ConvDecoder architecture. It is comprised of up-sampling, convolutional, ReLU, batch normalization, and linear combination layers.

Sample reconstructions for ConvDecoder, TV, U-net, and the end-to-end variational network (VarNet) for a validation image from multi-coil knee measurements (4x accelerated). The second row represents the zoomed-in version of the first row. ConvDecoder and the end-to-end variational network (VarNet) find the best reconstructions for this image (slightly better than U-net and significantly better than TV). The scores given below are averaged over 200 different mid-slice images from the FastMRI validation set.

Figure 2. Reconstructions in spatial domain under different acceleration factors with the zoom-in images of the enclosed part and the corresponding error maps.

Figure 1. Illustration of the proposed Learned DC. The architecture of each iteration is corresponding to Eq. (6), where the k-space sub-network has 2 layers and image domain sub-network has 3 layers. The numbers on the layers indicate the number of filters in that convolutional layer.

Comparisons between 2D GRAPPA, 2D RAKI, and eRAKI reconstructions on 3D ME-MPRAGE data using 24×24×256 ACS and 3×3 undersampling. We use Tikhonov regularization in GRAPPA and set parameter λ=1×e^-9. Our eRAKI benefits from 3D convolutions and can achieve comparable performance with high-speed reconstruction.

Comparisons between GRAPPA, RAKI, and eRAKI reconstructions on EPTI data using 32x80x16 ACS. All three methods are implemented in k_x-k_y-t space and eRAKI can achieve comparable performance with GRAPPA but uses only two models to perform the whole learning/reconstruction process.

Figure 1. Schematic of the unrolled ADMM for $$$\ell_{1}$$$-wavelet compressed sensing (CS). One unrolled iteration of ADMM with $$$\ell_{1}$$$-wavelet regularizers consists of regularizer (R), data consistency (DC) and dual update (DWT: Discrete wavelet transform). For T = 10 iterations, this leads to a total of 116 trainable parameters, as dual update is not needed for the last iteration.

Figure 3. Two representative slices from coronal PD–FS knee MRI, reconstructed using CG-SENSE, PG-DL, and the optimized -wavelet CS. CG-SENSE is visibly outperformed by PG-DL and -wavelet CS. Both PG-DL and the optimized -wavelet CS show comparable image quality and quantitative metrics.

General cross-domain networks architecture. Skip and residual connection are omitted for the sake of clarity. y are the under-sampled measurements,in our case the k-space measurements, Ω is the under-sampling scheme,Fis the measurement operator, in our case the Fourier Transform, and x is the recovered solution.

Magnitude reconstruction results for the different fastMRI contrasts at acceleration factor 4. The top row represents the ground truth, the middle one represents the reconstruction from a retrospectively under-sampled k-space,and the bottom row represents the absolute error when comparing the two. The average image quantitative metrics are given for 30 validation volumes.

Figure 1: A image obtained from the undersampled k-space data was used as input and a fully sampled image quantized to 8-bit (256 grey levels) was used as target image to the network for training. The DL Unet network classifies each pixel in the reconstructed image and the output of the network was probability for each pixel belonging to one of the 256 distinct quantized grey levels. A weighted linear combination of the predicted probability forms the reconstructed image and standard deviation of a Gaussian fitted curve to the predicted probability distribution from the uncertainty maps.

Figure 2: (a) Input image from undersampled (factor of 4) k-space data; (b, d) are the predicted probabilities at spatial location as pointed out by the red, blue and green dots in the reconstructed image respectively; (c) Reconstructed image obtained using weighted linear combination of the predicted probabilities; (e) fully sampled reference image; ; (f) error image obtained by subtracting reference and output image; (c) Uncertainty maps obtained from standard deviation of the fitted Gaussian curve to the predicted probabilities.

Fig. 2: Undersampled k-space data are reconstructed by zero-filling and then converted into blocks, which are decomposed using SVD to initialize $$$L_b$$$ and $$$R_b$$$. These are iteratively processed by DSLR+ by alternating conjugate gradient and CNN updates. Before each CNN, the basis functions are split into real/imaginary parts and concatenated along the featuredimension. For simplicity, 2-D and 1-D ResNets comprised of 6 convolutions each are used. At the end of the network, the basis functions are combined to form the output images.

