基于数值代数的图像复原问题研究
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摘要
在信号和图像处理中，期望将原始场景从观测到的降质数据中恢复出来。在数学上，这个过程就模型化为求解一个系数矩阵为模糊矩阵的线性代数系统。本学位论文研究基于数值代数的高性能正则化算法求解图像复原问题中的大规模线性代数系统。此类线性代数系统系数矩阵规模巨大、严重病态且在特定边界条件下具有特殊结构,例如在零边界条件下具有Toeplitz结构，Neumann边界条件下具有Toeplitz-plus-Hankel结构。研究分析结构矩阵的性质，设计稳定快速算法求解基于结构矩阵的线性方程组具有重要意义。本学位论文共有八章，主要研究内容分为六个部分。
求解系数矩阵为三对角M矩阵的线性代数方程组是求解许多积分微分方程中的核心问题之一。保证该系数矩阵的单调性常常是问题研究的关键。第一部分基于三对角M矩阵的特殊性质，采用新的分块法则以及递推使用克莱姆法，研究提出了扰动后矩阵单调性的充分必要条件，并设计了计算最大可达上界的快速算法。理论分析和数值实验均显示了新算法的有效性。
第二部分根据三角Toeplitz矩阵的特殊结构性质，使用循环矩阵逼近三角Toeplitz矩阵，设计了基于尺度参数和快速傅里叶变换的快速三角Toeplitz逆求取算法，即尺度Bini算法。尺度Bini算法在不增加原算法计算量的同时，提高了近似逆矩阵的精度。
第三部分基于可逆Toeplitz矩阵的逆运算公式，通过求解两个基础方程，研究构造了多右端向量Toeplitz线性系统的近似逆预条件子。讨论了近似逆预条件共轭梯度法的计算复杂度，并给出了收敛性分析。数值实验比较说明了新逆预条件子的优越性。
第四部分设计了图像复原问题中假设anti-reflective边界下的预条件技术。基于模糊矩阵的谱分解性质，设计了具有正则化性质的截断谱分解预条件子。截断谱分解预条件子改善了模糊矩阵的特征值分布，令较大特征值为1，较小特征值保持不变，加快了共轭梯度法求解图像复原问题的收敛速度，提高了复原图像质量。
第五部分针对图像复原问题，设计了更符合真实场景的移位反射边界条件，给出了该边界条件下的模糊矩阵。根据模糊矩阵的特殊结构性质，设计了对应的Kronecker积逼近形式，提出了基于Kronecker积逼近的SVD型正则化算法。数值实验说明了SVD型正则化算法的高效性。
采用CGLS和MRNSD等迭代正则化方法求解图像复原问题中的大规模严重病态线性系统，收敛速度虽然很快，但却出现了半收敛现象。第六部分通过分析CGLS和MRNSD方法中迭代向量的性质，结合具有高去噪性能的软阈值方法，设计了类CGLS和类MRNSD迭代算法求解图像复原问题。新算法克服了原算法的半收敛性质，提高了复原图像的质量。
In signal and image processing, one wants to recover a faithful representation of anoriginal scene from blurred noisy image data. This process can be transformed mathe-matically into solving a linear system with a blurring matrix. This dissertation will studynumerical linear algebra based high-performance regularized algorithms for large-scale,structured linear systems in image restoration problems. Particularly, the blurring matrixhas different special structures under different boundary conditions, i.e., block Toeplitzwith Toeplitz blocks under Dirichlet boundary conditions or block Toeplitz-plus-Hankelwith Toeplitz-plus-Hankel blocks under Neumann boundary conditions. In this case, itis important and necessary to analysis the special structures of these matrices and theirinverse matrices and moreover, to design fast and stable algorithms for structured matrixcomputations. This dissertation consists eight chapters with six main parts.
Ensuring monotonicity of perturbed triangular M-matrices is important in many ap-plications. Based on special properties of triangular M-matrices, Part I in this dissertationprovides sufficient and necessary conditions for maximum allowable of a single perturba-tion and higher rank perturbations. The key techniques include explicit formulas for theinverse of partitioned matrices and the use of Cramer’s rule. Perturbed Toeplitz tridiago-nal M-matrices are considered as a special case. The utility of these results is shown byconsidering an application: ensuring a nonnegative solution of a discrete analogue of anintegro-differential population model.
