Undersampled MR image reconstruction using an enhanced recursive residual network

Knowledge-driven deep learning for fast MR imaging: Undersampled MR image reconstruction from supervised to un-supervised learning

Deep learning (DL) has emerged as a leading approach in accelerating MRI. It employs deep neural networks to extract knowledge from available datasets and then applies the trained networks to reconstruct accurate images from limited measurements. Unlike natural image restoration problems, MRI involves physics-based imaging processes, unique data properties, and diverse imaging tasks. This domain knowledge needs to be integrated with data-driven approaches. Our review will introduce the significant challenges faced by such knowledge-driven DL approaches in the context of fast MRI along with several notable solutions, which include learning neural networks and addressing different imaging application scenarios. The traits and trends of these techniques have also been given which have shifted from supervised learning to semi-supervised learning, and finally, to unsupervised learning methods. In addition, MR vendors' choices of DL reconstruction have been provided along with some discussions on open questions and future directions, which are critical for the reliable imaging systems.

Dual-domain faster Fourier convolution based network for MR image reconstruction

Deep learning methods for fast MRI have shown promise in reconstructing high-quality images from undersampled multi-coil k-space data, leading to reduced scan duration. However, existing methods encounter challenges related to limited receptive fields in dual-domain (k-space and image domains) reconstruction networks, rigid data consistency operations, and suboptimal refinement structures, which collectively restrict overall reconstruction performance. This study introduces a comprehensive framework that addresses these challenges and enhances MR image reconstruction quality. Firstly, we propose Faster Inverse Fourier Convolution (FasterIFC), a frequency domain convolutional operator that significantly expands the receptive field of k-space domain reconstruction networks. Expanding the information extraction range to the entire frequency spectrum according to the spectral convolution theorem in Fourier theory enables the network to easily utilize richer redundant long-range information from adjacent, symmetrical, and diagonal locations of multi-coil k-space data. Secondly, we introduce a novel softer Data Consistency (softerDC) layer, which achieves an enhanced balance between data consistency and smoothness. This layer facilitates the implementation of diverse data consistency strategies across distinct frequency positions, addressing the inflexibility observed in current methods. Finally, we present the Dual-Domain Faster Fourier Convolution Based Network (D2F2), which features a centrosymmetric dual-domain parallel structure based on FasterIFC. This architecture optimally leverages dual-domain data characteristics while substantially expanding the receptive field in both domains. Coupled with the softerDC layer, D2F2 demonstrates superior performance on the NYU fastMRI dataset at multiple acceleration factors, surpassing state-of-the-art methods in both quantitative and qualitative evaluations.

Image Reconstruction for Accelerated MR Scan With Faster Fourier Convolutional Neural Networks

High quality image reconstruction from undersampled k -space data is key to accelerating MR scanning. Current deep learning methods are limited by the small receptive fields in reconstruction networks, which restrict the exploitation of long-range information, and impede the mitigation of full-image artifacts, particularly in 3D reconstruction tasks. Additionally, the substantial computational demands of 3D reconstruction considerably hinder advancements in related fields. To tackle these challenges, we propose the following: 1) A novel convolution operator named Faster Fourier Convolution (FasterFC), aims at providing an adaptable broad receptive field for spatial domain reconstruction networks with fast computational speed. 2) A split-slice strategy that substantially reduces the computational load of 3D reconstruction, enabling high-resolution, multi-coil, 3D MR image reconstruction while fully utilizing inter-layer and intra-layer information. 3) A single-to-group algorithm that efficiently utilizes scan-specific and data-driven priors to enhance k -space interpolation effects. 4) A multi-stage, multi-coil, 3D fast MRI method, called the faster Fourier convolution based single-to-group network (FAS-Net), comprising a single-to-group k -space interpolation algorithm and a FasterFC-based image domain reconstruction module, significantly minimizes the computational demands of 3D reconstruction through split-slice strategy. Experimental evaluations conducted on the NYU fastMRI and Stanford MRI Data datasets reveal that the FasterFC significantly enhances the quality of both 2D and 3D reconstruction results. Moreover, FAS-Net, characterized as a method that can achieve high-resolution (320, 320, 256), multi-coil, (8 coils), 3D fast MRI, exhibits superior reconstruction performance compared to other state-of-the-art 2D and 3D methods.