Fig. 5: Two prospectively undersampled scans are performed in a patient with premature ventricular contractions (PVC), which manifests as a double heartbeat motion. The first acquisition is 12X accelerated and reconstructed using MoDL, DSLR, and DSLR+ (left to right). A second acquisition is 2X accelerated and reconstructed with SENSE (rightmost). All show remarkable image quality despite severe acceleration, and never having seen a PVC during training. Both MoDL and DSLR+ faithfully demonstrate the double beat motion, although, DSLR+ depicts less residual aliasing (arrows).

GBCA dose-reduction model with T2W image and the generated UAD mask as attention mechanism. The model is trained with a combination of SSIM, perceptual, adversarial and UAD-mask weighted L1-loss.

Examples of cases where the proposed model has a better tumor/lesion enhancement pattern compared to the T1-only model.

Figure 5: Comparison of reconstructions for a knee image using the proposed BLIPS method versus strict supervised learning, blind dictionary learning, and zero-filled reconstruction for the 5-fold undersampling mask depicted in Fig. 2. Metrics listed below each reconstruction correspond to PSNR(in dB)/SSIM/HFEN respectively.

Figure 1: Proposed pipeline for combining blind dictionary learning and supervised learning-based MR image reconstruction.

Figure 1. For the case of moderate undersampling, the two reconstructions regularized by log-likelihood and $$$\ell_1$$$ are very close, and the structural similarity index between them is 0.95. The $$$\ell_1$$$ reconstruction has blocky artifacts in Region 1 introduced by the wavelet transform, especially for higher undersampling. Overall, the learned log-likelihood outperforms $$$\ell_1$$$ in noise suppression, especially in Region 2. The reconstructions regularized by the learned log-likelihood also better preserve the boundaries between tissues and have less noise.

Figure 2. Reconstructed coil sensitivities (grayscale magnitude and color-coded phase) after channel compression.

Figure 1. (a) Framework overview example in a supervised setting with a conditional GAN when fully-sampled datasets are available.

(b) Our proposed framework overview in an unsupervised setting. A sensing matrix comprised of coil sensitivity maps, an FFT and a randomized undersampling mask is applied to the generated image to simulate the imaging process. The discriminator takes simulated and observed measurements as inputs and tries to differentiate between them. The generator’s loss is based on the discriminator as well as reducing spatial and temporal variation.

Figure 5. (a) Representative DCE images. The leftmost column is the input zero-filled reconstruction, the middle column is our generator’s reconstruction, and the rightmost column is the CS reconstruction. The generator improves the input image quality by recovering sharpness and adding more structure to the input images.

(b) Comparison of DCE inference time per three-dimensional DCE volume (2D + time) between CS low-rank and our unsupervised GAN. Our method is approximately 2.98 times faster.

Figure 2. The proposed Manifold-Net for dynamic MR imaging.

Figure 1. Gradient descent on the low-rank tensor manifold.

Figure 1. The proposed plug-and-play LR network module. (a) The original ISTA-Net. (b) The original DC-CNN. (c) The original CRNN. (d) ISTA-LR-Net by embedding the LR network module into the original ISTA-Net. (e) DC-CNN-LR by embedding the LR network module into the original DC-CNN.(f) CRNN-LR by embedding the LR network module into the original CRNN. The numbers in the dotted box represent the locations where the LR module can be embedded.

Figure 2. The reconstruction results of the different methods (DC-CNN, DC-CNN-LR, CRNN, and CRNN-LR) at 8-fold acceleration. From left to right, the first row shows the ground truth, the reconstruction result of these methods. The second row shows the enlarged view of their respective heart regions framed by a yellow box. The third row shows the error map (display ranges [0, 0.07]). The y-t image (extraction of the 124th slice along the y and temporal dimensions) and the error of the y-t image are also given for each signal to show the reconstruction performance in the temporal dimension.

