基于压缩感知的图像重构技术研究
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摘要
压缩感知理论表明，若信号在某变换域具有稀疏表示，且采样矩阵与稀疏矩阵不相关，则可从远低于信号维度的少量非自适应测量值中精确恢复原信号。目前，压缩感知理论已被广泛用于各类磁共振成像中，以便在不降低成像质量的情况下减少采样点数，提高系统扫描速度。本文即研究从亚采样的磁共振数据中，怎样快速而有效地恢复目标图像。主要研究内容包括：
（1）为消除亚采样的磁共振成像重构时可能出现的过光滑（over-smoothed）和混叠伪影现象，将重构问题转化成含复合正则项的约束最小化问题，并提出一种高效的算法来求解。该算法首先利用Bregman迭代技术，将约束问题转化成一系列无约束问题。然后利用算子分裂技术，将各无约束问题分解成一个梯度问题和一个能使用修改的SBD（Splitting Bregman Denoising）算法来求解的复合正则项的去噪问题。最后再用加速方案对无约束问题的求解予以加速。本文将该算法称作BFSA（Bregman based Fast SBD Algorithm）。对非笛卡尔轨迹采样的重构，本文还提出了一种动态更新L的方法。实验结果表明，新算法能够获得比其他算法更好的重构质量。
（2）为了克服现有动态磁共振成像重构速度较慢的问题，本文基于BFSA算法框架，提出一种高效的动态磁共振成像重构算法ktBFSA。该算法利用SBD3D（Splitting Bregman Denoising for3D images）来求解含复合正则项的3D去噪问题。实验结果表明，ktBFSA在重构速度和重构质量上都有优势。
（3）SENSE（Sensitivity encoding）是常用的并行磁共振成像技术，引入压缩感知后重构质量可有较大提升。本文针对现有SENSE重构算法速度较慢的问题，基于BFSA算法框架，提出一种快速SENSE重构算法FSRA（Fast SENSEReconstruction Algorithm）。实验结果表明，新算法能极大地减少重构所需时间。自校准方案无需显式使用线圈灵敏度信息，因此避免了SENSE重构中的灵敏度估计的困难。为了提高基于自校准技术框架SPIRiT的重构质量，提出一种高效的算法ERAS（Efcient Reconstruction Algorithm for SPIRiT Based ParallelImaging）。该算法用算子分离算法将重构问题分解成一个梯度计算问题和一个能通过联合软阈值法求解的去噪问题，最后再用加速方案进行加速，并使用动态更新方法更新L。实验结果表明，新算法的重构图像质量好于POCS。
（4）对压缩感知在视频编码中的应用进行了初步研究，提出一种基于压缩感知的改进视频编码方案。该方案基于原始图像的梯度比残差图像的梯度更稀疏这一特点，利用像素域最小全变分法对图像块进行重构，并选择具有较小误差的方法作为最终重构方法。仿真实验表明，将该方案分别与MPEG-2和H.264视频编码标准相结合，可取得一定的编码增益。
The emerging Compressed Sensing (CS) theory suggests that it is likely to recoverthe signals from highly undersampled measurements, if the signal have a sparse repre-sentation under some transform domain and the sparse transform matrix and the sensingmatrix are incoherent. The researchers have introduced the CS theory to the MagneticResonance (MR) imaging, to speed up the imaging without degrading the image qual-ity. My research deals with how to efciently recover the target MR image from highlyundersampled k-space data. The major contribution of this thesis is summarized asfollow:
(1) In order to improve the quality of the reconstructed MR image, which may beover-smoothed or may sufer from aliasing artifacts, the reconstruction problem is for-mulated as a constrained minimization problem with compound regularization terms,which is then solved by an efcient algorithm proposed. In the proposed algorithm, theoriginal constrained problem is reformulated as a sequence of unconstrained problemsby using the Bregman iteration. Then, by using the operator splitting technique, eachunconstrained problem can be decomposed into a gradient problem and a denoisingproblem which can be solved by the Split Bregman Denoising (SBD) algorithm. Then,the solving process of unconstrained problems is accelerated by using an acceleratedscheme. We call the proposed algorithm as BFSA (Bregman based Fast SBD Algo-rithm). For non-Cartesian grid sampling, an updating method is proposed for the stepsize L. Comparisons with the previous algorithms indicate that the proposed algorithmsignificantly improves the quality of the reconstructed images.
