synthesis)两种形式。在Peaceman-Rachford算子分裂框架下对模型进行数值求解,对于综合形式下的恢复模型,问题变量为稀疏分解系数,稀疏性正则项通常具有可分离形式,直接采用原始(Primal) Peaceman-Rachford算法进行求解,迭代算法中的子问题通过共轭梯度法进行快速计算。而分解形式下的恢复模型,问题变量为图像自身,稀疏性正则项关于问题变量通常是非光滑且不可分离的,采用对偶(Dual)Peaceman-Rachford算法进行求解,对偶算法能够将原始问题解耦,解耦后的子问题通过FFT变换算法进行快速计算,简化了问题的求解,提高了运算效率。
(5)建立了稀疏性正则化的多帧图像超分辨重建凸变分模型,模型中的正则项刻画了理想图像在字典下的稀疏性先验约束,保真项度量其在退化模型下与观测信号的一致性,分析了该模型解的存在性、唯一性以及最优性条件。进一步,分别采用前向后向算子分裂与线性化Bregman迭代算法进行快速求解,两种数值算法均能够将每一次迭代分解为仅对保真项的前向(显式)步与仅对正则项的后向(隐式)步,从而大幅度降低了计算复杂性,并分析比较了两种数值算法各自的优缺点。实验结果表明该模型能够有效保持重建图像中的边缘轮廓结构。
(6)针对图像的卡通与纹理结构形态,分别建立符合类内强稀疏而类间强不相干的卡通和纹理分量子成分字典,形成了图像的多形态稀疏表示模型,进而提出了一种基于多形态稀疏性正则化的多帧图像超分辨重建凸变分模型,模型中的正则项刻画了理想图像在多成分字典下的稀疏性先验约束,保真项度量其在退化模型下与观测信号的一致性,采用交替迭代算法对该多变量优化问题进行数值求解,每一子问题采用前向后向算子分裂法进行快速求解。该模型能够同时保持重建图像中的边缘轮廓与纹理结构。
The sparse and overcomplete representations of images are a new image model, which can represent images in a compact and efficient way. Most atom coefficients are zero, only few coefficients are big, and the nonzero coefficients can reveal the intrinsic structures and essential properties of images. Besides, redundant systems are also robust to noise and error. For these reasons, sparse representations are beneficial to subsequent image processing applications. At the same time, sparse representation model can effectively match the sparse coding strategy in the primary visual cortex of mammal. Sparse representation theory has already attracted large numbers of international and domestic scholars. At present it is a research hotspot and difficult problem. This paper mainly revolves around the three aspects of sparse representation theory, which include the design of overcomplete dictionary, sparse decomposition (approximation) algorithms and the application of overcomplete and sparse representation model to inverse problems in image processing. The obtained achievements include:
(1) In terms of geometric properties of the image structures and the perception characters of human visual system, two dimensional Gabor function is adopted as the generating function of the dictionary and a multi-component Gabor perception dictionary matching various image structures is constructed, which includes smooth, edge and texture sub-dictionaries. Meanwhile, discretization sampling densities of all free parameters in the Gabor function are allocated according to the receptive field properties of neurons in visual cortex and geometric characters of the image structures.Thus, the number of atoms in the dictionary is reduced dramatically.Furthermore, an effective algorithm based on the matching pursuit method is proposed to obtain sparse decomposition of images with our dictionary. The experimental results indicate that the Gabor multi-component perception dictionary can adaptively provide a precise and complete characterization of local geometry structures, such as plain, edge and texture structures in images. In comparison with the anisotropic refinement-Gaussian (AR-Gauss) mixed dictionary, our dictionary has a much sparser representation of images.
(2) A structure adaptive matching pursuit subspace search algorithm is proposed to obtain effective sparse representations of images. Firstly, images are adaptively segmented into quad-tree blocks in terms of geometrical structure character. Then each block is classified as one of smooth, edge or texture structure types. When seeking for sparse decomposition of every quad-tree block, it is only to search in subspace of single component sub-dictionary with the same structure type as current block. Due to the reduction of dimension of image and complexity of search in the dictionary, our algorithm for sparse decomposition is effective and fast.