Figure 2: An example of the concatenation of U-net blocks which operate in either image domain or k-space domain with data consistency layer in between. The deformable convolution layer, which extracts the motion feature and generates offsets from a series of dynamic MRI, is at the beginning of the U-net block on the image domain.

Figure 1: The illustration of the generation offset fields and the performance of deformed sampling. The offsets are obtained by applying a convolution layer over the same input feature map. The channel dimension of the offset fields is twice as much as that of the input feature maps, given the 2D offsets.

Figure 1. The network structure with joint learning of sampling pattern and data sharing. The interconnections in the data sharing stage highlight the different data sharing used by each frame.

Figure 2. Exemplary reconstruction results of a testing frame. The (a) ground truth, (b, c) with temporal DoF and data sharing (model A) and its error map, (d, e) with temporal DoF and without data sharing (model B) and its error map, (f, g) without temporal DoF and data sharing (model C) and its error map, and (h, i) trained with pseudo golden angle radial sampling and without data sharing (model D) and its error map are shown. Model A shows better qualitative reconstruction quality.

Figure. 1 - COMNET consists of an unrolled cascade of sub-networks where each sub-network consists of a calibration consistency (CC) block fused with a network consistency (NC) block, both followed by a data consistency (DC) block.

Figure. 2 - Average PSNR values of a) cT1, b) T2, and c) FLAIR images of test subjects as a function of number of training subjects (upper x-axis), and training samples (lower x-axis). COMNET requires just a few training samples from a single subject to outperform L1-SPIRiT. On the other hand, DNN on average requires around 90 samples from 9 different subjects to start performing better than L1-SPIRiT.

Figure 1: Diagram of our two-step weakly supervised framework. Step 1 involves weakly labeled training data creation using either Self-training or ConvDecoder Weak Labeling. In Self-training, a "teacher model" is trained with fully sampled data to run inference on undersampled data, for creating weak labels. Alternatively, ConvDecoder can directly create weakly labeled data from undersampled data. Step 2 involves training a "student model" unrolled neural network using both supervised and weakly labeled pairs, which is then used at inference time.

Figure 5: Image quality comparison between Supervised Baseline, Self-training, and CD Weak Labeling. All methods use 1 supervised and 5 weakly labeled volumes during training with 4x acceleration. We observe that CD Weak Labeling provides clear improvement in reducing background noise and aliasing artifacts over both Supervised Baseline and Self-training.

Figure 1: Sample reconstructions for the different loss functions with 4-fold acceleration. Each row represents a different image contrast.

Table 1: 4-fold acceleration results averaged across the 16-coil only FastMRI validation dataset. For PSNR and SSIM, the higher the score the better, and vice-versa for NMSE.

Fig. 1. The architecture of EEDN. FLB represents feature learning sub-module, which is implemented with EAM [5]. Up and Down are the up and down sampling operation, respectively. For FLB modules, multiple inputs of each FLB module are first concatenated and then reduced for channels by 1x1 conv.

Fig. 2. Comparison of different models using five different metrics.

The proposed layer-wise training algorithm. At each training step, a two-layer network is trained to fit the ground-truth. After training has finished, the weights of the first layer are frozen, and the second training step is performed. This training algorithm is repeated k times to obtain a deep network with k+1 convolutional layers. L(y,$$$\hat{x}_i$$$) is the loss for i^th step where y is ground-truth, and $$$\hat{x}_i$$$ is the prediction of i^th layer. In the final network, each layer except the final layer have 7x7 kernel size and 256 channels, and the final layer has 1x1 kernel size.

Representative knee images from the test set for layer-wise training vs end-to-end training with Poisson disc sampling (top) and Cartesian sampling (bottom). End-to-end training and layer-wise training have similar perceptual quality.

Figure 3. A representative test slice from reconstructions using 2D and 3D unrolled networks. 2D processing suffers from residual artifacts (red arrows), which are suppressed with the 3D processing.

