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.’     

Sparsity-constrained SENSE reconstruction: An efficient implementation using a fast composite splitting algorithm

Parallel imaging and compressed sensing have been arguably the most successful and widely used techniques for fast magnetic resonance imaging (MRI). Recent studies have shown that the combination of these two techniques is useful for solving the inverse problem of recovering the image from highly under-sampled k-space data. In sparsity-enforced sensitivity encoding (SENSE) reconstruction, the optimization problem involves data fidelity (L2-norm) constraint and a number of L1-norm regularization terms (i.e. total variation or TV, and L1 norm). This makes the optimization problem difficult to solve due to the non-smooth nature of the regularization terms. In this paper, to effectively solve the sparsity-regularized SENSE reconstruction, we utilize a new optimization method, called fast composite splitting algorithm (FCSA), which was developed for compressed sensing MRI. By using a combination of variable splitting and operator splitting techniques, the FCSA algorithm decouples the large optimization problem into TV and L1 sub-problems, which are then, solved efficiently using existing fast methods. The operator splitting separates the smooth terms from the non-smooth terms, so that both terms are treated in an efficient manner. The final solution to the SENSE reconstruction is obtained by weighted solutions to the sub-problems through an iterative optimization procedure. The FCSA-based parallel MRI technique is tested on MR brain image reconstructions at various acceleration rates and with different sampling trajectories. The results indicate that, for sparsity-regularized SENSE reconstruction, the FCSA-based method is capable of achieving significant improvements in reconstruction accuracy when compared with the state-of-the-art reconstruction method.     

Evaluation of motion effects on parallel MR imaging with precalibration

Several parallel imaging techniques such as SMASH, SENSE, k-space inherited parallel acquisition (KIPA) and others use reference (calibration) scans to find the parameters required for image reconstruction. Reference data is used to estimate coil sensitivity profiles for image domain techniques such as SENSE or reconstruction coefficients for k-space domain methods such as SMASH and KIPA. Any motion between the reference and accelerated imaging scans can make the reconstruction coefficients determined from the reference scan data suboptimal, resulting in an artifactual reconstruction. This work aims at comparing the effects of motion on the performance of three parallel imaging methods: SENSE, variable-density SENSE and KIPA, which all require one or more reference scans for calibration.     

Calibration-less multi-coil MR image reconstruction

In parallel magnetic resonance imaging (MRI), the problem is to reconstruct an image given the partial K-space scans from all the receiver coils. Depending on its position within the scanner, each coil has a different sensitivity profile. All existing parallel MRI techniques require estimation of certain parameters pertaining to the sensitivity profile, e.g., the sensitivity map needs to be estimated for the SENSE and SMASH and the interpolation weights need to be calibrated for GRAPPA and SPIRiT. The assumption is that the estimated parameters are applicable at the operational stage. This assumption does not always hold, consequently the reconstruction accuracies of existing parallel MRI methods may suffer. We propose a reconstruction method called Calibration-Less Multi-coil (CaLM) MRI. As the name suggests, our method does not require estimation of any parameters related to the sensitivity maps and hence does not require a calibration stage. CaLM MRI is an image domain method that produces a sensitivity encoded image for each coil. These images are finally combined by the sum-of-squares method to yield the final image. It is based on the theory of Compressed Sensing (CS). During reconstruction, the constraint that "all the coil images should appear similar" is introduced within the CS framework. This leads to a CS optimization problem that promotes group-sparsity. The results from our proposed method are comparable (at least for the data used in this work) with the best results that can be obtained from state-of-the-art methods.     

Phase-constrained parallel MR image reconstruction

A generalized method for phase-constrained parallel MR image reconstruction is presented that combines and extends the concepts of partial-Fourier reconstruction and parallel imaging. It provides a framework for reconstructing images employing either or both techniques and for comparing image quality achieved by varying k-space sampling schemes. The method can be used as a parallel image reconstruction with a partial-Fourier reconstruction built in. It can also be used with trajectories not readily handled by straightforward combinations of partial-Fourier and SENSE-like parallel reconstructions, including variable-density, and non-Cartesian trajectories. The phase constraint specifies a better-conditioned inverse problem compared to unconstrained parallel MR reconstruction alone. This phase-constrained parallel MRI reconstruction offers a one-step alternative to the standard combination of homodyne and SENSE reconstructions with the added benefit of flexibility of sampling trajectory. The theory of the phase-constrained approach is outlined, and its calibration requirements and limitations are discussed. Simulations, phantom experiments, and in vivo experiments are presented.     