IMJENSE: Scan-Specific Implicit Representation for Joint Coil Sensitivity and Image Estimation in Parallel MRI

Parallel imaging is a commonly used technique to accelerate magnetic resonance imaging (MRI) data acquisition. Mathematically, parallel MRI reconstruction can be formulated as an inverse problem relating the sparsely sampled k-space measurements to the desired MRI image. Despite the success of many existing reconstruction algorithms, it remains a challenge to reliably reconstruct a high-quality image from highly reduced k-space measurements. Recently, implicit neural representation has emerged as a powerful paradigm to exploit the internal information and the physics of partially acquired data to generate the desired object. In this study, we introduced IMJENSE, a scan-specific implicit neural representation-based method for improving parallel MRI reconstruction. Specifically, the underlying MRI image and coil sensitivities were modeled as continuous functions of spatial coordinates, parameterized by neural networks and polynomials, respectively. The weights in the networks and coefficients in the polynomials were simultaneously learned directly from sparsely acquired k-space measurements, without fully sampled ground truth data for training. Benefiting from the powerful continuous representation and joint estimation of the MRI image and coil sensitivities, IMJENSE outperforms conventional image or k-space domain reconstruction algorithms. With extremely limited calibration data, IMJENSE is more stable than supervised calibrationless and calibration-based deep-learning methods. Results show that IMJENSE robustly reconstructs the images acquired at 5× and 6× accelerations with only 4 or 8 calibration lines in 2D Cartesian acquisitions, corresponding to 22.0% and 19.5% undersampling rates. The high-quality results and scanning specificity make the proposed method hold the potential for further accelerating the data acquisition of parallel MRI.

A spatially adaptive regularization based three-dimensional reconstruction network for quantitative susceptibility mapping

Objective.Quantitative susceptibility mapping (QSM) is a new imaging technique for non-invasive characterization of the composition and microstructure ofin vivotissues, and it can be reconstructed from local field measurements by solving an ill-posed inverse problem. Even for deep learning networks, it is not an easy task to establish an accurate quantitative mapping between two physical quantities of different units, i.e. field shift in Hz and susceptibility value in ppm for QSM.Approach. In this paper, we propose a spatially adaptive regularization based three-dimensional reconstruction network SAQSM. A spatially adaptive module is specially designed and a set of them at different resolutions are inserted into the network decoder, playing a role of cross-modality based regularization constraint. Therefore, the exact information of both field and magnitude data is exploited to adjust the scale and shift of feature maps, and thus any information loss or deviation occurred in previous layers could be effectively corrected. The network encoding has a dynamic perceptual initialization, which enables the network to overcome receptive field intervals and also strengthens its ability to detect features of various sizes.Main results. Experimental results on the brain data of healthy volunteers, clinical hemorrhage and simulated phantom with calcification demonstrate that SAQSM can achieve more accurate reconstruction with less susceptibility artifacts, while perform well on the stability and generalization even for severe lesion areas.Significance. This proposed framework may provide a valuable paradigm to quantitative mapping or multimodal reconstruction.

Model-based Deep Learning Reconstruction Using a Folded Image Training Strategy for Abdominal 3D T1-weighted Imaging

PURPOSE

To evaluate the feasibility of folded image training strategy (FITS) and the quality of images reconstructed using the improved model-based deep learning (iMoDL) network trained with FITS (FITS-iMoDL) for abdominal MR imaging.

METHODS

This retrospective study included abdominal 3D T1-weighted images of 122 patients. In the experimental analyses, peak SNR (PSNR) and structure similarity index (SSIM) of images reconstructed with FITS-iMoDL were compared with those with the following reconstruction methods: conventional model-based deep learning (conv-MoDL), MoDL trained with FITS (FITS-MoDL), total variation regularized compressed sensing (CS), and parallel imaging (CG-SENSE). In the clinical analysis, SNR and image contrast were measured on the reference, FITS-iMoDL, and CS images. Three radiologists evaluated the image quality using a 5-point scale to determine the mean opinion score (MOS).

