部分K空间采样MR快速重建与相位解缠绕
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摘要
磁共振成像属无创伤、无射线检查,具有高对比分辨率,并能根据需要灵活选择成像参数与成像层面,而且对人体没有电离辐射等伤害。如今,磁共振系统已经成为现代医学影像领域中最先进,最昂贵的诊断设备,并广泛应用于临床,是当前临床医学影像学的重要检查手段之一,正在人类疾病的诊断中发挥着无与伦比的作用。
然而,磁共振成像的最大缺点是数据采集的时间比较长。缩短成像时间不仅可以提高效率和病人的舒适度、减少时间依赖性伪影,还是实现心血管检查、功能信息获取、实时温度检测与介入手术成像等动态成像的关键所在。因此,缩短成像时间一直以来都是磁共振成像技术发展的重要目标之一。
压缩传感理论(Compressed Sensing,CS)是近年来新兴的一个很有意思的研究方向,被誉为信号处理领域的“一个大想法”(A Big Idea)。压缩传感理论指出,利用图像在适当变换域中的稀疏特性,仅利用部分采样数据(采样频率远小于奈奎斯特频率)就可以重建出高质量的图像。因此,基于CS理论就可以设计随机测量矩阵在K空间中稀疏采样,再经非线性重建方法重建出高质量的MR图像。在本文中将基于CS理论的MR快速重建简称为CS-MRI。
目前,CS-MRI的研究工作尚处于起步阶段,很多关键问题亟需解决,主要包括:K空间稀疏采样方式的确定,各种序列图像稀疏性质的表达,图像重建中的约束模型,从医学临床的角度对重建图像质量进行评价,以及相关的快速优化算法等等。
本文对CS的基本理论特别是它的三个基本要素,即稀疏性(Sparsity)、不一致性(Incoherency)和基于稀疏性的重建算法作了透彻的阐述和分析。稀疏性是图像本身的一个内在属性,有的图像本身就是稀疏的(如血管造影图像),而大部分图像经适当的变换(如有限差分变换、小波变换等)后是稀疏的。将图像本身的稀疏性或它在适当变换域中的稀疏性作为先验知识构建重建模型是当前CS-MRI研究中的主流方法。从数学的角度看,由少量数据重建图像是一个病态问题。不一致性是就降采样的方式而言的,CS-MRI要求对K空间数据随机降采样而不是等间隔采样,它是CS理论用于少量数据快速重建的关键。如果对K空间数据随机降采样,那么在重建图像中的混叠伪影将呈现出不一致性(看起来很象噪声)。因此,通过随机降采样,就可以将上述病态问题转化为稀疏信号的去噪问题。稳健的快速优化算法是CS-MRI应用于临床的一个必备要素,本文研究的重点就是设计适用于大规模数据的高效优化算法。
目前,CS-MRI采用的模型主要是含有L_1范数的先验项和数据保真项的L_2范数。由于模型中先验项是不可微的,而数据保真项中的变量不是独立的,因此既不能用传统的梯度类的优化方法求解,也不能给出其闭解形式,而只能用迭代法进行求解。
从稀疏K空间数据重建二维图像的另一个模型是基于图像梯度的稀疏性,即将图像的全变分(Total Variation,TV)最小作为先验构造重建模型。Candes等人用基于全变分最小的模型处理图像恢复问题,并将求解图像恢复模型的最优化问题转化为二阶锥规划问题。他们用对数障碍法(Logarithmic barriermethod)进行数值求解,但是,他们的算法中对数障碍参数(Logarithmic barrierparameter)的选取是非适应的,没有利用迭代过程中图像的任何信息。
在部分K空间采样MR快速重建方面,本文的主要创新点有:
1)在深入研究当前国际上最新算法的基础上,本文将迫近算子的思想用于求解CS-MRI重建模型,给出了一个基于不动点迭代的MR快速重建算法,并证明了该算法的收敛性。该算法的主要优点是不需要解大规模的线性方程组,也不用矩阵分解的方法,算法中仅用到向量的运算以及矩阵与向量的乘法运算。实验结果表明,用本文算法重建的图像质量显著提高,重建时间大大缩短,且具有很好的鲁棒性。
2)对Candes等人的方法作了改进,主要是利用迭代过程中图像的全变分信息自适应地决定对数障碍参数。我们将改进的算法用来求解部分K空间数据的MR重建模型,取得了很好的效果。
呈现在医生面前的MR图像是能量(大小)图像,有经验的影像医生通过这种图像中灰度的分布、对比度等信息就可以准确地进行诊断,制定医疗计划。然而,在成像过程中的相位信息却是一种更为重要的信息,它决定着图像中的结构信息,它可以被用来进行实时温度检测、油水分离、测定血流速度等,基于相位的脂肪抑制技术可以通过抑制脂肪信号,达到去除化学位移伪影,提高图像对比度等目的。另外,近年来刚刚发展起来的磁敏感加权成像技术(Susceptibility Weighted Imaging,SWI)也需要对相位进行精确的校正。
然而,磁共振成像系统能够检测到的相位值都属于区间(-π,π],该范围之外的真实相位值又被缠绕到这个区间内,通常称这种现象为相位缠绕,即我们检测到的相位存在着不同程度的周期模糊。因此,基于相位信息的磁共振新技术迫切需要精确、高效的相位解缠绕算法。
