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

Compressed sensing based dynamic MR image reconstruction by using 3D-total generalized variation and tensor decomposition: k-t TGV-TD

PURPOSE

Compressed Sensing Magnetic Resonance Imaging (CS-MRI) is a promising technique to accelerate dynamic cardiac MR imaging (DCMRI). For DCMRI, the CS-MRI usually exploits image signal sparsity and low-rank property to reconstruct dynamic images from the undersampled k-space data. In this paper, a novel CS algorithm is investigated to improve dynamic cardiac MR image reconstruction quality under the condition of minimizing the k-space recording.

METHODS

The sparse representation of 3D cardiac magnetic resonance data is implemented by synergistically integrating 3D total generalized variation (3D-TGV) algorithm and high order singular value decomposition (HOSVD) based Tensor Decomposition, termed k-t TGV-TD method. In the proposed method, the low rank structure of the 3D dynamic cardiac MR data is performed with the HOSVD method, and the localized image sparsity is achieved by the 3D-TGV method. Moreover, the Fast Composite Splitting Algorithm (FCSA) method, combining the variable splitting with operator splitting techniques, is employed to solve the low-rank and sparse problem. Two different cardiac MR datasets (cardiac perfusion and cine MR datasets) are used to evaluate the performance of the proposed method.

RESULTS

Compared with the state-of-art methods, such as k-t SLR, 3D-TGV, HOSVD based tensor decomposition and low-rank plus sparse method, the proposed k-t TGV-TD method can offer improved reconstruction accuracy in terms of higher peak SNR (PSNR) and structural similarity index (SSIM). The proposed k-t TGV-TD method can achieve significantly better and stable reconstruction results than state-of-the-art methods in terms of both PSNR and SSIM, especially for cardiac perfusion MR dataset.

CONCLUSIONS

This work proved that the k-t TGV-TD method was an effective sparse representation way for DCMRI, which was capable of significantly improving the reconstruction accuracy with different acceleration factors.

Dynamic MRI Reconstruction via Weighted Tensor Nuclear Norm Regularizer

In this paper, we propose a novel multi-dimensional reconstruction method based on the low-rank plus sparse tensor (L+S) decomposition model to reconstruct dynamic magnetic resonance imaging (dMRI). The multi-dimensional reconstruction method is formulated using a non-convex alternating direction method of multipliers (ADMM), where the weighted tensor nuclear norm (WTNN) and l₁-norm are used to enforce the low-rank in L and the sparsity in S, respectively. In particular, the weights used in the WTNN are sorted in a non-descending order, and we obtain a closed-form optimal solution of the WTNN minimization problem. The theoretical properties provided guarantee the weak convergence of our reconstruction method. In addition, a fast inexact reconstruction method is proposed to increase imaging speed and efficiency. Experimental results demonstrate that both of our reconstruction methods can achieve higher reconstruction quality than the state-of-the-art reconstruction methods.

Sliding motion compensated low-rank plus sparse (SMC-LS) reconstruction for high spatiotemporal free-breathing liver 4D DCE-MRI

Liver dynamic contrast-enhanced MRI (DCE-MRI) requires high spatiotemporal resolution and large field of view to clearly visualize all relevant enhancement phases and detect early-stage liver lesions. The low-rank plus sparse (L + S) reconstruction outperforms standard sparsity-only-based reconstruction through separation of low-rank background component (L) and sparse dynamic components (S). However, the L + S decomposition is sensitive to respiratory motion so that image quality is compromised when breathing occurs during long time data acquisition. To enable high quality reconstruction for free-breathing liver 4D DCE-MRI, this paper presents a novel method called SMC-LS, which incorporates Sliding Motion Compensation into the standard L + S reconstruction. The global superior-inferior displacement of the internal abdominal organs is inferred directly from the undersampled raw data and then used to correct the breathing induced sliding motion which is the dominant component of respiratory motion. With sliding motion compensation, the reconstructed temporal frames are roughly registered before applying the standard L + S decomposition. The proposed method has been validated using free-breathing liver 4D MRI phantom data, free-breathing liver 4D DCE-MRI phantom data, and in vivo free breathing liver 4D MRI dataset. Results demonstrated that SMC-LS reconstruction can effectively reduce motion blurring artefacts and preserve both spatial structures and temporal variations at a sub-second temporal frame rate for free-breathing whole-liver 4D DCE-MRI.