Using a circulant matrix to approximate a given Toeplitz matrix, Part II in the dis-sertation studies scale parameter and fast Fourier transform based algorithms, scaling andrevised scaling Bini’s algorithms for fast inversion of triangular Toeplitz matrices. Inparticular, comparing with the original Bini’s algorithm, the scaling one improves theaccuracy of the numerical solution without additional computational cost.
According to the inverse formula for Toeplitz matrices given by Lv and Huang, anapproximate inverse preconditioner is proposed for symmetric positive definite, but ill-conditioned, Toeplitz systems with multiple right-hand sides in Part III. The constructionof the preconditioner is of low computational cost, requires only the entries of Toeplitzmatrices and does not require explicit knowledge of the generating function f of Toeplitz matrices. Its convergence is proved. Numerical results are given to illustrate the efficiencyof the proposed inverse preconditioner.
Enforcing anti-reflective boundary conditions, a truncated spectral decompositionpreconditioner is proposed for image restoration problems in Part IV. The preconditioneris based on the spectral decomposition of the blurring matrix with anti-reflective boundaryconditions. It clusters the large eigenvalues around one, but leaves the small eigenvaluesalone as well. Hence, conjugate gradient type methods, when applied to solving thesepreconditioned systems, converge very fast. Numerical examples are given to demonstratethe effectiveness of the proposed preconditioner.
Shifting reflective boundary conditions are proposed in Part V for preserving the con-tinuity at the boundaries and therefore reducing ringing effects in the restored image. AKronecker product approximation of the corresponding blurring matrix is then provided,regardless of symmetry requirement of the PSF. The efficiency of the approximation in anSVD-based regularization method is demonstrated by several numerical examples.
Iterative regularization algorithms, such as the conjugate gradient algorithm for leastsquares problems (CGLS) and the modified residual norm steepest descent (MRNSD)algorithm, are popular tools for solving large-scale linear systems arising from imagerestoration problems. These algorithms, however, are hindered by a semi-convergencebehavior, in that the quality of the computed solution first increases and then decreases.Two soft-thresholding based iterative algorithms for image deblurring are proposed inPart VI. They combine CGLS and MRNSD with a denoising technique such as soft-thresholding on the residual vector at each iteration, respectively. The convergence ofMRNSD and soft-thresholding based algorithm is proved. Numerical results show that theproposed algorithms overcome the semi-convergence behavior and the deblurring resultsare slightly better than those of CGLS and MRNSD with their optimal iterations.
求解系数矩阵为三对角M矩阵的线性代数方程组是求解许多积分微分方程中的核心问题之一。保证该系数矩阵的单调性常常是问题研究的关键。第一部分基于三对角M矩阵的特殊性质，采用新的分块法则以及递推使用克莱姆法，研究提出了扰动后矩阵单调性的充分必要条件，并设计了计算最大可达上界的快速算法。理论分析和数值实验均显示了新算法的有效性。
第二部分根据三角Toeplitz矩阵的特殊结构性质，使用循环矩阵逼近三角Toeplitz矩阵，设计了基于尺度参数和快速傅里叶变换的快速三角Toeplitz逆求取算法，即尺度Bini算法。尺度Bini算法在不增加原算法计算量的同时，提高了近似逆矩阵的精度。
第三部分基于可逆Toeplitz矩阵的逆运算公式，通过求解两个基础方程，研究构造了多右端向量Toeplitz线性系统的近似逆预条件子。讨论了近似逆预条件共轭梯度法的计算复杂度，并给出了收敛性分析。数值实验比较说明了新逆预条件子的优越性。
第四部分设计了图像复原问题中假设anti-reflective边界下的预条件技术。基于模糊矩阵的谱分解性质，设计了具有正则化性质的截断谱分解预条件子。截断谱分解预条件子改善了模糊矩阵的特征值分布，令较大特征值为1，较小特征值保持不变，加快了共轭梯度法求解图像复原问题的收敛速度，提高了复原图像质量。
第五部分针对图像复原问题，设计了更符合真实场景的移位反射边界条件，给出了该边界条件下的模糊矩阵。根据模糊矩阵的特殊结构性质，设计了对应的Kronecker积逼近形式，提出了基于Kronecker积逼近的SVD型正则化算法。数值实验说明了SVD型正则化算法的高效性。
采用CGLS和MRNSD等迭代正则化方法求解图像复原问题中的大规模严重病态线性系统，收敛速度虽然很快，但却出现了半收敛现象。第六部分通过分析CGLS和MRNSD方法中迭代向量的性质，结合具有高去噪性能的软阈值方法，设计了类CGLS和类MRNSD迭代算法求解图像复原问题。新算法克服了原算法的半收敛性质，提高了复原图像的质量。
In signal and image processing, one wants to recover a faithful representation of anoriginal scene from blurred noisy image data. This process can be transformed mathe-matically into solving a linear system with a blurring matrix. This dissertation will studynumerical linear algebra based high-performance regularized algorithms for large-scale,structured linear systems in image restoration problems. Particularly, the blurring matrixhas different special structures under different boundary conditions, i.e., block Toeplitzwith Toeplitz blocks under Dirichlet boundary conditions or block Toeplitz-plus-Hankelwith Toeplitz-plus-Hankel blocks under Neumann boundary conditions. In this case, itis important and necessary to analysis the special structures of these matrices and theirinverse matrices and moreover, to design fast and stable algorithms for structured matrixcomputations. This dissertation consists eight chapters with six main parts.