目前,相位解缠绕算法主要可归并为四大类,分别为积分(Integration-Based)法,最小范数(Minimum-Norm)法,基于模型(Model-Based)的方法和基于Bayesian理论的方法。这些方法的共同之处是估计解缠相位图像的梯度场。Bioucas-Dias和Valadao于2007年首次将网络流理论用于相位解缠绕问题,给出基于网络流的相位解缠绕基本算法框架。该算法的优点是收敛速度快,缺点是能量函数不具一般性,在利用网络流理论进行优化时,网络的构造方法不具普适性,而且其模型是在不考虑噪声的前提下构建的,未对相位解缠绕中的去噪问题进行研究。
在相位解缠绕方面,本文提出一新的基于标记理论的相位解缠绕算法。新算法有如下三个方面的创新:
1)为便于用网络流算法求解模型,首次将相位解缠绕看作计算机视觉中的标记问题,给出的能量函数更具一般性;
2)为消除噪声导致的相位解缠绕的不一致性,提出了一种对相位图像进行滤波的新方法;
3)提出了一种新的模糊质量图,以更好地引导相位解缠绕。实验表明,本文算法相位解缠绕精度更高,且计算速度快。
Magnetic Resonance Imaging (MRI) is a non-invasive imaging modality. Unlike Computed Tomography (CT), MRI does not use ionizing radiation. In addition, MRI provides a large number of flexible contrast parameters. These provide excellent soft tissue contrast and spatial resolution. Therefore, MRI has been widely applied in clinics and maybe the most promising non-invasive diagnostic tool in medicine.
However, the main drawback of MRI is the long data acquisition time. Reducing scan time can improve the imaging efficiency, make the patient feel more comfortable and mitigate the time-depending motion artifacts. Moreover, imaging speed is essential to many of the MRI applications such as cardio-vascular examination, acquiring functional information, real time temperature monitoring, dynamic imaging in interventional therapy etc. Therefore, seeking for methods to reduce the scan time is one of the important goals of the MRI development.
Compressed sensing, a big idea in signal processing community, is an interesting new area of research which has gained enormous popularity due to its ability to reconstruct perfect images from a limited number of samples by making advantage of the sparse nature of the image in a proper transform domain. MR image data are often highly redundant, which can be exploited to reduce the amount of acquired data, and hence the scan time. Specifically, we can design a proper measurement matrix and collect data very sparsely in k-space, and then obtain a perfect MR image through nonlinear reconstruction algorithm. The rapid MR reconstruction based on CS theory is called CS-MRI in this paper hereafter.