Figure 2. Schematic of the supervised training on the small slabs generated from the volumetric datasets. Unrolled neural networks comprise a series of data consistency and regularizer units, arising from conventional iterative optimization algorithms. In this work, we use a 3D unrolled network. A ResNet architecture with 5 residual blocks, consisting of 2 convolution layers with former followed by ReLU and latter followed by a scaling layer is used for implicit regularization. All layers of this ResNet use 3×3×3 kernels and 64 channels.

Figure 1. GPU memory limitations for high-dimensional unrolled DL recons: a) Conventionally, in order to reconstruct a 3D volume, each slice in the readout dimension is independently passed through a 2D unrolled network and then re-joined into a 3D volume. In contrast, 3D unrolled networks require more memory but use 3D slabs during the training, which leverages more data redundancy. b) Typically, cardiac cine DL recons are often performed with a small number of unrolls due to memory limitations. More unrolls are able to better learn the finer spatial and temporal textures.

Figure 2. a) Unrolled networks are formed by unrolling the iterations of an image reconstruction optimization. Each unroll consists of a CNN-based regularization layer and a DC layer. Conventionally, gradients of all layers are computed simultaneously, which requires a significant GPU memory cost. b) Memory-efficient learning procedure for a single layer: 1) Recalculate the layer’s input $$$\mathbf{x}^{(n-1)}$$$, from the known output $$$\mathbf{x}^{(n)}$$$. 2) Reform the AD graph for that layer. 3) Backpropagate gradients $$$q^{(n)}$$$ through the layer’s AD graph.

FIG_3 SSA-FARY-gated compressed sensing recontruction of the heart. All three SMS-slices are depicted. The respiratory motion is resolved into 5 bins using SSA-FARY amplitud-binning. The white lines serve as reference to appreciate the different respiratory states. Bin 1: End-expiration, Bin 5: End-inspiration. The slightly blurred appearance of the fifth bin is a result of the limited number of spokes which were aquired in the typically shorter end-inspiration state.

FIG_2 Respiratory gating with SSA-FARY. A) The background shows the diaphragm-motion extracted from a real-time reconstruction of the entire time-series. On top, EOF 3 and EOF 4, i.e. the third and fourth column of the SVD-Matrix $$$U$$$, are plotted. B) For each EOF, four representative filters extracted from the$$$V^H$$$-matrix are depicted. Note, that the symmetric filters (green) correspond to EOF 4, which is in-phase with the diaphragm and which can therefore be used for amplitude-binning.

Figure 2: Schematic illustration of the pre-trained network for the Proposed NUFFT TL-Net framework at AF=7 & 13

Figure 4: Middle slice (short axis) End-diastole and End-Systole reconstructed images of a patient for whole heart cine Radial MR at acceleration factor 13 with 24 radial lines per image. Reference cine: Fully sampled image: Undersampled Image: NUFFT with iFFT image. NUFFT U-Net: Reconstructed image of the fully trained NUFFT U-Net at AF=13, NUFFT TL-Net: Reconstructed image of NUFFT LT-Net at AF=13. Corresponding edge images provide reconstructed image quality assessment. Arrows show the myocardial wall distortion that is less pronounced in the NUFFT TL-Net.

Fig. 2. Demonstration of the framework of the proposed scheme on the first dataset. We plot the latent variables of 150 frames in time series on the first dataset. We showed four different phases from 4 different slices that are reconstructed in the time series: systole in End-Expiration (E-E), systole in End-Inspiration (E-I), diastole in End-Expiration (E-E) and diastole in End-Inspiration (E-I). The latent vectors corresponding to the four different phases are indicated in the plot of the latent vectors.

Fig. 3. Illustration of the framework of the proposed scheme on the second dataset. We plot the latent variables of 150 frames in time series on the first dataset. We showed four different phases from 4 different slices that are reconstructed in the time series: systole in End-Expiration (E-E), systole in End-Inspiration (E-I), diastole in End-Expiration (E-E) and diastole in End-Inspiration (E-I). The latent vectors corresponding to the four different phases are indicated in the plot of the latent vectors.