Ensuring monotonicity of perturbed triangular M-matrices is important in many ap-plications. Based on special properties of triangular M-matrices, Part I in this dissertationprovides sufficient and necessary conditions for maximum allowable of a single perturba-tion and higher rank perturbations. The key techniques include explicit formulas for theinverse of partitioned matrices and the use of Cramer’s rule. Perturbed Toeplitz tridiago-nal M-matrices are considered as a special case. The utility of these results is shown byconsidering an application: ensuring a nonnegative solution of a discrete analogue of anintegro-differential population model.
Using a circulant matrix to approximate a given Toeplitz matrix, Part II in the dis-sertation studies scale parameter and fast Fourier transform based algorithms, scaling andrevised scaling Bini’s algorithms for fast inversion of triangular Toeplitz matrices. Inparticular, comparing with the original Bini’s algorithm, the scaling one improves theaccuracy of the numerical solution without additional computational cost.
According to the inverse formula for Toeplitz matrices given by Lv and Huang, anapproximate inverse preconditioner is proposed for symmetric positive definite, but ill-conditioned, Toeplitz systems with multiple right-hand sides in Part III. The constructionof the preconditioner is of low computational cost, requires only the entries of Toeplitzmatrices and does not require explicit knowledge of the generating function f of Toeplitz matrices. Its convergence is proved. Numerical results are given to illustrate the efficiencyof the proposed inverse preconditioner.
Enforcing anti-reflective boundary conditions, a truncated spectral decompositionpreconditioner is proposed for image restoration problems in Part IV. The preconditioneris based on the spectral decomposition of the blurring matrix with anti-reflective boundaryconditions. It clusters the large eigenvalues around one, but leaves the small eigenvaluesalone as well. Hence, conjugate gradient type methods, when applied to solving thesepreconditioned systems, converge very fast. Numerical examples are given to demonstratethe effectiveness of the proposed preconditioner.
Shifting reflective boundary conditions are proposed in Part V for preserving the con-tinuity at the boundaries and therefore reducing ringing effects in the restored image. AKronecker product approximation of the corresponding blurring matrix is then provided,regardless of symmetry requirement of the PSF. The efficiency of the approximation in anSVD-based regularization method is demonstrated by several numerical examples.
Iterative regularization algorithms, such as the conjugate gradient algorithm for leastsquares problems (CGLS) and the modified residual norm steepest descent (MRNSD)algorithm, are popular tools for solving large-scale linear systems arising from imagerestoration problems. These algorithms, however, are hindered by a semi-convergencebehavior, in that the quality of the computed solution first increases and then decreases.Two soft-thresholding based iterative algorithms for image deblurring are proposed inPart VI. They combine CGLS and MRNSD with a denoising technique such as soft-thresholding on the residual vector at each iteration, respectively. The convergence ofMRNSD and soft-thresholding based algorithm is proved. Numerical results show that theproposed algorithms overcome the semi-convergence behavior and the deblurring resultsare slightly better than those of CGLS and MRNSD with their optimal iterations.
引文
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