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.     

GPU based parallel framework for receiver coil sensitivity estimation in SENSE reconstruction

Magnetic Resonance Imaging (MRI) uses non-ionizing radiations and is safer as compared to CT and X-ray imaging. MRI is broadly used around the globe for medical diagnostics. One main limitation of MRI is its long data acquisition time. Parallel MRI (pMRI) was introduced in late 1990's to reduce the MRI data acquisition time. In pMRI, data is acquired by under-sampling the Phase Encoding (PE) steps which introduces aliasing artefacts in the MR images. SENSitivity Encoding (SENSE) is a pMRI based method that reconstructs fully sampled MR image from the acquired under-sampled data using the sensitivity information of receiver coils. In SENSE, precise estimation of the receiver coil sensitivity maps is vital to obtain good quality images. Eigen-value method (a recently proposed method in literature for the estimation of receiver coil sensitivity information) does not require a pre-scan image unlike other conventional methods of sensitivity estimation. However, Eigen-value method is computationally intensive and takes a significant amount of time to estimate the receiver coil sensitivity maps. This work proposes a parallel framework for Eigen-value method of receiver coil sensitivity estimation that exploits its inherent parallelism using Graphics Processing Units (GPUs). We evaluated the performance of the proposed algorithm on in-vivo and simulated MRI datasets (i.e. human head and simulated phantom datasets) with Peak Signal-to-Noise Ratio (PSNR) and Artefact Power (AP) as evaluation metrics. The results show that the proposed GPU implementation reduces the execution time of Eigen-value method of receiver coil sensitivity estimation (providing up to 30 times speed up in our experiments) without degrading the quality of the reconstructed image.     

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.     

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.     

FPGA implementation of real-time SENSE reconstruction using pre-scan and Emaps sensitivities

Sensitivity Encoding (SENSE) is a widely used technique in Parallel Magnetic Resonance Imaging (MRI) to reduce scan time. Reconfigurable hardware based architecture for SENSE can potentially provide image reconstruction with much less computation time. Application specific hardware platform for SENSE may dramatically increase the power efficiency of the system and can decrease the execution time to obtain MR images. A new implementation of SENSE on Field Programmable Gate Array (FPGA) is presented in this study, which provides real-time SENSE reconstruction right on the receiver coil data acquisition system with no need to transfer the raw data to the MRI server, thereby minimizing the transmission noise and memory usage. The proposed SENSE architecture can reconstruct MR images using receiver coil sensitivity maps obtained using pre-scan and eigenvector (E-maps) methods. The results show that the proposed system consumes remarkably less computation time for SENSE reconstruction, i.e., 0.164 ms @ 200 MHz, while maintaining the quality of the reconstructed images with good mean SNR (29 + dB), less RMSE (< 5 × 10^− 2) and comparable artefact power (< 9 × 10^− 4) to conventional SENSE reconstruction. A comparison of the center line profiles of the reconstructed and reference images also indicates a good quality of the reconstructed images. Furthermore, the results indicate that the proposed architectural design can prove to be a significant tool for SENSE reconstruction in modern MRI scanners and its low power consumption feature can be remarkable for portable MRI scanners.     

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.     

The multiple-parameter discrete fractional Hadamard transform

Discrete fractional Hadamard transform (DFrHaT) is a generalization of the Hadamard transform, which has been widely used in signal processing. In this paper, we present the multiple-parameter discrete fractional Hadamard transform (MPDFrHaT), which has multiple order parameters instead of only one in DFrHaT. The proposed MPDFrHaT is shown to possess all of the desired properties of DFrHaT. In fact, it will reduce to DFrHaT when all of its order parameters are the same. We also propose a novel encryption technique, double random amplitude (DRA) encoding scheme, by cascading twofold random amplitude filtering. As a primary application, we exploit the multiple-parameter feature of MPDFrHaT and double random amplitude encoding scheme for digital image encryption in the MPDFrHaT domain. Results show that this method can enhance data security.     

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.     