CS-MRI is still in its infancy. Many crucial issues remain unsettled. These include: optimizing sampling trajectories, expressions of the sparsity in various sequence images, developing improved sparse transforms that are incoherent to the sampling operator, studying reconstruction quality in terms of clinical significance, and improving the speed of reconstruction algorithm.
In this paper we explore some basic elements of compressed sensing: Sparsity, Incoherency and Sparsity based reconstruction algorithm. Many image are inherently sparse. For example, angiograms are extremely sparse in the pixel representation. More complex medical images may not be sparse in the pixel representation, but they do exhibit transform sparsity, since they have a sparse representation in terms of a proper transform (such as spatial finite difference transform, wavelet transform etc.). Sparsity is a powerful constraint and is used as prior knowledge in most of the current CS-MRI models. Mathematically speaking, reconstruction image from significantly few data is an ill-conditioned problem. The theory of CS suggests random undersampling rather than equispaced undersampling. In the random undersampling case, the aliasing artifacts due to k-space undersampling is incoherent (noise like). By random undersampling, we can turn the ill-conditioned problem into a sparse signal denoising problem. To apply CS-MRI in clinical practice, a robust rapid reconstruction algorithm is indispensable. In our work, we focus on designing an efficient algorithm which can deal with large scale data.
The model currently adopted in the CS-MRI includes two terms: a prior term and a data fidelity term. Since the prior term is not differential and the variables in fidelity term are not independent, the model can neither be solved through traditional gradient method nor has a closed form solution.
An alternate model for partial k-space data reconstruction is that the gradient is sparse. Therefore, we can use the total variation as the prior in the reconstruction model. Candes et al. used this model to recover image and obtained perfect result. However, in their algorithm the selection of barrier parameter is non-adaptive. In other words, the information in the process of iteration was not used.
The contributions in our MR reconstruction work mainly include:
1) Based on several recent algorithms, we apply the proximal forward-backward splitting idea into the CS-MRI and propose an efficient algorithm based on fixed point iteration. Furthermore, we give the proof of the algorithm's convergency. The calculation are accelerated by avoiding any linear system solvers or matrix factorizations~we restrict ourselves to vector operations and matrix-vector multiplications. Experiments show that faithful MR images can be reconstructed efficiently through our algorithm and our algorithm is robust and convergent.
2) We use the total variation of the reconstructed images in the process of iteration to adaptively define the barrier parameter. We apply this improved algorithm into partial k-space sampling rapid MR reconstruction and obtain perfect result.
Generally speaking, the MR image is the magnitude image. Based on the intensity distribution and tissue contrast, an experienced doctor can diagnose exactly and make a proper therapy plan. However, phase is an another important information in the process of imaging. Phase images can be used to monitor temperature changes during hyperthermic ablation procedure, to carry out the water/fat separation work, to measure the flow velocity, to eliminate chemical shift artifacts, to produce SWI image etc.
However, the observed phase data are wrapped principal values, which are restricted in a 2 n modulus, and they must be unwrapped to their true absolute phase values. Phase unwrapping is the process of recovering the absolute phase from the wrapped phase. Obviously, the new MRI applications based on phase information require exact and efficient phase unwrapping algorithm.
Phase unwrapping approaches belong mainly to one of the following categories: integration-based method, minimum norm, model-based method and Bayesian regularization. The common of these methods is to estimate the wrapped gradient field. Bioucas-Dias and Valadao proposed a new energy minimization framework based on network flow theory in 2007. However, they did not tackle the noise in the phase image. Furthermore, the energy function in their framework is too specific to be minimized conveniently using the network flow theory.
The main contributions in our phase unwrapping work are as following:
1) We regard the phase unwrapping problem as a labeling problem in computer vision. The corresponding energy function can be conveniently minimized by an efficient network flow algorithm.