Weak Fault Detection of Tapered Rolling Bearing Based on Penalty Regularization Approach

Aimed at the issue of estimating the fault component from a noisy observation, a novel detection approach based on augmented Huber non-convex penalty regularization (AHNPR) is proposed. The core objectives of the proposed method are that (1) it estimates non-zero singular values (i.e., fault component) accurately and (2) it maintains the convexity of the proposed objective cost function (OCF) by restricting the parameters of the non-convex regularization. Specifically, the AHNPR model is expressed as the L1-norm minus a generalized Huber function, which avoids the underestimation weakness of the L1-norm regularization. Furthermore, the convexity of the proposed OCF is proved via the non-diagonal characteristic of the matrix BTB, meanwhile, the non-zero singular values of the OCF is solved by the forward–backward splitting (FBS) algorithm. Last, the proposed method is validated by the simulated signal and vibration signals of tapered bearing. The results demonstrate that the proposed approach can identify weak fault information from the raw vibration signal under severe background noise, that the non-convex penalty regularization can induce sparsity of the singular values more effectively than the typical convex penalty (e.g., L1-norm fused lasso optimization (LFLO) method), and that the issue of underestimating sparse coefficients can be improved.

Low-Rank and Sparse Decomposition Model for Accelerating Dynamic MRI Reconstruction

The reconstruction of dynamic magnetic resonance imaging (dMRI) from partially sampled k-space data has to deal with a trade-off between the spatial resolution and temporal resolution. In this paper, a low-rank and sparse decomposition model is introduced to resolve this issue, which is formulated as an inverse problem regularized by robust principal component analysis (RPCA). The inverse problem can be solved by convex optimization method. We propose a scalable and fast algorithm based on the inexact augmented Lagrange multipliers (IALM) to carry out the convex optimization. The experimental results demonstrate that our proposed algorithm can achieve superior reconstruction quality and faster reconstruction speed in cardiac cine image compared to existing state-of-art reconstruction 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.

Accelerated cardiac cine MRI using locally low rank and finite difference constraints

PURPOSE

To evaluate the potential value of combining multiple constraints for highly accelerated cardiac cine MRI.

METHODS

A locally low rank (LLR) constraint and a temporal finite difference (FD) constraint were combined to reconstruct cardiac cine data from highly undersampled measurements. Retrospectively undersampled 2D Cartesian reconstructions were quantitatively evaluated against fully-sampled data using normalized root mean square error, structural similarity index (SSIM) and high frequency error norm (HFEN). This method was also applied to 2D golden-angle radial real-time imaging to facilitate single breath-hold whole-heart cine (12 short-axis slices, 9-13s single breath hold). Reconstruction was compared against state-of-the-art constrained reconstruction methods: LLR, FD, and k-t SLR.

RESULTS

At 10 to 60 spokes/frame, LLR+FD better preserved fine structures and depicted myocardial motion with reduced spatio-temporal blurring in comparison to existing methods. LLR yielded higher SSIM ranking than FD; FD had higher HFEN ranking than LLR. LLR+FD combined the complimentary advantages of the two, and ranked the highest in all metrics for all retrospective undersampled cases. Single breath-hold multi-slice cardiac cine with prospective undersampling was enabled with in-plane spatio-temporal resolutions of 2×2mm(2) and 40ms.

CONCLUSION

Highly accelerated cardiac cine is enabled by the combination of 2D undersampling and the synergistic use of LLR and FD constraints.

Accelerating patch-based directional wavelets with multicore parallel computing in compressed sensing MRI

Compressed sensing MRI (CS-MRI) is a promising technology to accelerate magnetic resonance imaging. Both improving the image quality and reducing the computation time are important for this technology. Recently, a patch-based directional wavelet (PBDW) has been applied in CS-MRI to improve edge reconstruction. However, this method is time consuming since it involves extensive computations, including geometric direction estimation and numerous iterations of wavelet transform. To accelerate computations of PBDW, we propose a general parallelization of patch-based processing by taking the advantage of multicore processors. Additionally, two pertinent optimizations, excluding smooth patches and pre-arranged insertion sort, that make use of sparsity in MR images are also proposed. Simulation results demonstrate that the acceleration factor with the parallel architecture of PBDW approaches the number of central processing unit cores, and that pertinent optimizations are also effective to make further accelerations. The proposed approaches allow compressed sensing MRI reconstruction to be accomplished within several seconds.

Multidimensional compressed sensing MRI using tensor decomposition-based sparsifying transform

Compressed Sensing (CS) has been applied in dynamic Magnetic Resonance Imaging (MRI) to accelerate the data acquisition without noticeably degrading the spatial-temporal resolution. A suitable sparsity basis is one of the key components to successful CS applications. Conventionally, a multidimensional dataset in dynamic MRI is treated as a series of two-dimensional matrices, and then various matrix/vector transforms are used to explore the image sparsity. Traditional methods typically sparsify the spatial and temporal information independently. In this work, we propose a novel concept of tensor sparsity for the application of CS in dynamic MRI, and present the Higher-order Singular Value Decomposition (HOSVD) as a practical example. Applications presented in the three- and four-dimensional MRI data demonstrate that HOSVD simultaneously exploited the correlations within spatial and temporal dimensions. Validations based on cardiac datasets indicate that the proposed method achieved comparable reconstruction accuracy with the low-rank matrix recovery methods and, outperformed the conventional sparse recovery methods.