Nuclear norm-regularized SENSE reconstruction

SENSitivity Encoding (SENSE) is a mathematically optimal parallel magnetic resonance (MRI) imaging technique when the coil sensitivities are known. In recent times, compressed sensing (CS)-based techniques are incorporated within the SENSE reconstruction framework to recover the underlying MR image. CS-based techniques exploit the fact that the MR images are sparse in a transform domain (e.g., wavelets). Mathematically, this leads to an l(1)-norm-regularized SENSE reconstruction. In this work, we show that instead of reconstructing the image by exploiting its transform domain sparsity, we can exploit its rank deficiency to reconstruct it. This leads to a nuclear norm-regularized SENSE problem. The reconstruction accuracy from our proposed method is the same as the l(1)-norm-regularized SENSE, but the advantage of our method is that it is about an order of magnitude faster.     

An algorithm for sparse MRI reconstruction by Schatten p-norm minimization

In recent years, there has been a concerted effort to reduce the MR scan time. Signal processing research aims at reducing the scan time by acquiring less K-space data. The image is reconstructed from the subsampled K-space data by employing compressed sensing (CS)-based reconstruction techniques. In this article, we propose an alternative approach to CS-based reconstruction. The proposed approach exploits the rank deficiency of the MR images to reconstruct the image. This requires minimizing the rank of the image matrix subject to data constraints, which is unfortunately a nondeterministic polynomial time (NP) hard problem. Therefore we propose to replace the NP hard rank minimization problem by its nonconvex surrogate — Schatten p-norm minimization. The same approach can be used for denoising MR images as well.Since there is no algorithm to solve the Schatten p-norm minimization problem, we derive an efficient first-order algorithm. Experiments on MR brain scans show that the reconstruction and denoising accuracy from our method is at par with that of CS-based methods. Our proposed method is considerably faster than CS-based methods.     

Exploiting the wavelet structure in compressed sensing MRI

Sparsity has been widely utilized in magnetic resonance imaging (MRI) to reduce k-space sampling. According to structured sparsity theories, fewer measurements are required for tree sparse data than the data only with standard sparsity. Intuitively, more accurate image reconstruction can be achieved with the same number of measurements by exploiting the wavelet tree structure in MRI. A novel algorithm is proposed in this article to reconstruct MR images from undersampled k-space data. In contrast to conventional compressed sensing MRI (CS-MRI) that only relies on the sparsity of MR images in wavelet or gradient domain, we exploit the wavelet tree structure to improve CS-MRI. This tree-based CS-MRI problem is decomposed into three simpler subproblems then each of the subproblems can be efficiently solved by an iterative scheme. Simulations and in vivo experiments demonstrate the significant improvement of the proposed method compared to conventional CS-MRI algorithms, and the feasibleness on MR data compared to existing tree-based imaging algorithms.     

Joint reconstruction of multiecho MR images using correlated sparsity

This works addresses the problem of reconstructing multiple T1- or T2-weighted images of the same anatomical cross section from partially sampled K-space data. Previous studies in reconstructing magnetic resonance (MR) images from partial samples of the K-space used compressed sensing (CS) techniques to exploit the spatial correlation of the images (leading to sparsity in wavelet domain). Such techniques can be employed to reconstruct the individual T1- or T2-weighted images. However, in the current context, the different images are not really independent; they are images of the same cross section and, hence, are highly correlated. We exploit the correlation between the images, along with the spatial correlation within the images to achieve better reconstruction results than exploiting spatial correlation only.For individual MR images, CS-based techniques lead to a sparsity-promoting optimization problem in the wavelet domain. In this article, we show that the same framework can be extended to incorporate correlation between images leading to group/row sparsity-promoting optimization. Algorithms for solving such optimization problems have already been developed in the CS literature. We show that significant improvement in reconstruction accuracy can be achieved by considering the correlation between different T1- and T2-weighted images. If the reconstruction accuracy is considered to be constant, our proposed group sparse formulation can yield the same result with 33% less K-space samples compared with simple sparsity-promoting reconstruction. Moreover, the reconstruction time by our proposed method is about two to four times less than the previous method.     