基于Hilbert变换实现数字全息高精度相位重建

为了提高数字全息相位重建精度,研究了基于Hilbert变换重建相位的理论和方法,并通过像面数字全息实验对该方法的有效性进行了验证,同时与数字全息常规重建方法得到的结果进行了比较.结果表明:Hilbert变换本身具有消除直流项的作用,对利用频域滤波滤除零级谱后的全息图进行Hilbert变换,能够彻底消除零级衍射项的干扰,从而提高相位重建精度.与常规重建结果相比,基于Hilbert变换的相位重建结果的标准偏差降低了14.0%.本文结果对提高数字全息相位重建精度具有重要的指导意义.     

Phase correction-based singularity function analysis for partial k-space reconstruction

Partial k-space acquisition is a conventional method in magnetic resonance imaging (MRI) for reducing imaging time while maintaining image quality. In this field, image reconstruction from partial k-space is a key issue. This paper proposes an approach fundamentally different from traditional techniques for reconstructing magnetic resonance (MR) images from partial k-space. It uses a so-called singularity function analysis (SFA) model based on phase correction. With such a reconstruction approach, some nonacquired negative spatial frequencies are first recovered by means of phase correction and Hermitian symmetry property, and then the other nonacquired negative and/or positive spatial frequencies are estimated using the mathematical SFA model. The method is particularly suitable for asymmetrical partial k-space acquisition owing to its ability of overcoming reconstruction limitations due to k-space truncations. The performance of this approach is evaluated using both simulated and real MR brain images, and compared with existing techniques. The results demonstrate that the proposed SFA based on phase correction achieves higher image quality than the initial SFA or the projection-onto-convex sets (POCS) method.     

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.     

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.     

Rapid MR imaging by sensitivity profile indexing and deconvolution reconstruction (SPID)

A new parallel MR imaging technique, which uses localized information from the elements of a multi-coil array to accelerate imaging, is described. The technique offers an alternative reconstruction approach to currently available techniques (e.g., SMASH and SENSE). Following a partial k-space data acquisition, image reconstruction in this approach proceeds in two steps: first, fitting the measured coil sensitivities to a set of partially localized target functions, a blurred intermediate image of the studied object is produced. Blurring is obtained in a systematic manner, forming images of the studied object convolved with a known convolution kernel. Full spatial resolution is then recovered by deconvolution of the blurred images with the known kernel function. The technique offers flexibility in the arrangement of the acquired signal data k-lines, and a mechanism for controlling reconstruction quality through the convolution the deconvolution procedure. The technique was validated in phantom and in vivo imaging experiments demonstrating high time reduction factors.     

Improved matrix inversion in image plane parallel MRI

A new 3D parallel magnetic resonance imaging (MRI) method named Generalized Unaliasing Incorporating Support constraint and sensitivity Encoding (GUISE) is presented. GUISE allows direct image recovery from arbitrary Cartesian k-space trajectories. However, periodic k-space sampling patterns are considered for reconstruction efficiency. Image recovery methods such as 2D SENSE (SENSitivity Encoding) and 2D CAIPIRINHA (Controlled Aliasing In Parallel Imaging Results IN Higher Acceleration) are special instances of GUISE where specific restrictions are placed on the k-space sampling patterns used. It is shown that the sampling pattern has large impacts on the image reconstruction error due to noise. An efficient sampling pattern design method that incorporates prior knowledge of object support and coil sensitivity profile is proposed. It requires no experimental trials and could be used in clinical imaging. Comparison of the proposed sampling pattern design method with 2D SENSE and 2D CAIPIRINHA are made based on both simulation and experiment results. It is seen that this new adaptive sampling pattern design method results in a lower noise level in reconstructions due to better exploitation of the coil sensitivity variation and object support constraint. In addition, elimination of the non-object region from reconstruction potentially allows an acceleration factor higher than the number of receiver coils used.     

一种基于多级小波域分解的时域扩散光学层析成像方法的研究

面向基于有限差分光扩散模型的平板乳腺扩散光学层析反演问题,提出了一种有效提高图像重建质量的方法.与基于全空间域离散像点的传统重建算法不同,本方法采用小波函数对吸收系数和散射系数在空间域进行多尺度分解,在减少低分辨率重建中光学参量的个数、有效改善重建过程病态性的同时,通过逐级细化分解尺度,最终提高了图像的量化度和空间分辨率.数值模拟表明,该方法可实现目标体边对边4 mm的空间分辨率,量化度明显高于传统算法.