2) To eliminate the residuals caused by noise, a new phase filtering method is proposed to reduce the noise in the phase data.
3) We propose a new fuzzy gradient field as quality map to guide the unwrapping process.
The proposed method has been tested with experimental data, yielding better results than some of the state-of-the-art methods.
然而,磁共振成像的最大缺点是数据采集的时间比较长。缩短成像时间不仅可以提高效率和病人的舒适度、减少时间依赖性伪影,还是实现心血管检查、功能信息获取、实时温度检测与介入手术成像等动态成像的关键所在。因此,缩短成像时间一直以来都是磁共振成像技术发展的重要目标之一。
压缩传感理论(Compressed Sensing,CS)是近年来新兴的一个很有意思的研究方向,被誉为信号处理领域的“一个大想法”(A Big Idea)。压缩传感理论指出,利用图像在适当变换域中的稀疏特性,仅利用部分采样数据(采样频率远小于奈奎斯特频率)就可以重建出高质量的图像。因此,基于CS理论就可以设计随机测量矩阵在K空间中稀疏采样,再经非线性重建方法重建出高质量的MR图像。在本文中将基于CS理论的MR快速重建简称为CS-MRI。
目前,CS-MRI的研究工作尚处于起步阶段,很多关键问题亟需解决,主要包括:K空间稀疏采样方式的确定,各种序列图像稀疏性质的表达,图像重建中的约束模型,从医学临床的角度对重建图像质量进行评价,以及相关的快速优化算法等等。
本文对CS的基本理论特别是它的三个基本要素,即稀疏性(Sparsity)、不一致性(Incoherency)和基于稀疏性的重建算法作了透彻的阐述和分析。稀疏性是图像本身的一个内在属性,有的图像本身就是稀疏的(如血管造影图像),而大部分图像经适当的变换(如有限差分变换、小波变换等)后是稀疏的。将图像本身的稀疏性或它在适当变换域中的稀疏性作为先验知识构建重建模型是当前CS-MRI研究中的主流方法。从数学的角度看,由少量数据重建图像是一个病态问题。不一致性是就降采样的方式而言的,CS-MRI要求对K空间数据随机降采样而不是等间隔采样,它是CS理论用于少量数据快速重建的关键。如果对K空间数据随机降采样,那么在重建图像中的混叠伪影将呈现出不一致性(看起来很象噪声)。因此,通过随机降采样,就可以将上述病态问题转化为稀疏信号的去噪问题。稳健的快速优化算法是CS-MRI应用于临床的一个必备要素,本文研究的重点就是设计适用于大规模数据的高效优化算法。
目前,CS-MRI采用的模型主要是含有L_1范数的先验项和数据保真项的L_2范数。由于模型中先验项是不可微的,而数据保真项中的变量不是独立的,因此既不能用传统的梯度类的优化方法求解,也不能给出其闭解形式,而只能用迭代法进行求解。
从稀疏K空间数据重建二维图像的另一个模型是基于图像梯度的稀疏性,即将图像的全变分(Total Variation,TV)最小作为先验构造重建模型。Candes等人用基于全变分最小的模型处理图像恢复问题,并将求解图像恢复模型的最优化问题转化为二阶锥规划问题。他们用对数障碍法(Logarithmic barriermethod)进行数值求解,但是,他们的算法中对数障碍参数(Logarithmic barrierparameter)的选取是非适应的,没有利用迭代过程中图像的任何信息。
在部分K空间采样MR快速重建方面,本文的主要创新点有:
1)在深入研究当前国际上最新算法的基础上,本文将迫近算子的思想用于求解CS-MRI重建模型,给出了一个基于不动点迭代的MR快速重建算法,并证明了该算法的收敛性。该算法的主要优点是不需要解大规模的线性方程组,也不用矩阵分解的方法,算法中仅用到向量的运算以及矩阵与向量的乘法运算。实验结果表明,用本文算法重建的图像质量显著提高,重建时间大大缩短,且具有很好的鲁棒性。
2)对Candes等人的方法作了改进,主要是利用迭代过程中图像的全变分信息自适应地决定对数障碍参数。我们将改进的算法用来求解部分K空间数据的MR重建模型,取得了很好的效果。
呈现在医生面前的MR图像是能量(大小)图像,有经验的影像医生通过这种图像中灰度的分布、对比度等信息就可以准确地进行诊断,制定医疗计划。然而,在成像过程中的相位信息却是一种更为重要的信息,它决定着图像中的结构信息,它可以被用来进行实时温度检测、油水分离、测定血流速度等,基于相位的脂肪抑制技术可以通过抑制脂肪信号,达到去除化学位移伪影,提高图像对比度等目的。另外,近年来刚刚发展起来的磁敏感加权成像技术(Susceptibility Weighted Imaging,SWI)也需要对相位进行精确的校正。
然而,磁共振成像系统能够检测到的相位值都属于区间(-π,π],该范围之外的真实相位值又被缠绕到这个区间内,通常称这种现象为相位缠绕,即我们检测到的相位存在着不同程度的周期模糊。因此,基于相位信息的磁共振新技术迫切需要精确、高效的相位解缠绕算法。