(3) Adopting Bayesian-MAP estimation framework and using the sparse representations of the underlying image in an overcomplete dictionary, a sparsity regularized convex functional model is proposed to deconvolve (denosie) Poisson noisy image. The negative-log Poisson likelihood functional is used for data fidelity term and non-smooth regularization term constrains the sparse image representation over the dictionary. An additional term is also added in the functional to ensure the non-negative of the restored image. Inspired form the Split Bergman iteration method, a multi-step fast iterative algorithm is proposed to the model above numerically. By introducing an intermediate variable and Bergman distance, the original problem is transformed into solving two simple sub-problems iteratively, thus decreases the computational complexity rapidly. Experimental results demonstrate the effectiveness of our recovery model and numerical iteration algorithm.
(4) Making use of the prior knowledge of image sparse representation, a general variational model is proposed for image recovery with sparsity regularization. The objective functional can be formulated as minimizing the sum of two lower semicontinuous convex functions (not necessarily differentiable) in a real Hilbert space. According to the difference of sparsity regularization term, the general model can be classified as decomposition prior or synthesis prior types. Further, a peceman-rachford operator splitting method is proposed to solve this general recovery model numerically. With regard to the synthesis type, the variable is the sparse representation coefficients and it is usually separable in the sparsity regularization term. The primal peceman-rachford operator splitting method is adopted to solve it directly and conjugate cradient method is employed to rapidly solve the subproblem in the iteratioa With respect to the decomposition type, the problem variable is image-self and it is usually unseparable in the non-smooth sparisty regularization term, thus it is necessary to employ dual peceman-rachford operator splitting method. Dual method can decouple the original recovery model. At the same time, FFT transform method is used to fast solve the subporlem in the iteration, which can simplify the solving of problem and improve operation efficiency.
(5) A convex variational model is proposed for multi-frame image super-resolution with sparse representation regularization. The regularization term of the model constrains the underlying image to have a sparse representation in a proper dictionary. The fidelity term of the model restricts the consistency with the measured image in terms of the data degradation model. The existence, uniqueness and character of solution to the model are analyzed. Furthermore, forward and backward operator splitting method and linearized bregman iteration are adopted to numerically solve the above model respectively. Both the algorithms can decompose each iteration into the forward (explicit) gradient sub-step only for the fidelity term and the backward (implicit) sub-step only for regularization term, thus decreasing the computational complexity rapidly. The advantages and disadvantages of the two algorithms are analysed and compared. Numerical results demonstrate that our super-resolution model can efficiently keep the edge and contour structures in the reconstructed image.
(6) Two incoherent geometry and texture sub-dictionaries are constructed, which can provide sparse representation of cartoon and texture structures respectively, thus constructs an image multi-morphology sparse representation model. Furthermore, a convex variational model is proposed for multi-frame image super-resolution with multi-morphology sparsity regularization. The regularization term of the model constrains the underlying image to have a sparse representation in a multi-component dictionary. The fidelity term of the model restricts the consistency with the measured image in terms of the data degradation model. An alternate minimization iteration algorithm is proposed to solve it numerically and adopts proximal forward-backward operator splitting method for each sub-problem. Our super-resolution model can efficiently keep the edge, contour and texture structures in the reconstructed image synchronously.