Singular Value Decomposition Using Jacobi Algorithm in pMRI and CS

Parallel magnetic resonance imaging (pMRI) and compressed sensing (CS) have been recently used to accelerate data acquisition process in MRI. Matrix inversion (for rectangular matrices) is required to reconstruct images from the acquired under-sampled data in various pMRI algorithms (e.g., SENSE, GRAPPA) and CS. Singular value decomposition (SVD) provides a mechanism to accurately estimate pseudo-inverse of a rectangular matrix. This work proposes the use of Jacobi SVD algorithm to reconstruct MR images from the acquired under-sampled data both in pMRI and in CS. The use of Jacobi SVD algorithm is proposed in advance MRI reconstruction algorithms, including SENSE, GRAPPA, and low-rank matrix estimation in L + S model for matrix inversion and estimation of singular values. Experiments are performed on 1.5T human head MRI data and 3T cardiac perfusion MRI data for different acceleration factors. The reconstructed images are analyzed using artifact power and central line profiles. The results show that the Jacobi SVD algorithm successfully reconstructs the images in SENSE, GRAPPA, and L + S algorithms. The benefit of using Jacobi SVD algorithm for MRI image reconstruction is its suitability for parallel computation on GPUs, which may be a great help in reducing the image reconstruction time.     

Visually weighted reconstruction of compressive sensing MRI

Compressive sensing (CS) enables the reconstruction of a magnetic resonance (MR) image from undersampled data in k-space with relatively low-quality distortion when compared to the original image. In addition, CS allows the scan time to be significantly reduced. Along with a reduction in the computational overhead, we investigate an effective way to improve visual quality through the use of a weighted optimization algorithm for reconstruction after variable density random undersampling in the phase encoding direction over k-space. In contrast to conventional magnetic resonance imaging (MRI) reconstruction methods, the visual weight, in particular, the region of interest (ROI), is investigated here for quality improvement. In addition, we employ a wavelet transform to analyze the reconstructed image in the space domain and fully utilize data sparsity over the spatial and frequency domains. The visual weight is constructed by reflecting the perceptual characteristics of the human visual system (HVS), and then applied to ?₁ norm minimization, which gives priority to each coefficient during the reconstruction process. Using objective quality assessment metrics, it was found that an image reconstructed using the visual weight has higher local and global quality than those processed by conventional methods.     

Dynamic MRI reconstruction from highly undersampled (k,t)-space data using weighted Schatten p-norm regularizer of tensor

Combination of the low-rankness and sparsity has been successfully used to reconstruct desired dynamic magnetic resonance image (MRI) from highly-undersampled (k, t)-space data. However, nuclear norm, as a convex relaxation of the rank function, can cause the solution deviating from the original solution of low-rank problem. Moreover, equally treating different rank component is not flexible to deal with real applications. In this paper, an efficient reconstruction model is proposed to efficiently reconstruct dynamic MRI. First, we treat dynamic MRI as a 3rd-order tensor, and formulate the low-rankness via non-convex Schatten p-norm of matrices unfolded from the tensor. Secondly, we assign different weight for each rank component in Schatten p-norm. Furthermore, we combine the proposed weighted Schatten p-norm of a tensor as low-rank regularizer, and spatiotemporal total variation as sparse regularizer to formulate the reconstruction model for dynamic MRI. Thirdly, to efficiently solve the formulated reconstruction model, we derive an algorithm based on Bregman iterations with alternating direction multiplier. Over two public data sets of dynamic MRI, experiments demonstrate that the proposed method achieves much better quality.     

Fast multi-contrast MRI reconstruction

Multi-contrast magnetic resonance imaging (MRI) is a useful technique to aid clinical diagnosis. This paper proposes an efficient algorithm to jointly reconstruct multiple T1/T2-weighted images of the same anatomical cross section from partially sampled k-space data. The joint reconstruction problem is formulated as minimizing a linear combination of three terms, corresponding to a least squares data fitting, joint total variation (TV) and group wavelet-sparsity regularization. It is rooted in two observations: 1) the variance of image gradients should be similar for the same spatial position across multiple contrasts; 2) the wavelet coefficients of all images from the same anatomical cross section should have similar sparse modes. To efficiently solve this problem, we decompose it into joint TV regularization and group sparsity subproblems, respectively. Finally, the reconstructed image is obtained from the weighted average of solutions from the two subproblems, in an iterative framework. Experiments demonstrate the efficiency and effectiveness of the proposed method compared to existing multi-contrast MRI methods.     

Image registration guided,sparsity constrained reconstructions for dynamic MRI

It is generally a challenging task to reconstruct dynamic magnetic resonance (MR) images with high spatial and high temporal resolutions, especially with highly incomplete k-space sampling. In this work, a novel method that combines a non-rigid image registration technique with sparsity-constrained image reconstruction is introduced. Employing a multi-resolution free-form deformation technique with B-spline interpolations, the non-rigid image registration accurately models the complex deformations of the physiological dynamics, and provides artifact-suppressed high spatial-resolution predictions. Based on these prediction images, the sparsity-constrained data fidelity-enforced image reconstruction further improves the reconstruction accuracy. When compared with the k-t FOCUSS with motion estimation/motion compensation (MEMC) technique on volunteer scans, the proposed method consistently outperforms in both the spatial and the temporal accuracy with variously accelerated k-space sampling. High fidelity reconstructions for dynamic systolic phases with reduction factor of 10 and cardiac perfusion series with reduction factor of 3 are presented.     