RESULTS

The PSNR of FITS-iMoDL was significantly higher than that of FITS-MoDL, conv-MoDL, CS, and CG-SENSE (P < 0.001). The SSIM of FITS-iMoDL was significantly higher than those of the others (P < 0.001), except for FITS-MoDL (P = 0.056). In the clinical analysis, the SNR of FITS-iMoDL was significantly higher than that of the reference and CS (P < 0.0001). Image contrast was equivalent within an equivalence margin of 10% among these three image sets (P < 0.0001). MOS was significantly improved in FITS-iMoDL (P < 0.001) compared with CS images in terms of liver edge and vessels conspicuity, lesion depiction, artifacts, blurring, and overall image quality.

CONCLUSION

The proposed method, FITS-iMoDL, allowed a deeper MoDL reconstruction network without increasing memory consumption and improved image quality on abdominal 3D T1-weighted imaging compared with CS images.

Emerging Trends in Fast MRI Using Deep-Learning Reconstruction on Undersampled k-Space Data: A Systematic Review

Magnetic Resonance Imaging (MRI) is an essential medical imaging modality that provides excellent soft-tissue contrast and high-resolution images of the human body, allowing us to understand detailed information on morphology, structural integrity, and physiologic processes. However, MRI exams usually require lengthy acquisition times. Methods such as parallel MRI and Compressive Sensing (CS) have significantly reduced the MRI acquisition time by acquiring less data through undersampling k-space. The state-of-the-art of fast MRI has recently been redefined by integrating Deep Learning (DL) models with these undersampled approaches. This Systematic Literature Review (SLR) comprehensively analyzes deep MRI reconstruction models, emphasizing the key elements of recently proposed methods and highlighting their strengths and weaknesses. This SLR involves searching and selecting relevant studies from various databases, including Web of Science and Scopus, followed by a rigorous screening and data extraction process using the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) guidelines. It focuses on various techniques, such as residual learning, image representation using encoders and decoders, data-consistency layers, unrolled networks, learned activations, attention modules, plug-and-play priors, diffusion models, and Bayesian methods. This SLR also discusses the use of loss functions and training with adversarial networks to enhance deep MRI reconstruction methods. Moreover, we explore various MRI reconstruction applications, including non-Cartesian reconstruction, super-resolution, dynamic MRI, joint learning of reconstruction with coil sensitivity and sampling, quantitative mapping, and MR fingerprinting. This paper also addresses research questions, provides insights for future directions, and emphasizes robust generalization and artifact handling. Therefore, this SLR serves as a valuable resource for advancing fast MRI, guiding research and development efforts of MRI reconstruction for better image quality and faster data acquisition.

Joint optimization of Cartesian sampling patterns and reconstruction for single-contrast and multi-contrast fast magnetic resonance imaging

BACKGROUND AND OBJECTIVE

Compressed sensing (CS) has gained increased attention in magnetic resonance imaging (MRI), leveraging its efficacy to accelerate image acquisition. Incoherence measurement and non-linear reconstruction are the most crucial guarantees of accurate restoration. However, the loose link between measurement and reconstruction hinders the further improvement of reconstruction quality, i.e., the default sampling pattern is not adaptively tailored to the downstream reconstruction method. When single-contrast reconstruction (SCR) has been upgraded to its multi-contrast reconstruction (MCR) variant, the identical morphologic information as a priori source could be integrated into the reconstruction procedure. How to measure less and reconstruct effectively by using the shareable morphologic information of various contrast images is an attractive topic.

METHODS

An adaptive sampling (AS) based end-to-end framework (ASSCR or ASMCR) is proposed to address this issue, which simultaneously optimizes sampling patterns and reconstruction from under-sampled data in SCR or MCR scenarios. Several deep probabilistic subsampling (DPS) modules are used in AS network to construct a sampling pattern generator. In SCR and MCR, a convolution block and a data consistency layer are iteratively applied in the reconstruction network. Specifically, the learned optimal sampling pattern output from the trained AS sub-net is used for under-sampling. Incoherence measurement for single-contrast images and the combination of sampling patterns for multi-contrast data are guided by the SCR/MCR sub-net.

RESULTS

Experiments were conducted on two single-contrast and one multi-contrast public MRI datasets. Compared with several state-of-the-art reconstruction methods, SCR results show that a learned sampling pattern brings the quality of the reconstructed image closer to the fully-sampled reference. With the addition of different contrast images, under-sampled images with higher acceleration factors could be well recovered.

CONCLUSION

The proposed method could improve the reconstruction quality of under-sampled images by using adaptive sampling patterns and learning-based reconstruction.