目前,相位解缠绕算法主要可归并为四大类,分别为积分(Integration-Based)法,最小范数(Minimum-Norm)法,基于模型(Model-Based)的方法和基于Bayesian理论的方法。这些方法的共同之处是估计解缠相位图像的梯度场。Bioucas-Dias和Valadao于2007年首次将网络流理论用于相位解缠绕问题,给出基于网络流的相位解缠绕基本算法框架。该算法的优点是收敛速度快,缺点是能量函数不具一般性,在利用网络流理论进行优化时,网络的构造方法不具普适性,而且其模型是在不考虑噪声的前提下构建的,未对相位解缠绕中的去噪问题进行研究。
在相位解缠绕方面,本文提出一新的基于标记理论的相位解缠绕算法。新算法有如下三个方面的创新:
1)为便于用网络流算法求解模型,首次将相位解缠绕看作计算机视觉中的标记问题,给出的能量函数更具一般性;
2)为消除噪声导致的相位解缠绕的不一致性,提出了一种对相位图像进行滤波的新方法;
3)提出了一种新的模糊质量图,以更好地引导相位解缠绕。实验表明,本文算法相位解缠绕精度更高,且计算速度快。
Magnetic Resonance Imaging (MRI) is a non-invasive imaging modality. Unlike Computed Tomography (CT), MRI does not use ionizing radiation. In addition, MRI provides a large number of flexible contrast parameters. These provide excellent soft tissue contrast and spatial resolution. Therefore, MRI has been widely applied in clinics and maybe the most promising non-invasive diagnostic tool in medicine.
However, the main drawback of MRI is the long data acquisition time. Reducing scan time can improve the imaging efficiency, make the patient feel more comfortable and mitigate the time-depending motion artifacts. Moreover, imaging speed is essential to many of the MRI applications such as cardio-vascular examination, acquiring functional information, real time temperature monitoring, dynamic imaging in interventional therapy etc. Therefore, seeking for methods to reduce the scan time is one of the important goals of the MRI development.
Compressed sensing, a big idea in signal processing community, is an interesting new area of research which has gained enormous popularity due to its ability to reconstruct perfect images from a limited number of samples by making advantage of the sparse nature of the image in a proper transform domain. MR image data are often highly redundant, which can be exploited to reduce the amount of acquired data, and hence the scan time. Specifically, we can design a proper measurement matrix and collect data very sparsely in k-space, and then obtain a perfect MR image through nonlinear reconstruction algorithm. The rapid MR reconstruction based on CS theory is called CS-MRI in this paper hereafter.
CS-MRI is still in its infancy. Many crucial issues remain unsettled. These include: optimizing sampling trajectories, expressions of the sparsity in various sequence images, developing improved sparse transforms that are incoherent to the sampling operator, studying reconstruction quality in terms of clinical significance, and improving the speed of reconstruction algorithm.
