基于数值线性代数与稀疏优化的图像复原问题研究
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摘要
图像在形成、记录、处理和传输过程中，由于成像系统、记录设备和传输介质等不完善，从而导致图像质量下降即图像降质。例如，大气湍流的扰动效应，环境条件的变化和传感元器件自身的质量原因产生的噪声干扰，被摄物与成像设备间的相对运动造成的运动模糊等。图像复原是利用图像的某些先验知识来重建图像从而改善图像质量的技术。然而随着对图像质量和图像分辨率要求的提高，图像复原算法的代价和硬件实现的复杂度显著地增加，且实际应用中对图像复原算法通常有实时性要求。这对现有数字图像复原技术提出了更大的挑战也预示了其广阔的应用前景。本文研究内容涉及图像复原问题中的两个方面:其一，主要发展了求解图像复原问题中大规模线性代数系统的高效正则化方法;其二，研究更加符合实际的图像复原模型。具体来说，本文研究内容组织为方法篇和模型篇，其中前四部分归于正则化方法研究篇，后两部分属于模型研究篇，具体内容如下
基于Toeplitz-plus-Hankel矩阵的分裂，提出了求解Toeplitz-plus-Hankel线性系统的分裂迭代方法并进一步讨论了分裂迭代方法的计算复杂度，收敛性质以及(拟)最优参数的选取问题。
利用新均值边界条件下模糊矩阵的特殊结构，给出了其模糊矩阵的最优Kronecker积逼近算法。并进一步提出了基于Kronecker积逼近的TSVD型正则化方法来求解新均值边界条件下图像复原问题。
Landweber方法具有简单的迭代格式，但缓慢的收敛速度限制了Landweber方法的广泛应用。提出了基于向量外推的Landweber方法并进一步考虑了更加实用的重启型基于向量外推的Landweber方法。数值实验说明基于向量外推的Landweber方法的收敛速度优于其它Landweber型方法的收敛速度。
受基于向量外推的截断奇异值分解(TSVD)方法在中小规模问题上优异表现启发，提出适用于大规模线性离散不适定问题的高效杂交正则化方法。该杂交正则化方法将基于向量外推的TSVD方法用于求解Krylov子空间方法产生的中小规模问题，从而发挥了Krylov子空间方法和基于向量外推的TSVD方法各自的优势。
由于各种原因，实际情况中点扩散函数通常不能精确给定。我们考虑涉及复原图像和点扩散函数噪声两个变量的模型来求解点扩散函数被噪声污染情况下的图像复原问题。并提出求解该模型的交替极小化方法和讨论交替极小化方法的收敛性质。大量的数值实验说明了建立模型的合理性和交替极小化方法的高效性。
如何有效地解译混合像元(unmixing)和去除造成遥感图像降质的模糊是高光谱遥感应用的关键问题。建立了高光谱图像的解译混合像元和去除模糊模型并提出了求解模型的交替迭代方法。大量的数值实验表明建立模型的合理性和交替迭代方法的高效性。
During the formation, recording, processing and transmitting, imperfect imagingsystems, recording equipments and transmission medium will result in the degradationof images. For examples, effects of atmospheric turbulence, noise caused by sensors andchanges of environment condition, motion blurs caused by the relative motion betweenthe camera and the entire scene. Image restoration is the reconstruction of the original im-age from a degraded observation with a priori knowledge of images to improve the qualityof images. With rising requirements of image resolution and image quality, and real-timerequirements in practice, the computational burdens increase significantly. This providesgrand challenges and also a promising future for the development of image restorationtechniques. The contents of this thesis are organized into regularization methods studyparts and models study parts, among which, the following first four parts are attributedto the first kind of study parts while the latter successive two parts belong to the secondkind. They are illustrated in detail as follows:
Based on the matrix splitting of a Toeplitz-plus-Hankel matrix, an efficient splittingiteration method for Toeplitz-plus-Hankel systems has been developed. The convergenceproperties and the computational cost of the proposed splitting iteration method are dis-cussed. Moreover, we have studied the problem of choosing optimal or quasi-optimalparameters of the proposed splitting iteration method. The performance of this splittingiteration method is illustrated by numerical experiments.
As alternatives to classic boundary conditions, new mean boundary conditions havebeen recently introduced for image restoration problems. We have proposed an efficien-t scheme of computing optimal Kronecker product approximations of blurring matriceswith new mean boundary conditions. Based on Kronecker product approximations, trun-cated singular value decomposition (TSVD) type regularization method is developed forimage restoration problems with new mean boundary conditions.
Landweber method is one of classical iterative regularization methods for solvinglinear discrete ill-posed problems. However, the slow convergence limits its availabilityfor widespread applications. We have presented the vector extrapolation based Landwebermethod which exhibits a fast and stable convergence behavior. Moreover, the restarted version of the vector extrapolation based Landweber method is considered for practicalconsiderations. Numerical results are given to illustrate the performance of the vectorextrapolation based Landweber method.
Motivated by the excellent performance of the vector extrapolation enhanced TSVDmethod on small-to-medium sized problems, we have proposed an efficient hybrid methodfor large-scale linear discrete ill-posed problems, which applies the vector extrapolationenhanced TSVD method to small-to-medium sized problems generated by Krylov sub-space methods. Numerical experiments are reported to illustrate the performance of theproposed hybrid method for large-scale linear discrete ill-posed problems.
In practice, one is often faced with imprecise knowledge of the point spread function-s. We studied the functional model by minimizing two variables: the restored image andthe noise on the data-fitting term, the magnitude of the noise in the point spread function,and the total variation regularization term. By making use of the structure of the func-tional, an efficient alternating minimization scheme is developed to solve the proposedmodel. The existence of minimizers of the proposed model and the convergence of theproposed scheme are established. Numerical examples are also presented to demonstratethe effectiveness of the proposed model and the efficiency of the numerical scheme.