Pyramid Convolutional RNN for MRI Image Reconstruction

Fast and accurate MRI image reconstruction from undersampled data is crucial in clinical practice. Deep learning based reconstruction methods have shown promising advances in recent years. However, recovering fine details from undersampled data is still challenging. In this paper, we introduce a novel deep learning based method, Pyramid Convolutional RNN (PC-RNN), to reconstruct images from multiple scales. Based on the formulation of MRI reconstruction as an inverse problem, we design the PC-RNN model with three convolutional RNN (ConvRNN) modules to iteratively learn the features in multiple scales. Each ConvRNN module reconstructs images at different scales and the reconstructed images are combined by a final CNN module in a pyramid fashion. The multi-scale ConvRNN modules learn a coarse-to-fine image reconstruction. Unlike other common reconstruction methods for parallel imaging, PC-RNN does not employ coil sensitive maps for multi-coil data and directly model the multiple coils as multi-channel inputs. The coil compression technique is applied to standardize data with various coil numbers, leading to more efficient training. We evaluate our model on the fastMRI knee and brain datasets and the results show that the proposed model outperforms other methods and can recover more details. The proposed method is one of the winner solutions in the 2019 fastMRI competition.

Scan-specific artifact reduction in k-space (SPARK) neural networks synergize with physics-based reconstruction to accelerate MRI

PURPOSE

To develop a scan-specific model that estimates and corrects k-space errors made when reconstructing accelerated MRI data.

METHODS

Scan-specific artifact reduction in k-space (SPARK) trains a convolutional-neural-network to estimate and correct k-space errors made by an input reconstruction technique by back-propagating from the mean-squared-error loss between an auto-calibration signal (ACS) and the input technique's reconstructed ACS. First, SPARK is applied to generalized autocalibrating partially parallel acquisitions (GRAPPA) and demonstrates improved robustness over other scan-specific models, such as robust artificial-neural-networks for k-space interpolation (RAKI) and residual-RAKI. Subsequent experiments demonstrate that SPARK synergizes with residual-RAKI to improve reconstruction performance. SPARK also improves reconstruction quality when applied to advanced acquisition and reconstruction techniques like 2D virtual coil (VC-) GRAPPA, 2D LORAKS, 3D GRAPPA without an integrated ACS region, and 2D/3D wave-encoded imaging.

RESULTS

SPARK yields SSIM improvement and 1.5 - 2× root mean squared error (RMSE) reduction when applied to GRAPPA and improves robustness to ACS size for various acceleration rates in comparison to other scan-specific techniques. When applied to advanced reconstruction techniques such as residual-RAKI, 2D VC-GRAPPA and LORAKS, SPARK achieves up to 20% RMSE improvement. SPARK with 3D GRAPPA also improves RMSE performance by ~2×, SSIM performance, and perceived image quality without a fully sampled ACS region. Finally, SPARK synergizes with non-Cartesian, 2D and 3D wave-encoding imaging by reducing RMSE between 20% and 25% and providing qualitative improvements.

CONCLUSION

SPARK synergizes with physics-based acquisition and reconstruction techniques to improve accelerated MRI by training scan-specific models to estimate and correct reconstruction errors in k-space.

A simultaneous multi-slice T2 mapping framework based on overlapping-echo detachment planar imaging and deep learning reconstruction

PURPOSE

Quantitative MRI (qMRI) is of great importance to clinical medicine and scientific research. However, most qMRI techniques are time-consuming and sensitive to motion, especially when a large 3D volume is imaged. To accelerate the acquisition, a framework is proposed to realize reliable simultaneous multi-slice T₂ mapping.

METHODS

The simultaneous multi-slice T₂ mapping framework is based on overlapping-echo detachment (OLED) planar imaging (dubbed SMS-OLED). Multi-slice overlapping-echo signals were generated by multiple excitation pulses together with echo-shifting gradients. The signals were excited and acquired with a single-channel coil. U-Net was used to reconstruct T₂ maps from the acquired overlapping-echo image.

RESULTS

Single-shot double-slice and two-shot triple-slice SMS-OLED scan schemes were designed according to the framework for evaluation. Simulations, water phantom, and in vivo rat brain experiments were carried out. Overlapping-echo signals were acquired, and T₂ maps were reconstructed and compared with references. The results demonstrate the superior performance of our method.