Accelerated magnetic resonance imaging using the sparsity of multi-channel coil images

Joint estimation of coil sensitivities and output image (JSENSE) is a promising approach that improves the reconstruction of parallel magnetic resonance imaging (pMRI). However, when acceleration factor increases, the signal to noise ratio (SNR) of JSENSE reconstruction decreases as quickly as that of the conventional pMRI. Although sparse constraints have been used to improve the JSENSE reconstruction in recent years, these constraints only use the sparsity of the output image, which cannot fully exploit the prior information of pMRI. In this paper, we use the sparsity of coil images, instead of the output image, to exploit more prior information for JSENSE. Numerical simulation, phantom and in vivo experiments demonstrate that the proposed method has better performance than the SparseSENSE method and the constrained JSENSE method using the sparsity of the output image only.     

Incorporating prior knowledge via volumetric deep residual network to optimize the reconstruction of sparsely sampled MRI

For sparse sampling that accelerates magnetic resonance (MR) image acquisition, non-linear reconstruction algorithms have been developed, which incorporated patient specific a prior information. More generic a prior information could be acquired via deep learning and utilized for image reconstruction. In this study, we developed a volumetric hierarchical deep residual convolutional neural network, referred to as T-Net, to provide a data-driven end-to-end mapping from sparsely sampled MR images to fully sampled MR images, where cartilage MR images were acquired using an Ultra-short TE sequence and retrospectively undersampled using pseudo-random Cartesian and radial acquisition schemes. The network had a hierarchical architecture that promoted the sparsity of feature maps and increased the receptive field, which were valuable for signal synthesis and artifact suppression. Relatively dense local connections and global shortcuts were established to facilitate residual learning and compensate for details lost in hierarchical processing. Additionally, volumetric processing was adopted to fully exploit spatial continuity in three-dimensional space. Data consistency was further enforced. The network was trained with 336 three-dimensional images (each consisting of 32 slices) and tested by 24 images. The incorporation of a priori information acquired via deep learning facilitated high acceleration factors (as high as 8) while maintaining high image fidelity (quantitatively evaluated using the structural similarity index measurement). The proposed T-Net had an improved performance as compared to several state-of-the-art networks.     

Smoothly Clipped Absolute Deviation (SCAD) regularization for compressed sensing MRI Using an augmented Lagrangian scheme

PurposeCompressed sensing (CS) provides a promising framework for MR image reconstruction from highly undersampled data, thus reducing data acquisition time. In this context, sparsity-promoting regularization techniques exploit the prior knowledge that MR images are sparse or compressible in a given transform domain. In this work, a new regularization technique was introduced by iterative linearization of the non-convex smoothly clipped absolute deviation (SCAD) norm with the aim of reducing the sampling rate even lower than it is required by the conventional l₁ norm while approaching an l₀ norm.Materials and MethodsThe CS-MR image reconstruction was formulated as an equality-constrained optimization problem using a variable splitting technique and solved using an augmented Lagrangian (AL) method developed to accelerate the optimization of constrained problems. The performance of the resulting SCAD-based algorithm was evaluated for discrete gradients and wavelet sparsifying transforms and compared with its l₁-based counterpart using phantom and clinical studies. The k-spaces of the datasets were retrospectively undersampled using different sampling trajectories. In the AL framework, the CS-MRI problem was decomposed into two simpler sub-problems, wherein the linearization of the SCAD norm resulted in an adaptively weighted soft thresholding rule with a sparsity enhancing effect.ResultsIt was demonstrated that the proposed regularization technique adaptively assigns lower weights on the thresholding of gradient fields and wavelet coefficients, and as such, is more efficient in reducing aliasing artifacts arising from k-space undersampling, when compared to its l₁-based counterpart.ConclusionThe SCAD regularization improves the performance of l₁-based regularization technique, especially at reduced sampling rates, and thus might be a good candidate for some applications in CS-MRI.     