U-Dream jointly trains image reconstruction and image registration modules. Inputs are unregistered measurement pairs from the same subject. The zero-filled images are passed through the reconstruction CNN to remove artifacts due to noise and undersampling. The output images are then used in the registration CNN to obtain the motion field characterizing the directional mapping between their coordinates. We register one of the reconstructed images to the other by the Spatial Transform Network.

Illustration of several reconstruction methods on synthetic undersampled brain MRI images. The top-right corner of each image provides the PSNR and SSIM values with respect to the groundtruth image. Unregistered N2N is directly trained on unregistered data, while SyN+N2N trains the CNN on pre-registered images that are corrupted with artifacts. U-Dream achieves a significant improvement relative to both methods by jointly addressing the reconstruction and the registration problems.

Fig. 1: These are the experiments in 2D settings at 6-fold acceleration with a noise of standard deviation of 0.02. Here, we trained the MoDL only with mask $$$M_0$$$ as shown in (b). During testing, we utilized the same mask $$$M_0$$$ as well as a different mask $$$M_1$$$ (d). The reconstruction results in (c) and (e) show that MoDL architecture is robust to small changes in the sampling mask.

Fig. 2: (a) The Cartesian mask $$$M_0$$$ used during training. (b) The regridding reconstruction of the test image using $$$M_0$$$. (c) MoDL results in good reconstruction since $$$M_0$$$ is used here during testing. (d) This is a different mask $$$M_1$$$ to test the robustness of MoDL architecture. (e) is the corresponding regridding reconstruction. The reconstruction (f) has lower PSNR than (c), as expected since $$$M_1$$$ was not used during training. (h) is the result of model adaptation on (f) using MSE (i) is the reconstructed image using the G-SURE-based model adaption.

Fig. 1: The CNN architecture used in this work. The input layer takes in the subsampled, zerofilled k-space data of the coil array with N_c independent choils. Real- and imaginary part of the k-space data are passed to separate channels, resulting in 2 x N_c input-channels. Two hidden layers are assigned 256 and 128 channels, respectively (n₂=256 and n₃=128). The output layer predicts all missing points across all coils simultaneously, and thus has 2 x (R-1) x N_c channels, with R denoting the undersampling rate.

Fig. 2: 2D image reconstructions of brain dataset (32 coils, T1-weighted) for retrospective undersampling rates in range 4-6 (denoted as R4-R6). ACS data (30 central phase lines) were re-inserted into reconstructed k-spaces. GRAPPA kernel size was optimized for each undersampling rate: 11 x 4 (R4-5), and 15 x 2 (R6) in read- and phase direction. RAKI performs better than GRAPPA in terms of noise resilience, but suffers from blurring artifacts, which have pronounced appearance at 6-fold undersampling.

Figure 1. The network structure of DLNUFFT. (a) Under-sampled k-space data Xu is processed by Block Layer to obtain Xp with multiple patches. Xp goes through the Density Compensation and Reordering Layer, which is initialized using the position map. After Adaptive Interpolation Layer, the fully sampled Cartesian patch data Xkf are obtained, followed by ReBlock Layer to generate the fully sampled k-space data Xf. Finally, the inverse FFT is applied on Xf to get the reconstructed image If. (b) Illustration of real acquired k-space data processed by DLNUFFT and the output after each block.

Figure 4. Reconstruction results for DLNUFFT and other methods on three datasets with radial trajectory (R=4). DLNUFFT can achieve relatively high PSNR (36.65dB) and the highest SSIM (93.21%).

Figure 1 (A) Illustration of proposed sampling pattern for incomplete single-channel FSE acquisition with partial shots (TRs). It acquires partial shots, e.g., by skipping 7 out of total 20 shots at 1/3/5/14/16/18/20 for each segment. (B) Proposed ResNet for image reconstruction, consisting of 2 convolutional layers with downsampling, 16 residual blocks (RBs), and 2 convolutional layers with upsampling. Each RB contains 2 convolutional layers with a rectified linear unit. The inputs are images directly reconstructed from incomplete k-space data with missing shots zero-filled.