CONCLUSION

Two slices of T₂ maps can be obtained in a single shot within hundreds of milliseconds. Higher quality multi-slice T₂ maps can be obtained via multiple shots. SMS-OLED provides a lower specific absorption rate scheme compared with conventional SMS methods with a coil with only a single receiver channel. The new method is of potential in dynamic qMRI and functional qMRI where temporal resolution is vital.

Systematic Analysis and Review of Magnetic Resonance Imaging (MRI) Reconstruction Techniques

BACKGROUND

Magnetic Resonance Imaging (MRI) plays an important role in the field of medical diagnostic imaging as it poses non-invasive acquisition and high soft-tissue contrast. However, a huge time is needed for the MRI scanning process that results in motion artifacts, degrades image quality, misinterprets the data, and may cause discomfort to the patient. Thus, the main goal of MRI research is to accelerate data acquisition processing without affecting the quality of the image.

INTRODUCTION

This paper presents a survey based on distinct conventional MRI reconstruction methodologies. In addition, a novel MRI reconstruction strategy is proposed based on weighted Compressive Sensing (CS), Penalty-aided minimization function, and Meta-heuristic optimization technique.

METHODS

An illustrative analysis is done concerning adapted methods, datasets used, execution tools, performance measures, and values of evaluation metrics. Moreover, the issues of existing methods and the research gaps considering conventional MRI reconstruction schemes are elaborated to obtain improved contribution for devising significant MRI reconstruction techniques.

RESULTS

The proposed method will reduce conventional aliasing artifact problems, may attain lower Mean Square Error (MSE), higher Peak Signal-to-Noise Ratio (PSNR), and Structural SIMilarity (SSIM) index.

CONCLUSION

The issues of existing methods and the research gaps considering conventional MRI reconstruction schemes are elaborated to devising an improved significant MRI reconstruction technique.

Projection-Based cascaded U-Net model for MR image reconstruction

BACKGROUND AND OBJECTIVE

Background and Objective: Recent studies in deep learning reveal that the U-Net stands out among the diverse set of deep models as an effective network structure, especially for imaging inverse problems. Initially, the U-Net model was developed to solve segmentation problems for biomedical images while using an annotated dataset. In this paper, we will study a novel application of the U-Net structure for the important inverse problem of MRI reconstruction. Deep networks are particularly efficient for the speed-up of the MR image reconstruction process by decreasing the data acquisition time, and they can significantly reduce the aliasing artifacts caused by the undersampling in the k-space. Our aim is to develop a novel and efficient cascaded U-Net framework for reconstructing MR images from undersampled k-space data. The new framework should have improved reconstruction performance when compared to competing methodologies.

METHODS

In this paper, a novel cascaded framework utilizing the U-Net as a sub-block is being proposed. The introduced U-Net cascade structure is applied to the magnetic resonance image reconstruction problem. The connection between the cascaded U-Nets is realized in the form of a recently developed projection-based updated data consistency layer. The novel structure is implemented in the PyTorch environment, which is one of the standards for deep learning implementations. The recently created fastMRI dataset which forms an important benchmark for MRI reconstruction is used for training and testing purposes.

RESULTS

We present simulation results comparing the novel method with a variety of competitive deep networks. The new cascaded U-Net structures PSNR performance stands on average 1.28 dB higher than the baseline U-Net. The improvement, when compared to the standard CNN, is on average 3.32 dB.

CONCLUSIONS

The proposed cascaded U-Net configuration results in an improved reconstruction performance when compared to the CNN, the cascaded CNN, and also the singular U-Net structures, where the singular U-Net forms the baseline reconstruction method from the fastMRI package. The use of the projection-based updated data consistency layer also leads to improved quantitative (including SSIM, PSNR, and NMSE results) and qualitative results when compared to the use of the conventional data consistency layer.

SR-Net: A sequence offset fusion net and refine net for undersampled multislice MR image reconstruction

BACKGROUND AND OBJECTIVE

The study of deep learning-based fast magnetic resonance imaging (MRI) reconstruction methods has become popular in recent years. However, there is still a challenge when MRI results undersample large acceleration factors. The objective of this study was to improve the reconstruction quality of undersampled MR images by exploring data redundancy among slices.