Improved dynamic MRI reconstruction by exploiting sparsity and rank-deficiency

In this paper we address the problem of dynamic MRI reconstruction from partially sampled K-space data. Our work is motivated by previous studies in this area that proposed exploiting the spatiotemporal correlation of the dynamic MRI sequence by posing the reconstruction problem as a least squares minimization regularized by sparsity and low-rank penalties. Ideally the sparsity and low-rank penalties should be represented by the l₀-norm and the rank of a matrix; however both are NP hard penalties. The previous studies used the convex l₁-norm as a surrogate for the l₀-norm and the non-convex Schatten-q norm (0 < q ≤ 1) as a surrogate for the rank of matrix. Following past research in sparse recovery, we know that non-convex l_p-norm (0 < p ≤ 1) is a better substitute for the NP hard l₀-norm than the convex l₁-norm. Motivated by these studies, we propose improvements over the previous studies by replacing the l₁-norm sparsity penalty by the l_p-norm. Thus, we reconstruct the dynamic MRI sequence by solving a least squares minimization problem regularized by l_p-norm as the sparsity penalty and Schatten-q norm as the low-rank penalty. There are no efficient algorithms to solve the said problems. In this paper, we derive efficient algorithms to solve them. The experiments have been carried out on Dynamic Contrast Enhanced (DCE) MRI datasets. Both quantitative and qualitative analysis indicates the superiority of our proposed improvement over the existing methods.     

Remarks on q-space MR propagator in partially restricted,axially-symmetric,and isotropic environments

The problem of reconstruction of an apparent propagator from a series of diffusion-attenuated magnetic resonance (MR) signals is revisited. In nonimaging acquisitions, the inverse Fourier transform of the MR signal attenuation is consistent with the notion of an ensemble average propagator. However, in image acquisitions where one is interested in quantifying a displacement distribution in every voxel of the image, the propagator derived in the traditional way may lead to a counter-intuitive profile when it is nonsymmetric, which could be a problem in partially restricted environments. By exploiting the reciprocity of the diffusion propagator, an alternative is introduced, which implies a forward Fourier transform of the MR signal attenuations yielding a propagator reflected around the origin. Two simple problems were considered as examples. In the case of diffusion in the proximity of a restricting barrier, the reflected propagator yields a more meaningful result, whereas in the case of curving fibers, the original propagator is more intuitive. In the final section of the article, two more one-dimensional transformations are introduced, which enable the reconstruction of two- and three-dimensional propagators in, respectively, axially symmetric and isotropic environments — in both cases, from one-dimensional q-space MR data.     

Compressed Sensing MRI Using Sparsity Averaging and FISTA

Magnetic resonance imaging (MRI) is widely adopted for clinical diagnosis due to its non-invasively detection. However, acquisition of full k-space data limits its imaging speed. Compressed sensing (CS) provides a new technique to significantly reduce the measurements with high-quality MR image reconstruction. The sparsity of the MR images is one of the crucial bases of CS-MRI. In this paper, we present to use sparsity averaging prior for CS-MRI reconstruction in the basis of that MR images have average sparsity over multiple wavelet frames. The problem is solved using a Fast Iterative Shrinkage Thresholding Algorithm (FISTA), each iteration of which includes a shrinkage step. The performance of the proposed method is evaluated for several types of MR images. The experiment results illustrate that our approach exhibits a better performance than those methods that using redundant frame or a single orthonormal basis to promote sparsity.     

Motion correction based reconstruction method for compressively sampled cardiac MR imaging

Respiratory motion during Magnetic Resonance (MR) acquisition causes strong blurring artifacts in the reconstructed images. These artifacts become more pronounced when used with the fast imaging reconstruction techniques like compressed sensing (CS). Recently, an MR reconstruction technique has been done with the help of compressed sensing (CS), to provide good quality sparse images from the highly under-sampled k-space data. In order to maximize the benefits of CS, it is obvious to use CS with the motion corrected samples. In this paper, we propose a novel CS based motion corrected image reconstruction technique. First, k-space data have been assigned to different respiratory state with the help of frequency domain phase correlation method. Then, multiple sparsity constraints has been used to provide good quality reconstructed cardiac cine images with the highly under-sampled k-space data. The proposed method exploits the multiple sparsity constraints, in combination with demon based registration technique and a novel reconstruction technique to provide the final motion free images. The proposed method is very simple to implement in clinical settings as compared to existing motion corrected methods. The performance of the proposed method is examined using simulated data and clinical data. Results show that this method performs better than the reconstruction of CS based method of cardiac cine images. Different acceleration rates have been used to show the performance of the proposed method.     