Figure 2 (A) Demonstration of the MR reconstruction results by the proposed ResNet from the incomplete single-channel FSE data (containing 13 out of total 20 shots with ETL=15). (B) Typical MR reconstruction results with 13/12/11 out of total 20 shots, respectively. Quantitative measures PSNR and SSIM are also shown. These results demonstrated that the trained ResNet model could effectively remove the aliasing artifacts and recover the high frequency information without noticeable blurring at various undersampling levels.

Figure 4. Comparison of untrained method against fully supervised baseline on a 3D qDESS axial slice. The k-space was downsampled using a 2D Poisson disc at acceleration factors of 4x and 8x for the top and bottom rows, respectively. The image quality of the reconstruction with the same undersampling factors is higher for 3D qDESS than 2D fastMRI data, likely because the undersampling aliasing is spread across two dimensions ($$$k_y$$$ and $$$k_z$$$).

Figure 1. Network architecture for our untrained method.

Figure 1: (a) Pretraining of the style-generative model. A fully connected mapper to generate intermediate latent vectors w, a synthesizer to generate images, a discriminator for adversarial training and noise n. w and n are defined for each synthesizer block separately, where block cover resolution from 4x4 to 256x256 pixels. (b) Testing phase of ZSL-Net. w* and n* correspond to optimized latent vector and noise components for the synthesizer. Optimization is performed to minimize partial k-space loss between masked Fourier coefficients of reconstructed and undersampled images.

Figure 2: Demonstrations of the proposed and competing methods on IXI for T1-contrast image reconstruction when acceleration rate R is 8. Reconstructed images are shown along with the error maps which are absolute differences between reconstructed and reference images. Error map corresponds to ZSL-Net, appears to be darker compared to the competing methods and most of the error concentrated on skull.

Figure 3. A representative test brain MRI slice showing reconstruction results using CG-SENSE, SSDU PG-DL and proposed multi-mask SSDU PG-DL at R=8, as well as CG-SENSE at acquisition acceleration R=2. CG-SENSE suffers from major noise amplification at R=8, whereas SSDU PG-DL at R=8 achieves similar reconstruction quality to CG-SENSE at R=2. The proposed multi-mask SSDU PG-DL further improves the reconstruction quality compared to SSDU PG-DL.

Figure 2. a) Reconstruction results on a representative test slice at R = 8 using CG-SENSE, supervised PG-DL, SSDU PG-DL and proposed multi-mask SSDU PG-DL. CG-SENSE suffers from major noise amplification and artifacts. SSDU PG-DL also shows residual artifacts (red arrows) at this high acceleration rate. Proposed multi-mask SSDU PG-DL further suppresses these artifacts and achieve artifact-free reconstruction, removing artifacts that are still visible in supervised PG-DL.

Figure 1. Schema of the proposed parallel imaging and generative adversarial network (PIC-GAN) reconstruction network.

Figure 2. Representative reconstructed abdominal images with acceleration AF= 6. The 1^st and 2^nd rows depict reconstruction results for regular Cartesian sampling, the 3^rd and 4^th row depict the same for variable density random sampling. The PIC-GAN reconstruction shows reduced artifacts compared to other methods.

Figure-1. A residual Encoder-Decoder network of CNNs (top and middle), enhanced by frequency-attention and channel-attention layers (bottom), for image reconstruction from undersampled k-space data.

Figure-2. Visualization of representative cases of the Stanford knee dataset (top row) and the fastMRI brain dataset (bottom row), including images reconstructed from the fully sampled data and from under-sampled data without CNN processing. The difference images were amplified 5 times for Stanford cases and 10 times for fastMRI dataset. Yellow and red boxes indicate the zoomed-in and difference images, respectively.

Figure 1. MoDern: The proposed Model-inspired Deep Learning framework for NMR spectra reconstruction. The recursive MoDern framework that alternates between the data consistency (DC), which is same to Eq. (1a), and the learnable adaptive soft-thresholding (LS) inspired by Eq. (1b). With the increase of iterations, artifacts are gradually removed, and finally a high-quality reconstructed spectrum can be obtained. Note: “FT” is the Fourier transform. A data consistency followed by a learnable adaptive soft-thresholding constitutes an iteration.