METHODS

There are two aspects of redundancy in multislice MR images including correlations inside a single slice and correlations among slices. Thus, we built two subnets for the two kinds of redundancy. For correlations among slices, we built a bidirectional recurrent convolutional neural network, named Sequence Offset Fusion Net (S-Net). In S-Net, we used a deformable convolution module to construct a neighbor slice feature extractor. For the correlation inside a single slice, we built a Refine Net (R-Net), which has 5 layers of 2D convolutions. In addition, we used a data consistency (DC) operation to maintain data fidelity in k-space. Finally, we treated the reconstruction task as a dealiasing problem in the image domain, and S-Net and R-Net are applied alternately and iteratively to generate the final reconstructions.

RESULTS

The proposed algorithm was evaluated using two online public MRI datasets. Compared with several state-of-the-art methods, the proposed method achieved better reconstruction results in terms of dealiasing and restoring tissue structure. Moreover, with over 14 slices per second reconstruction speed on 256x256 pixel images, the proposed method can meet the need for real-time processing.

CONCLUSION

With spatial correlation among slices as additional prior information, the proposed method dramatically improves the reconstruction quality of undersampled MR images.

Robust water-fat separation based on deep learning model exploring multi-echo nature of mGRE

PURPOSE

To design a new deep learning network for fast and accurate water-fat separation by exploring the correlations between multiple echoes in multi-echo gradient-recalled echo (mGRE) sequence and evaluate the generalization capabilities of the network for different echo times, field inhomogeneities, and imaging regions.

METHODS

A new multi-echo bidirectional convolutional residual network (MEBCRN) was designed to separate water and fat images in a fast and accurate manner for the mGRE data. This new MEBCRN network contains 2 main modules, the first 1 is the feature extraction module, which learns the correlations between consecutive echoes, and the other one is the water-fat separation module that processes the feature information extracted from the feature extraction module. The multi-layer feature fusion (MLFF) mechanism and residual structure were adopted in the water-fat separation module to increase separation accuracy and robustness. Moreover, we trained the network using in vivo abdomen images and tested it on the abdomen, knee, and wrist images.

RESULTS

The results showed that the proposed network could separate water and fat images accurately. The comparison of the proposed network and other deep learning methods shows the advantage in both quantitative metrics and robustness for different TEs, field inhomogeneities, and images acquired for various imaging regions.

CONCLUSION

The proposed network could learn the correlations between consecutive echoes and separate water and fat images effectively. The deep learning method has certain generalization capabilities for TEs and field inhomogeneity. Although the network was trained only in vivo abdomen images, it could be applied for different imaging regions.

Triple-D network for efficient undersampled magnetic resonance images reconstruction

Compressed sensing (CS) theory can help accelerate magnetic resonance imaging (MRI) by sampling partial k-space measurements. However, conventional optimization-based CS-MRI methods are often time-consuming and are based on fixed transform or shallow image dictionaries, which limits modeling capabilities. Recently, deep learning models have been used to solve the CS-MRI problem. However, recent researches have focused on modeling in image domain, and the potential of k-space modeling capability has not been utilized seriously. In this paper, we propose a deep model called Dual Domain Dense network (Triple-D network), which consisted of some k-space and image domain sub-network. These sub-networks are connected with dense connections, which can utilize feature maps at different levels to enhance performance. To further promote model capabilities, we use two strategies: multi-supervision strategies, which can avoid loss of supervision information; channel-wise attention layer (CA layer), which can adaptively adjust the weight of the feature map. Experimental results show that the proposed Triple-D network provides promising performance in CS-MRI, and it can effectively work on different sampling trajectories and noisy settings.

pFISTA-SENSE-ResNet for parallel MRI reconstruction

Magnetic resonance imaging has been widely applied in clinical diagnosis. However, it is limited by its long data acquisition time. Although the imaging can be accelerated by sparse sampling and parallel imaging, achieving promising reconstructed images with a fast computation speed remains a challenge. Recently, deep learning methods have attracted a lot of attention for encouraging reconstruction results, but they are lack of proper interpretability for neural networks. In this work, in order to enable high-quality image reconstruction for the parallel magnetic resonance imaging, we design the network structure from the perspective of sparse iterative reconstruction and enhance it with the residual structure. Experimental results on a public knee dataset indicate that, as compared with the state-of-the-art deep learning-based and optimization-based methods, the proposed network achieves lower error in reconstruction and is more robust under different samplings.