(2) In order to speed up the reconstruction of the dynamic MR Imaging (dMRI), anefcient algorithm called ktBFSA is proposed based on the BFSA framework. ktBFSAuses the SBD3D (Split Bregman Denoising for3D images) to solve the3D image de-noising problem with compound regularization terms. Comparisons with the previousalgorithms indicate that the proposed algorithm not only significantly speeds up thereconstruction, but also improves the quality of the reconstructed images.
(3) In order to speed up the Sensitivity encoding (SENSE) reconstruction for par- allel MR imaging, an efcient algorithm called FSRA (Fast SENSE ReconstructionAlgorithm) is proposed based on the BFSA framework. The experimental results showthat the proposed algorithm significantly reduces the reconstruction time. Autocalibrat-ing methods like SPIRiT (Iterative Self-consistent Parallel Imaging Reconstruction)implicitly use the sensitivity information for reconstruction and avoid some of the dif-ficulties associated with explicit estimation of the sensitivities. In order to improve thereconstruction quality, an efcient algorithm called ERAS (Efcient Reconstruction Al-gorithm for SPIRiT Based Parallel Imaging) is proposed. The proposed algorithm usesthe operator splitting technique to decompose the unconstrained problem into a gradientproblem and a denoising problem which can be solved by the joint-sparsity promotingalgorithm. And then the proposed algorithm is accelerated by using an acceleratedscheme. The experimental results show that the proposed algorithm significantly im-proves the reconstruction quality.
(4) In addition, in order to seek for applications of compressed sensing in videocoding and improve the coding efciency of video coding, a compressed sensing basedimproved video coding scheme is proposed. This scheme choose the method produc-ing an image block with smaller sum of squared diference as the final reconstructionmethod between the standard method and the Total Variation (TV) minimization in thepixel domain, which is based on the fact that the original image has sparser gradientthan the residual image. The improved scheme is then integrated into the standardMPEG-2and H.264encoder. The experimental results show that the new encoder canimprove the coding efciency to some degree.
（1）为消除亚采样的磁共振成像重构时可能出现的过光滑（over-smoothed）和混叠伪影现象，将重构问题转化成含复合正则项的约束最小化问题，并提出一种高效的算法来求解。该算法首先利用Bregman迭代技术，将约束问题转化成一系列无约束问题。然后利用算子分裂技术，将各无约束问题分解成一个梯度问题和一个能使用修改的SBD（Splitting Bregman Denoising）算法来求解的复合正则项的去噪问题。最后再用加速方案对无约束问题的求解予以加速。本文将该算法称作BFSA（Bregman based Fast SBD Algorithm）。对非笛卡尔轨迹采样的重构，本文还提出了一种动态更新L的方法。实验结果表明，新算法能够获得比其他算法更好的重构质量。
（2）为了克服现有动态磁共振成像重构速度较慢的问题，本文基于BFSA算法框架，提出一种高效的动态磁共振成像重构算法ktBFSA。该算法利用SBD3D（Splitting Bregman Denoising for3D images）来求解含复合正则项的3D去噪问题。实验结果表明，ktBFSA在重构速度和重构质量上都有优势。
（3）SENSE（Sensitivity encoding）是常用的并行磁共振成像技术，引入压缩感知后重构质量可有较大提升。本文针对现有SENSE重构算法速度较慢的问题，基于BFSA算法框架，提出一种快速SENSE重构算法FSRA（Fast SENSEReconstruction Algorithm）。实验结果表明，新算法能极大地减少重构所需时间。自校准方案无需显式使用线圈灵敏度信息，因此避免了SENSE重构中的灵敏度估计的困难。为了提高基于自校准技术框架SPIRiT的重构质量，提出一种高效的算法ERAS（Efcient Reconstruction Algorithm for SPIRiT Based ParallelImaging）。该算法用算子分离算法将重构问题分解成一个梯度计算问题和一个能通过联合软阈值法求解的去噪问题，最后再用加速方案进行加速，并使用动态更新方法更新L。实验结果表明，新算法的重构图像质量好于POCS。
（4）对压缩感知在视频编码中的应用进行了初步研究，提出一种基于压缩感知的改进视频编码方案。该方案基于原始图像的梯度比残差图像的梯度更稀疏这一特点，利用像素域最小全变分法对图像块进行重构，并选择具有较小误差的方法作为最终重构方法。仿真实验表明，将该方案分别与MPEG-2和H.264视频编码标准相结合，可取得一定的编码增益。
The emerging Compressed Sensing (CS) theory suggests that it is likely to recoverthe signals from highly undersampled measurements, if the signal have a sparse repre-sentation under some transform domain and the sparse transform matrix and the sensingmatrix are incoherent. The researchers have introduced the CS theory to the MagneticResonance (MR) imaging, to speed up the imaging without degrading the image qual-ity. My research deals with how to efciently recover the target MR image from highlyundersampled k-space data. The major contribution of this thesis is summarized asfollow:
(1) In order to improve the quality of the reconstructed MR image, which may beover-smoothed or may sufer from aliasing artifacts, the reconstruction problem is for-mulated as a constrained minimization problem with compound regularization terms,which is then solved by an efcient algorithm proposed. In the proposed algorithm, theoriginal constrained problem is reformulated as a sequence of unconstrained problemsby using the Bregman iteration. Then, by using the operator splitting technique, eachunconstrained problem can be decomposed into a gradient problem and a denoisingproblem which can be solved by the Split Bregman Denoising (SBD) algorithm. Then,the solving process of unconstrained problems is accelerated by using an acceleratedscheme. We call the proposed algorithm as BFSA (Bregman based Fast SBD Algo-rithm). For non-Cartesian grid sampling, an updating method is proposed for the stepsize L. Comparisons with the previous algorithms indicate that the proposed algorithmsignificantly improves the quality of the reconstructed images.
(2) In order to speed up the reconstruction of the dynamic MR Imaging (dMRI), anefcient algorithm called ktBFSA is proposed based on the BFSA framework. ktBFSAuses the SBD3D (Split Bregman Denoising for3D images) to solve the3D image de-noising problem with compound regularization terms. Comparisons with the previousalgorithms indicate that the proposed algorithm not only significantly speeds up thereconstruction, but also improves the quality of the reconstructed images.
(3) In order to speed up the Sensitivity encoding (SENSE) reconstruction for par- allel MR imaging, an efcient algorithm called FSRA (Fast SENSE ReconstructionAlgorithm) is proposed based on the BFSA framework. The experimental results showthat the proposed algorithm significantly reduces the reconstruction time. Autocalibrat-ing methods like SPIRiT (Iterative Self-consistent Parallel Imaging Reconstruction)implicitly use the sensitivity information for reconstruction and avoid some of the dif-ficulties associated with explicit estimation of the sensitivities. In order to improve thereconstruction quality, an efcient algorithm called ERAS (Efcient Reconstruction Al-gorithm for SPIRiT Based Parallel Imaging) is proposed. The proposed algorithm usesthe operator splitting technique to decompose the unconstrained problem into a gradientproblem and a denoising problem which can be solved by the joint-sparsity promotingalgorithm. And then the proposed algorithm is accelerated by using an acceleratedscheme. The experimental results show that the proposed algorithm significantly im-proves the reconstruction quality.
(4) In addition, in order to seek for applications of compressed sensing in videocoding and improve the coding efciency of video coding, a compressed sensing basedimproved video coding scheme is proposed. This scheme choose the method produc-ing an image block with smaller sum of squared diference as the final reconstructionmethod between the standard method and the Total Variation (TV) minimization in thepixel domain, which is based on the fact that the original image has sparser gradientthan the residual image. The improved scheme is then integrated into the standardMPEG-2and H.264encoder. The experimental results show that the new encoder canimprove the coding efciency to some degree.
引文
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