In this paper we explore some basic elements of compressed sensing: Sparsity, Incoherency and Sparsity based reconstruction algorithm. Many image are inherently sparse. For example, angiograms are extremely sparse in the pixel representation. More complex medical images may not be sparse in the pixel representation, but they do exhibit transform sparsity, since they have a sparse representation in terms of a proper transform (such as spatial finite difference transform, wavelet transform etc.). Sparsity is a powerful constraint and is used as prior knowledge in most of the current CS-MRI models. Mathematically speaking, reconstruction image from significantly few data is an ill-conditioned problem. The theory of CS suggests random undersampling rather than equispaced undersampling. In the random undersampling case, the aliasing artifacts due to k-space undersampling is incoherent (noise like). By random undersampling, we can turn the ill-conditioned problem into a sparse signal denoising problem. To apply CS-MRI in clinical practice, a robust rapid reconstruction algorithm is indispensable. In our work, we focus on designing an efficient algorithm which can deal with large scale data.
The model currently adopted in the CS-MRI includes two terms: a prior term and a data fidelity term. Since the prior term is not differential and the variables in fidelity term are not independent, the model can neither be solved through traditional gradient method nor has a closed form solution.
An alternate model for partial k-space data reconstruction is that the gradient is sparse. Therefore, we can use the total variation as the prior in the reconstruction model. Candes et al. used this model to recover image and obtained perfect result. However, in their algorithm the selection of barrier parameter is non-adaptive. In other words, the information in the process of iteration was not used.
The contributions in our MR reconstruction work mainly include:
1) Based on several recent algorithms, we apply the proximal forward-backward splitting idea into the CS-MRI and propose an efficient algorithm based on fixed point iteration. Furthermore, we give the proof of the algorithm's convergency. The calculation are accelerated by avoiding any linear system solvers or matrix factorizations~we restrict ourselves to vector operations and matrix-vector multiplications. Experiments show that faithful MR images can be reconstructed efficiently through our algorithm and our algorithm is robust and convergent.
2) We use the total variation of the reconstructed images in the process of iteration to adaptively define the barrier parameter. We apply this improved algorithm into partial k-space sampling rapid MR reconstruction and obtain perfect result.
Generally speaking, the MR image is the magnitude image. Based on the intensity distribution and tissue contrast, an experienced doctor can diagnose exactly and make a proper therapy plan. However, phase is an another important information in the process of imaging. Phase images can be used to monitor temperature changes during hyperthermic ablation procedure, to carry out the water/fat separation work, to measure the flow velocity, to eliminate chemical shift artifacts, to produce SWI image etc.
However, the observed phase data are wrapped principal values, which are restricted in a 2 n modulus, and they must be unwrapped to their true absolute phase values. Phase unwrapping is the process of recovering the absolute phase from the wrapped phase. Obviously, the new MRI applications based on phase information require exact and efficient phase unwrapping algorithm.
Phase unwrapping approaches belong mainly to one of the following categories: integration-based method, minimum norm, model-based method and Bayesian regularization. The common of these methods is to estimate the wrapped gradient field. Bioucas-Dias and Valadao proposed a new energy minimization framework based on network flow theory in 2007. However, they did not tackle the noise in the phase image. Furthermore, the energy function in their framework is too specific to be minimized conveniently using the network flow theory.
The main contributions in our phase unwrapping work are as following:
1) We regard the phase unwrapping problem as a labeling problem in computer vision. The corresponding energy function can be conveniently minimized by an efficient network flow algorithm.
2) To eliminate the residuals caused by noise, a new phase filtering method is proposed to reduce the noise in the phase data.
3) We propose a new fuzzy gradient field as quality map to guide the unwrapping process.
The proposed method has been tested with experimental data, yielding better results than some of the state-of-the-art methods.
引文
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