Figure 2. 2D ¹H-¹⁵N HSQC spectrum of the cytosolic domain of CD79b protein. (a) The fully sampled spectrum. (b)-(c) are reconstructed spectra using DLNMR and MoDern from 20% data, respectively. (d) is zoomed out 1D ¹⁵N traces. (e) is the peak intensity correlation obtained with two methods under different NUS levels. The insets of (b)-(c) show the corresponding peak intensity correlation between fully sampled spectrum and reconstructed spectrum. Note: The average and standard deviations of correlations in (e) are computed over 100 NUS trials.

Figure 4: An example of results and corresponding point-wise error-images obtained for an acceleration factor of $$$R=18$$$. From left to right: The initial NUFFT-reconstruction obtained from $$$N_{\theta}=560$$$ radial spokes, the CNN-regularized solution¹⁰, the DL+SC-regularized solution⁷, the proposed method and the $$$kt$$$-SENSE reconstruction obtained from $$$N_{\theta}=3400$$$ radial spokes from which the $$$k$$$-space data was retrospectively simulated.

Figure 2: Convergence results for the proposed method (red) compared to DL and SC (blue) for the two acceleration factors $$$R=18$$$ and $$$R=9$$$ given by $$$N_{\theta}=560$$$ and $$$N_{\theta}=1130$$$ radial spokes, respectively. The solid lines correspond to the mean value of the respective mesure obtained over the entire test set. The dashed lines correspond to the mean $$$\pm$$$ the standard deviation obtained over the test set.

Note that the values differ from the ones shown in the Table because here, no masks were used to restrict the calculations to a region of interest.

Figure 2. Cardiac functional analysis based on DL-cine, CS-cine and retro-cine. Difference with statistical significance (p<0.05) is indicated by the star.

Figure 1. Examples of reconstructed images from DL-cine, CS-cine and retro-cine MRI. Data were acquired from the same subject but three different scans.

Figure 1: Schematic overview of the proposed LapMRI framework.

Figure 2: The inputs and outputs of LapMRI(D5C5, Λ=3) at three pyramid levels. For each pyramid level, the two images in the left column are the inputs, and the two images in the right column are the output and the respective ground-truth image, respectively. For visual understanding, the output of each pyramid level is displayed in the image domain, while the respective ground-truth image is framed with a red dotted box.

Figure 2. Image quality scores averaged over 5 subjects for ARC (net R=2.1) and 3 variants of DCI-Net (net R=10) in 8 scoring categories. The error bars denote the sample standard deviation.

Figure 3. Example image slices from subject D, showing the cerebellar vermis indicated by the yellow arrow, reconstructed by (A) ARC (net R=2.1), (B) 2D DCI-Net (net R=10), (C) alternating 2D DCI-Net (net R=10), and (D) 3D DCI-Net (net R=10).

Figure 4. Results of reconstructing spatially phase varied images. Figure (a) and (b) show the phase map and magnitude of fully scanned image. Figure (c) and (d) are reconstructed images with EsUS and RaUS, respectively. Enlarged images corresponding to (a) through (d) are shown in (e) through (h).

Figure 2. Comparison of PSNR and SSIM between PSFT-CS-Net and FT-CS-Net for real-value images. PSNR results using sampling pattern of Fig.1 (a) are shown. Comparison among CNN reconstruction using PSFT or FT imaging and conventional iterative reconstruction were made.

Fig. 1: ProvoGAN performs a series of cross-sectional subtasks optimally-ordered across individual rectilinear orientations (Axial$$$\rightarrow$$$Sagittal$$$\rightarrow$$$Coronal illustrated here) to handle the aimed volumetric recovery task. Within a given subtask, source-contrast volume is divided into cross-sections across the longitudinal dimension, and a cross-sectional mapping is learned to recover target cross-sections from source cross-sections, where the previous subtask's (if available) output is further incorporated to leverage contextual priors.