(1)根据图像的几何结构特性,从人类视觉系统感知特性出发,选取二维Gabor函数作为字典的生成函数,建立了匹配各层面图像结构的Gabor感知多成分字典,包含平滑、边缘与纹理三种结构类型的子成分字典,同时依据视觉皮层中神经元感受野的结构特性与图像几何结构特征来分配生成函数中自由参数的采样密度,大幅度缩减了原子个数,进而提出了一种高效的基于匹配追踪的图像稀疏分解算法。实验结果表明Gabor感知多成分字典具有对图像中平滑、边缘与纹理结构的自适应性,与Anisotropic refinement-Gauss混合字典相比能够以较少的原子实现对图像更为高效的稀疏逼近。
(2)提出了一种图像结构自适应的子空间匹配追踪稀疏分解快速算法,首先对待分解图像进行结构自适应的四叉树区域剖分,并将剖分后的每一子块分类为平滑、边缘或纹理三种结构类型之一,进一步将每一子块只在与其结构类型一致的单一子成分字典中进行低维子空间的匹配追踪搜索,从而降低了图像维数与字典搜索复杂度,大幅度提高了稀疏分解效率。
(3)基于图像在过完备字典下的稀疏表示,在Bayesian-MAP框架下,建立了针对泊松噪声的稀疏性正则化图像去噪与恢复凸变分模型,采用负log的泊松似然函数作为数据保真项,模型中非光滑的正则性约束图像表示系数的稀疏性,并附加非负性约束保证恢复图像的非负性。进一步基于分裂Bregman方法,提出了数值求解该模型的多步迭代快速算法,通过引入辅助变量与Bregman距离可将原问题转化为两个简单子问题的迭代求解,降低了计算复杂性。实验结果验证了本文模型与数值算法的有效性。
(4)利用图像的稀疏性先验知识,建立了稀疏性正则化的图像恢复模型,目标泛函建模为实希尔伯特空间中两个下半连续凸泛函之和,并无光滑性要求。根据恢复模型中稀疏性正则项的不同,分为分解(Decompsition)与综合(Synthesis)两种形式。在Peaceman-Rachford算子分裂框架下对模型进行数值求解,对于综合形式下的恢复模型,问题变量为稀疏分解系数,稀疏性正则项通常具有可分离形式,直接采用原始(Primal) Peaceman-Rachford算法进行求解,迭代算法中的子问题通过共轭梯度法进行快速计算。而分解形式下的恢复模型,问题变量为图像自身,稀疏性正则项关于问题变量通常是非光滑且不可分离的,采用对偶(Dual)Peaceman-Rachford算法进行求解,对偶算法能够将原始问题解耦,解耦后的子问题通过FFT变换算法进行快速计算,简化了问题的求解,提高了运算效率。
(5)建立了稀疏性正则化的多帧图像超分辨重建凸变分模型,模型中的正则项刻画了理想图像在字典下的稀疏性先验约束,保真项度量其在退化模型下与观测信号的一致性,分析了该模型解的存在性、唯一性以及最优性条件。进一步,分别采用前向后向算子分裂与线性化Bregman迭代算法进行快速求解,两种数值算法均能够将每一次迭代分解为仅对保真项的前向(显式)步与仅对正则项的后向(隐式)步,从而大幅度降低了计算复杂性,并分析比较了两种数值算法各自的优缺点。实验结果表明该模型能够有效保持重建图像中的边缘轮廓结构。
(6)针对图像的卡通与纹理结构形态,分别建立符合类内强稀疏而类间强不相干的卡通和纹理分量子成分字典,形成了图像的多形态稀疏表示模型,进而提出了一种基于多形态稀疏性正则化的多帧图像超分辨重建凸变分模型,模型中的正则项刻画了理想图像在多成分字典下的稀疏性先验约束,保真项度量其在退化模型下与观测信号的一致性,采用交替迭代算法对该多变量优化问题进行数值求解,每一子问题采用前向后向算子分裂法进行快速求解。该模型能够同时保持重建图像中的边缘轮廓与纹理结构。
The sparse and overcomplete representations of images are a new image model, which can represent images in a compact and efficient way. Most atom coefficients are zero, only few coefficients are big, and the nonzero coefficients can reveal the intrinsic structures and essential properties of images. Besides, redundant systems are also robust to noise and error. For these reasons, sparse representations are beneficial to subsequent image processing applications. At the same time, sparse representation model can effectively match the sparse coding strategy in the primary visual cortex of mammal. Sparse representation theory has already attracted large numbers of international and domestic scholars. At present it is a research hotspot and difficult problem. This paper mainly revolves around the three aspects of sparse representation theory, which include the design of overcomplete dictionary, sparse decomposition (approximation) algorithms and the application of overcomplete and sparse representation model to inverse problems in image processing. The obtained achievements include:
(1) In terms of geometric properties of the image structures and the perception characters of human visual system, two dimensional Gabor function is adopted as the generating function of the dictionary and a multi-component Gabor perception dictionary matching various image structures is constructed, which includes smooth, edge and texture sub-dictionaries. Meanwhile, discretization sampling densities of all free parameters in the Gabor function are allocated according to the receptive field properties of neurons in visual cortex and geometric characters of the image structures.Thus, the number of atoms in the dictionary is reduced dramatically.Furthermore, an effective algorithm based on the matching pursuit method is proposed to obtain sparse decomposition of images with our dictionary. The experimental results indicate that the Gabor multi-component perception dictionary can adaptively provide a precise and complete characterization of local geometry structures, such as plain, edge and texture structures in images. In comparison with the anisotropic refinement-Gaussian (AR-Gauss) mixed dictionary, our dictionary has a much sparser representation of images.
