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.     

基于L1范数和正交梯度算子的超分辨率重建

针对超分辨率图像重建的病态问题,设计了一种新的自适应超分辨率图像序列重建算法。该算法在L1范数重建框架下,利用金字塔算法与Lucas-Kanade算法相结合的方法实现图像配准,获得亚像素的运动估计;通过引入移位算子给出了基于正交梯度算子的正则项的实现方法,并从自适应的角度选择正则化参数,最后通过最速下降法求解模型的目标泛函最小值。结果表明:对于模拟实验和真实序列实验,该方法相比于样条插值算法、Tikhonov正则化算法、双边全变差重建算法都有一定的优势,能够取得更好的复原效果,并且由于正则项较为简单,重建所需时间相对减少。     

Exploiting rank deficiency and transform domain sparsity for MR image reconstruction

The reconstruction of magnetic resonance (MR) images from the partial samples of their k-space data using compressed sensing (CS)-based methods has generated a lot of interest in recent years. To reconstruct the MR images, these techniques exploit the sparsity of the image in a transform domain (wavelets, total variation, etc.). In a recent work, it has been shown that it is also possible to reconstruct MR images by exploiting their rank deficiency. In this work, it will be shown that, instead of exploiting the sparsity of the image or rank deficiency alone, better reconstruction results can be achieved by combining transform domain sparsity with rank deficiency.To reconstruct an MR image using its transform domain sparsity and its rank deficiency, this work proposes a combined l₁-norm (of the transform coefficients) and nuclear norm (of the MR image matrix) minimization problem. Since such an optimization problem has not been encountered before, this work proposes and derives a first-order algorithm to solve it.The reconstruction results show that the proposed approach yields significant improvements, in terms of both visual quality as well as the signal to noise ratio, over previous works that reconstruct MR images either by exploiting rank deficiency or by the standard CS-based technique popularly known as the ‘Sparse MRI.’     

双树小波变换与小波树稀疏联合的低场CS-MRI算法

压缩感知理论常用在磁共振快速成像上,仅采样少量的K空间数据即可重建出高质量的磁共振图像.压缩感知磁共振成像技术的原理是将磁共振图像重建问题建模成一个包含数据保真项、稀疏先验项和全变分项的线性组合最小化问题,显著减少磁共振扫描时间.稀疏表示是压缩感知理论的一个关键假设,重建结果很大程度上依赖于稀疏变换.本文将双树复小波变换和小波树稀疏联合作为压缩感知磁共振成像中的稀疏变换,提出了基于双树小波变换和小波树稀疏的压缩感知低场磁共振图像重建算法.实验表明,本文所提算法可以在某些磁共振图像客观评价指标中表现出一定的优势.     

Novel Fourier-based iterative reconstruction for sparse fan projection using alternating direction total variation minimization

Sparse-view x-ray computed tomography(CT) imaging is an interesting topic in CT field and can efficiently decrease radiation dose. Compared with spatial reconstruction, a Fourier-based algorithm has advantages in reconstruction speed and memory usage. A novel Fourier-based iterative reconstruction technique that utilizes non-uniform fast Fourier transform(NUFFT) is presented in this work along with advanced total variation(TV) regularization for a fan sparse-view CT. The proposition of a selective matrix contributes to improve reconstruction quality. The new method employs the NUFFT and its adjoin to iterate back and forth between the Fourier and image space. The performance of the proposed algorithm is demonstrated through a series of digital simulations and experimental phantom studies. Results of the proposed algorithm are compared with those of existing TV-regularized techniques based on compressed sensing method, as well as basic algebraic reconstruction technique. Compared with the existing TV-regularized techniques, the proposed Fourier-based technique significantly improves convergence rate and reduces memory allocation, respectively.     

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.     

基于联合正则化及压缩传感的MRI图像重构

基于压缩传感的MRI图像重构利用图像稀疏的先验知识能从很少的投影值重构原图像。目前MRI重构算法只利用MRI图像稀疏性表示或只利用基于其局部光滑性的先验知识,重构效果不理想。针对此问题,结合两种先验知识,提出一种基于联合正则化及压缩传感的MRI图像重构方法。利用块坐标下降法将求解联合正则化问题转化为交替求解二次凸优化、稀疏正则化和全变差正则化三个简单的优化问题。并提出分别采用共轭梯度法、二元自适应收缩法以及梯度下降法对以上优化问题求解。实验结果表明,该算法重构效果比现有算法有明显地提高。     

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.     