总变差约束的数据分离最小图像重建模型及其Chambolle-Pock求解算法

基于优化的迭代法,可以结合压缩感知和低秩矩阵等稀疏优化技术高精度地重建图像.其中,总变差最小（total variation minimization,TV）模型是一种简单有效的优化模型.传统的约束TV模型,使用数据保真项为约束项,TV正则项为目标函数.本文研究TV约束的、数据分离最小（TV constrained,data divergence minimization,TVcDM）新型TV模型及其求解算法.详细推导了TVcDM模型的Chambolle-Pock（CP）算法,验证了模型及算法的正确性;分析了算法的收敛行为;评估了模型的稀疏重建能力;分析了模型参数的选择对重建的影响及算法参数对收敛速率的影响.研究表明,TVcDM模型有高精度稀疏重建能力;TVcDM-CP算法确保收敛,但迭代过程中有振荡现象;TV限对重建有重要影响,参数值过大会引入噪声而过小会模糊图像细节;算法参数的不同选取会导致不同的收敛速率.     

选择性双向顺序压缩感知重建动态磁共振成像

压缩感知是一种新兴技术,该技术能够用远低于奈奎斯特采样频率采集的信号恢复出原始信号. 压缩感知成像方法大大提高了心脏磁共振成像的采集速度,已有的方法主要利用动态图像时间相关及心脏的周期性运动特征,如采用在时间维做傅立叶变换或求解每帧数据跟参考帧数据的差异获取稀疏数据,满足压缩感知重建的要求. 该文提出了选择性双向顺序压缩感知重建算法,利用相邻帧的差异更小的特点,获取更加稀疏的差异数据,同时利用动态图像的周期性,以目标函数积分为判据,在时间顺序和时间逆序两个方向选择效果更好的方向进行数据重建,降低图像伪影和噪声. 该选择算法,可以在不增加重建时间的情况下,选择双向顺序重建中最佳的结果. 该文对心脏磁共振图像数据进行了数据处理实验,并且跟传统压缩感知算法、参考帧差异方法及匙孔成像方法进行了比较. 结果表明：该方法无论从视觉效果还是从统计结果上,都有很大的改善.     

Adaptive fixed-point iterative shrinkage/thresholding algorithm for MR imaging reconstruction using compressed sensing

Recently compressed sensing (CS) has been applied to under-sampling MR image reconstruction for significantly reducing signal acquisition time. To guarantee the accuracy and efficiency of the CS-based MR image reconstruction, it necessitates determining several regularization and algorithm-introduced parameters properly in practical implementations. The regularization parameter is used to control the trade-off between the sparsity of MR image and the fidelity measures of k-space data, and thus has an important effect on the reconstructed image quality. The algorithm-introduced parameters determine the global convergence rate of the algorithm itself. These parameters make CS-based MR image reconstruction a more difficult scheme than traditional Fourier-based method while implemented on a clinical MR scanner. In this paper, we propose a new approach that reveals that the regularization parameter can be taken as a threshold in a fixed-point iterative shrinkage/thresholding algorithm (FPIST) and chosen by employing minimax threshold selection method. No extra parameter is introduced by FPIST. The simulation results on synthetic and real complex-valued MRI data show that the proposed method can adaptively choose the regularization parameter and effectively achieve high reconstruction quality. The proposed method should prove very useful for practical CS-based MRI applications.     

Referenceless PRFS MR Thermometry Using Partial Separability Model

Many areas of magnetic resonance (MR)-guided thermal therapy research would benefit from temperature maps with high spatial and temporal resolution. Conventional thermometry relies on the subtraction of baseline images, which makes it sensitive to tissue motion and frequency drift during the course of treatment. For another case is the limit of magnetic resonance imaging sampling speed, it is hard to accurately achieve MR thermometry with high spatiotemporal resolution especially for dynamic organs. To address these issues, a novel method for MR thermometry is presented by exploiting the data redundancy based on partial separability (PS) model and the referenceless thermometry. The PS model highly sparse sample two datasets in the (k–t) space for image reconstruction, which respectively determine the spatial and temporal resolutions. After the phase information is extracted from the images reconstructed by the PS model, the background phase outside the heated region from each acquired phase image through a polynomial fitting is estimated. Extrapolation of the polynomial to the heated region serves as the background phase estimate, which is then subtracted from the actual phase. The thermometry results showed that this method could accurately capture the dynamic change of MR thermometric images with 1.5 mm × 1.5 mm spatial resolution and 250 ms temporal resolution, respectively. The in vivo experiment of MR-guided high intensity focused ultrasound research and the cardiac dynamic MR thermometry are shown to demonstrate the benefits of the proposed method in high spatiotemporal resolution MR thermometry.