Unmixing and deblurring are key issues of hyperspectral imaging. We have proposedan unmixing and deblurring model of hyperspactral data and developed an alternatingdirection method to solve the proposed model. Extensive numerical results are reportedto demonstrate the effectiveness of the proposed model and the efficiency of the proposedscheme.
基于Toeplitz-plus-Hankel矩阵的分裂，提出了求解Toeplitz-plus-Hankel线性系统的分裂迭代方法并进一步讨论了分裂迭代方法的计算复杂度，收敛性质以及(拟)最优参数的选取问题。
利用新均值边界条件下模糊矩阵的特殊结构，给出了其模糊矩阵的最优Kronecker积逼近算法。并进一步提出了基于Kronecker积逼近的TSVD型正则化方法来求解新均值边界条件下图像复原问题。
Landweber方法具有简单的迭代格式，但缓慢的收敛速度限制了Landweber方法的广泛应用。提出了基于向量外推的Landweber方法并进一步考虑了更加实用的重启型基于向量外推的Landweber方法。数值实验说明基于向量外推的Landweber方法的收敛速度优于其它Landweber型方法的收敛速度。
受基于向量外推的截断奇异值分解(TSVD)方法在中小规模问题上优异表现启发，提出适用于大规模线性离散不适定问题的高效杂交正则化方法。该杂交正则化方法将基于向量外推的TSVD方法用于求解Krylov子空间方法产生的中小规模问题，从而发挥了Krylov子空间方法和基于向量外推的TSVD方法各自的优势。
由于各种原因，实际情况中点扩散函数通常不能精确给定。我们考虑涉及复原图像和点扩散函数噪声两个变量的模型来求解点扩散函数被噪声污染情况下的图像复原问题。并提出求解该模型的交替极小化方法和讨论交替极小化方法的收敛性质。大量的数值实验说明了建立模型的合理性和交替极小化方法的高效性。
如何有效地解译混合像元(unmixing)和去除造成遥感图像降质的模糊是高光谱遥感应用的关键问题。建立了高光谱图像的解译混合像元和去除模糊模型并提出了求解模型的交替迭代方法。大量的数值实验表明建立模型的合理性和交替迭代方法的高效性。
During the formation, recording, processing and transmitting, imperfect imagingsystems, recording equipments and transmission medium will result in the degradationof images. For examples, effects of atmospheric turbulence, noise caused by sensors andchanges of environment condition, motion blurs caused by the relative motion betweenthe camera and the entire scene. Image restoration is the reconstruction of the original im-age from a degraded observation with a priori knowledge of images to improve the qualityof images. With rising requirements of image resolution and image quality, and real-timerequirements in practice, the computational burdens increase significantly. This providesgrand challenges and also a promising future for the development of image restorationtechniques. The contents of this thesis are organized into regularization methods studyparts and models study parts, among which, the following first four parts are attributedto the first kind of study parts while the latter successive two parts belong to the secondkind. They are illustrated in detail as follows:
Based on the matrix splitting of a Toeplitz-plus-Hankel matrix, an efficient splittingiteration method for Toeplitz-plus-Hankel systems has been developed. The convergenceproperties and the computational cost of the proposed splitting iteration method are dis-cussed. Moreover, we have studied the problem of choosing optimal or quasi-optimalparameters of the proposed splitting iteration method. The performance of this splittingiteration method is illustrated by numerical experiments.
As alternatives to classic boundary conditions, new mean boundary conditions havebeen recently introduced for image restoration problems. We have proposed an efficien-t scheme of computing optimal Kronecker product approximations of blurring matriceswith new mean boundary conditions. Based on Kronecker product approximations, trun-cated singular value decomposition (TSVD) type regularization method is developed forimage restoration problems with new mean boundary conditions.
Landweber method is one of classical iterative regularization methods for solvinglinear discrete ill-posed problems. However, the slow convergence limits its availabilityfor widespread applications. We have presented the vector extrapolation based Landwebermethod which exhibits a fast and stable convergence behavior. Moreover, the restarted version of the vector extrapolation based Landweber method is considered for practicalconsiderations. Numerical results are given to illustrate the performance of the vectorextrapolation based Landweber method.
Motivated by the excellent performance of the vector extrapolation enhanced TSVDmethod on small-to-medium sized problems, we have proposed an efficient hybrid methodfor large-scale linear discrete ill-posed problems, which applies the vector extrapolationenhanced TSVD method to small-to-medium sized problems generated by Krylov sub-space methods. Numerical experiments are reported to illustrate the performance of theproposed hybrid method for large-scale linear discrete ill-posed problems.
In practice, one is often faced with imprecise knowledge of the point spread function-s. We studied the functional model by minimizing two variables: the restored image andthe noise on the data-fitting term, the magnitude of the noise in the point spread function,and the total variation regularization term. By making use of the structure of the func-tional, an efficient alternating minimization scheme is developed to solve the proposedmodel. The existence of minimizers of the proposed model and the convergence of theproposed scheme are established. Numerical examples are also presented to demonstratethe effectiveness of the proposed model and the efficiency of the numerical scheme.
Unmixing and deblurring are key issues of hyperspectral imaging. We have proposedan unmixing and deblurring model of hyperspactral data and developed an alternatingdirection method to solve the proposed model. Extensive numerical results are reportedto demonstrate the effectiveness of the proposed model and the efficiency of the proposedscheme.
引文
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