The reconstruction of particle size distributions from dynamic light scattering data using particle swarm optimization techniques with different objective functions

In this paper, the reconstruction of particle size distributions (PSDs) using particle swarm optimization (PSO) techniques from dynamic light scattering (DLS) data was established. Three different objective functions containing non-smooth constrained objective function, smooth functional objective function of Tikhonov regularization and L objective function, were employed. Simulated results of unimodal, bimodal and bi-dispersed particles show that the PSO technique with non-smooth constrained objective function produces narrower PSDs focusing on peak position in the presence of random noise, the PSO technique with smooth functional of Tikhonov regularization creates relative smooth PSDs, which could be successfully applied to the broad particles inversion, and the PSO technique with L objective function yields smooth PSDs, which saves calculation amount. Experimental results as well as comparisons with CONTIN algorithm and Cumulants method demonstrate the performance of our algorithms. Therefore, the PSO techniques employing the three different objective functions, which only require objective function and need a few initial guesses, may be applied to the reconstruction of PSDs from DLS data.     

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.     

Non-convex algorithm for sparse and low-rank recovery: Application to dynamic MRI reconstruction

In this work we exploit two assumed properties of dynamic MRI in order to reconstruct the images from under-sampled K-space samples. The first property assumes the signal is sparse in the x-f space and the second property assumes the signal is rank-deficient in the x-t space. These assumptions lead to an optimization problem that requires minimizing a combined l_p-norm and Schatten-p norm. We propose a novel FOCUSS based approach to solve the optimization problem. Our proposed method is compared with state-of-the-art techniques in dynamic MRI reconstruction. Experimental evaluation carried out on three real datasets shows that for all these datasets, our method yields better reconstruction both in quantitative and qualitative evaluation.     

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.     

Improved l1-SPIRiT using 3D walsh transform-based sparsity basis

l1-SPIRiT is a fast magnetic resonance imaging (MRI) method which combines parallel imaging (PI) with compressed sensing (CS) by performing a joint l1-norm and l2-norm optimization procedure. The original l1-SPIRiT method uses two-dimensional (2D) Wavelet transform to exploit the intra-coil data redundancies and a joint sparsity model to exploit the inter-coil data redundancies. In this work, we propose to stack all the coil images into a three-dimensional (3D) matrix, and then a novel 3D Walsh transform-based sparsity basis is applied to simultaneously reduce the intra-coil and inter-coil data redundancies. Both the 2D Wavelet transform-based and the proposed 3D Walsh transform-based sparsity bases were investigated in the l1-SPIRiT method. The experimental results show that the proposed 3D Walsh transform-based l1-SPIRiT method outperformed the original l1-SPIRiT in terms of image quality and computational efficiency.     

Source-Space Compressive Matched Field Processing for Source Localization

Source localization by matched-field processing(MFP) can be accelerated by building a database of Green's functions which however requires a bulk-storage memory.According to the sparsity of the source locations in the search grids of MFP,compressed sensing inspires an approach to reduce the database by introducing a sensing matrix to compress the database.Compressed sensing is further used to estimate the source locations with higher resolution by solving the l_1-norm optimization problem of the compressed Green's function and the data received by a vertical/horizontal line array.The method is validated by simulation and is verified with the experimental data.     

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.     

并行MRI图像重建算法比较及软件实现

首先介绍了不加速的并行MRI图像重建方法,然后对加速的并行MRI的4种图像重建算法进行了比较,得出结论：加速因子相同时,重建质量上,GRAPPA和SENSE的重建质量最好,SMASH的重建质量次之, PILS算法对线圈位置要求极高,重建质量最差;重建速度上,SMASH的重建速度最快,其次是SENSE和PILS,GRAPPA的重建速度最慢. 当加速因子变大时,所有算法重建质量都变差. 最后介绍了算法实现软件,该软件可以读入原始数据,显示数据采集轨迹,计算线圈灵敏度,选择图像重建方法,分析和比较重建图像质量. 该软件为我国在MRI成像领域提供了一个学习和进一步研究图像重建算法的有力工具.     