(2) A structure adaptive matching pursuit subspace search algorithm is proposed to obtain effective sparse representations of images. Firstly, images are adaptively segmented into quad-tree blocks in terms of geometrical structure character. Then each block is classified as one of smooth, edge or texture structure types. When seeking for sparse decomposition of every quad-tree block, it is only to search in subspace of single component sub-dictionary with the same structure type as current block. Due to the reduction of dimension of image and complexity of search in the dictionary, our algorithm for sparse decomposition is effective and fast.
(3) Adopting Bayesian-MAP estimation framework and using the sparse representations of the underlying image in an overcomplete dictionary, a sparsity regularized convex functional model is proposed to deconvolve (denosie) Poisson noisy image. The negative-log Poisson likelihood functional is used for data fidelity term and non-smooth regularization term constrains the sparse image representation over the dictionary. An additional term is also added in the functional to ensure the non-negative of the restored image. Inspired form the Split Bergman iteration method, a multi-step fast iterative algorithm is proposed to the model above numerically. By introducing an intermediate variable and Bergman distance, the original problem is transformed into solving two simple sub-problems iteratively, thus decreases the computational complexity rapidly. Experimental results demonstrate the effectiveness of our recovery model and numerical iteration algorithm.
(4) Making use of the prior knowledge of image sparse representation, a general variational model is proposed for image recovery with sparsity regularization. The objective functional can be formulated as minimizing the sum of two lower semicontinuous convex functions (not necessarily differentiable) in a real Hilbert space. According to the difference of sparsity regularization term, the general model can be classified as decomposition prior or synthesis prior types. Further, a peceman-rachford operator splitting method is proposed to solve this general recovery model numerically. With regard to the synthesis type, the variable is the sparse representation coefficients and it is usually separable in the sparsity regularization term. The primal peceman-rachford operator splitting method is adopted to solve it directly and conjugate cradient method is employed to rapidly solve the subproblem in the iteratioa With respect to the decomposition type, the problem variable is image-self and it is usually unseparable in the non-smooth sparisty regularization term, thus it is necessary to employ dual peceman-rachford operator splitting method. Dual method can decouple the original recovery model. At the same time, FFT transform method is used to fast solve the subporlem in the iteration, which can simplify the solving of problem and improve operation efficiency.
(5) A convex variational model is proposed for multi-frame image super-resolution with sparse representation regularization. The regularization term of the model constrains the underlying image to have a sparse representation in a proper dictionary. The fidelity term of the model restricts the consistency with the measured image in terms of the data degradation model. The existence, uniqueness and character of solution to the model are analyzed. Furthermore, forward and backward operator splitting method and linearized bregman iteration are adopted to numerically solve the above model respectively. Both the algorithms can decompose each iteration into the forward (explicit) gradient sub-step only for the fidelity term and the backward (implicit) sub-step only for regularization term, thus decreasing the computational complexity rapidly. The advantages and disadvantages of the two algorithms are analysed and compared. Numerical results demonstrate that our super-resolution model can efficiently keep the edge and contour structures in the reconstructed image.
(6) Two incoherent geometry and texture sub-dictionaries are constructed, which can provide sparse representation of cartoon and texture structures respectively, thus constructs an image multi-morphology sparse representation model. Furthermore, a convex variational model is proposed for multi-frame image super-resolution with multi-morphology sparsity regularization. The regularization term of the model constrains the underlying image to have a sparse representation in a multi-component dictionary. The fidelity term of the model restricts the consistency with the measured image in terms of the data degradation model. An alternate minimization iteration algorithm is proposed to solve it numerically and adopts proximal forward-backward operator splitting method for each sub-problem. Our super-resolution model can efficiently keep the edge, contour and texture structures in the reconstructed image synchronously.
引文
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