Fig. 2: ProvoGAN is demonstrated on IXI for T₁-weighted image synthesis. Representative results from ProvoGAN, sGAN models (sGAN-A is trained axially, sGAN-C coronally, and sGAN-S sagittally), and vGAN are displayed for all rectilinear orientations (first row: axial, second row: coronal, third row: sagittal) together with reference images.

Figure 1 The flowchart of the proposed EMS-PF reconstruction method, with adjacent slices having complementary sampling patterns. Re and Im denote the real and imaginary parts of the reconstructed slice (Slice 2) and its adjacent slices (Slices 1 and 3); Re2’ and Im2’ denote the real and imaginary parts of the predicted residual image. The ResNet model has 16 RBs and each of them contains 2 convolutional layers followed by Rectified Linear Unit (ReLU) activation function. In each convolutional layer, 64 convolutional kernels of size 3×3 each are included.

Figure 3 Error maps of the reconstructed images in Figure 2 with enhanced brightness (×10) and corresponding peak signal-to-noise ratio (PSNR) / structural similarity (SSIM) at PF fraction=0.51. It was obvious that the proposed EMS-PF outperformed the other methods in terms of reduced residual error.

Fig 1. Structure of proposed Joint-ISTA-Net and a iterative Share Block. In Share Block, three 3x3 CNN layers with Relu stands forward sparse transformation. Figures in transform domain pass through both individual and joint soft threshold function to be denoised, and then decoded back to image domain. Here joint soft threshold function is implemented to exploit the group sparsity property of multi-contrast MR images.

Fig 2. Comparison of reconstruction method with R=10. Proposed Joint-ISTA-Net has advantages on feature preserving and provides sharper edge.

Figure 4. Calibrationless parallel imaging and compressed sensing on a T1-weighted brain image. All methods were able to successfully reconstruct the image at R=4.0. However, at R=8.5, only DNLINV was able to reconstruct the image without any loss of structure. Furthermore, DNLINV reconstructions have higher apparent SNR.

Figure 5. Autocalibrating parallel imaging with CAIPI sampling on a T1-weighted brain image. All methods were able to successfully reconstruct the image at R=16.0 with DNLINV having the highest apparent SNR. At R=25.0, residual aliasing artifacts remain on ESPIRiT and ENLIVE while these are largely suppressed in DNLINV.

Fig.1.(a) Framework of the proposed method.The generator of the network is connected with two residual autoencoder U-net. The discriminator is composed of 6 layers. (b) Composition of KS-Block.

Fig.2.The architecture of Residual SCAU-Net.The encode block is indicated by green, and the decode block is indicated by blue. The 4D tensor is used as input, and using the 2D convolution with filter_size of 3x3 and Stride of 2. The number of feature maps is defined as feature_num=64. The residual block is represented by orange, which is used to increase the depth of the network. SCA Block is indicated by yellow composed of spatial attention and channel attention.

Figure 1 The framework of the proposed fusion model in which the above network is MDN and the below network is SRLN. $$$N_f$$$ represents the number of convolutional kernels, DF represents the dilated factor of convolutional kernel. $$$c_n$$$ and $$$res_n$$$ represent n-th convolution layer and residual learning respectively.

Figure 2 Reconstruction results for 30% radial sampling. The first and third row include groundtruth and reconstruction results of CSMRI methods. The second and fourth row include radial sampling mask and errors. Values of PSNR and SSIM are shown in the upper left corner.

Figure2. Process of restoring the missing k-data in the specified kernel. Extract the acquired data in this kernel and all the data in the target region as a 1D vector respectively. The mapping function can be fitted based on the fully sampled calibration data.

Figure4. Reconstructed images with different (a)loss function: MSE, MAE, Huber (b)number of nodes in each hidden layer (c)multi-contrast and reference images by conventional linear algorithm.

Figure 1. a. the proposed network architecture for joint MRI reconstruction and segmentation with model-based deep learning and U-net. b. the reconstruction scheme using model-based deep learning for cartesian and wave-encodings.

Figure 5. Reconstructed images using SENSE, MoDL, wave-CAIPI and wave-MoDL at R_yxR_z=4x4 on 32-channel HCP data.