Joint Reconstruction of Multi-contrast Images and Multi-channel Coil Sensitivities

Magnetic resonance imaging (MRI) has an important feature that it provides multiple images with different contrasts for complementary diagnostic information. However, a large amount of data is needed for multi-contrast images depiction, and thus, the scan is time-consuming. Many methods based on parallel magnetic resonance imaging (pMRI) and compressed sensing (CS) are applied to accelerate multi-contrast MR imaging. Nevertheless, the image reconstructed by sophisticated pMRI methods contains residual aliasing artifact that degrades the quality of the image when the acceleration factor is high. Other methods based on CS always suffer the regularization parameter-selecting problem. To address these issues, a new method is presented for joint multi-contrast image reconstruction and coil sensitivity estimation. The coil sensitivities can be shared during the reconstruction due to the identity of coil sensitivity profiles of different contrast images for imaging stationary tissues. The proposed method uses the coil sensitivities as sharable information during the reconstruction to improve the reconstruction quality. As a result, the residual aliasing artifact can be effectively removed in the reconstructed multi-contrast images even if the acceleration factor is high. Besides, as there is no regularization term in the proposed method, the troublesome regularization parameter selection in the CS can also be avoided. Results from multi-contrast in vivo experiments demonstrated that multi-contrast images can be jointly reconstructed by the proposed method with effective removal of the residual aliasing artifact at a high acceleration factor.     

Compressed MRI reconstruction exploiting a rotation-invariant total variation discretization

Inspired by the first-order method of Malitsky and Pock, we propose a new variational framework for compressed MR image reconstruction which introduces the application of a rotation-invariant discretization of total variation functional into MR imaging while exploiting BM3D frame as a sparsifying transform. In the first step, we provide theoretical and numerical analysis establishing the exceptional rotation-invariance property of this total variation functional and observe its superiority over other well-known variational regularization terms in both upright and rotated imaging setups. Thereupon, the proposed MRI reconstruction model is presented as a constrained optimization problem, however, we do not use conventional ADMM-type algorithms designed for constrained problems to obtain a solution, but rather we tailor the linesearch-equipped method of Malitsky and Pock to our model, which was originally proposed for unconstrained problems. As attested by numerical experiments, this framework significantly outperforms various state-of-the-art algorithms from variational methods to adaptive and learning approaches and in particular, it eliminates the stagnating behavior of a previous work on BM3D-MRI which compromised the solution beyond a certain iteration.     

Nonlocal regularization based on DT-CWT for WMSN video image iterative reconstruction

The prior knowledge of signal is the previous condition of image compressed sensing reconstruction. In order to improve the quality of the priors except for image sparsity, this paper proposes a new model of video image reconstruction. The texture is the important visual feature of video image as a result of its repeat, leading to image global geometrical structures. The nonlocal idea comes from image self-familiar and can represent image detail features from the geometrical point of view. Therefore, the texture geometrical feature of video image is researched, and we take advantage of dual-tree complex wavelet transform to portray the sparsity representation regularization of the texture. What is more, global constrained regularization is constructed with the help of the nonlocal idea. On the basis of the two regularizations above, a new reconstruction model of video image compressed sensing is proposed, which not only preserves the sparsity prior knowledge of image but also improves the quality of prior knowledge of image by promoting geometrical structure. Iterative shrinkage thresholding algorithm is adopted to solve the model leading to a both simple and quick iterative algorithm. Numerical experiments show that our method is efficient for video image recovery, especially preserving the global details of the original video image.     

Optimizing Image Reconstruction in SENSE Using GPU

Parallel magnetic resonance imaging (MRI) (pMRI) uses multiple receiver coils to reduce the MRI scan time. To accelerate the data acquisition process in MRI, less amount of data is acquired from the scanner which leads to artifacts in the reconstructed images. SENSitivity Encoding (SENSE) is a reconstruction algorithm in pMRI to remove aliasing artifacts from the undersampled multi coil data and recovers fully sampled images. The main limitation of SENSE is computing inverse of the encoding matrix. This work proposes the inversion of encoding matrix using Jacobi singular value decomposition (SVD) algorithm for image reconstruction on GPUs to accelerate the reconstruction process. The performance of Jacobi SVD is compared with Gauss–Jordan algorithm. The simulations are performed on two datasets (brain and cardiac) with acceleration factors 2, 4, 6 and 8. The results show that the graphics processing unit (GPU) provides a speed up to 21.6 times as compared to CPU reconstruction. Jacobi SVD algorithm performs better in terms of acceleration in reconstructions on GPUs as compared to Gauss–Jordan method. The proposed algorithm is suitable for any number of coils and acceleration factors for SENSE reconstruction on real time processing systems.