Figure 2: Exemplary images (up-down: T2, T1POST, and FLAIR) from all tracks and finalist teams. The submissions were rated for overall appearance, artifacts, sharpness, and CNR. Overall, judged cases included both intra- and extra-axial tumors, strokes, microvascular ischemia, white matter lesions, edema, surgical cavities, as well as postsurgical changes and hardware including craniotomies and ventricular shunts.

Figure 3: Examples of failed reconstructions, i.e., resulting artifacts and hallucinations. In the 4x example, some non-winning submissions introduced vessel-like aliasing artifacts, originating from a susceptibility artifact. A similar aliasing-induced hallucination, resulting in incorrect sulcus depiction, is observed in the Transfer case. The particular shown 8x reconstruction suffered from a severe aliasing and hallucination behavior that renders the reconstruction unusable in more obvious fashion.

Figure 4. Illustration of using beamforming to steer the sensitivity of the virtual array to different spatial positions. The video shows ROVir results obtained from sweeping a circular-shaped ROI (i.e., everything interior to the green circle) through the FOV for a 32-channel brain MRI dataset.

Figure 3. Application of ROVir to reduce the size of the FOV in non-cartesian sagittal brain imaging, which enables highly-effective mitigation of aliasing artifacts when using a spiral k-space trajectory designed for a smaller FOV. The signal ROIs are marked in green, while the interference regions are marked in red. Reconstruction was based on simple gridding¹³.

Figure 2. Overview of Compact Maps in a dynamic MRI reconstruction pipeline. Maps are generally pre-computed for reconstructions with temporal constraints, not computed on-the-fly. Conventionally, maps are estimated (a) from time-averaged k-space data or (b) individually for each time frame, then stored in the spatial domain. (c) Instead, Compact Maps are linear combinations of temporal basis components. Memory estimates are for $$$N$$$=4 basis components, 24x24 map kernel size, 8 channels, and the listed matrix size. We see ~1000x savings in memory with Compact Maps over (b).

Figure 5. Comparison of image reconstruction quality using time-averaged maps and Compact Maps. Reconstruction has been performed for both time frames A and B using iterative SENSE with l₂ regularization of λ=0.001. Difference images (x10) with the fully sampled reference are also shown. Residual aliasing artifacts are visible in images reconstructed with time-averaged maps (red arrows) and in the magnitude difference images, but are mitigated when using Compact Maps. Normalized root-mean square error (NRMSE) is also reduced in the images reconstructed using Compact Maps.

Figure 1. Structure of Eigen-Sketching matrix $$$S^t$$$. Effectively, matrix $$$S^t$$$ sketches the sensitivity map operator $$$C$$$, yielding a reduced sensitivity map operator $$$C^t_S$$$ with less coils. Similar to coil compression^4-7(CC) the Eigen-Sketching matrix considers the high-energy virtual coils; but unlike CC, the matrix also considers a sketched coil. This sketched coil is formed by multiplying the remaining coils by “-1” or “+1” with probability 50% each and then summing them together. In sketching literature, this scheme is denoted as CountSketch.^10-12

Figure 5. Image comparison of 3D Cones reconstruction for (A) reconstruction with all $$$c=20$$$ coils, (B) coil compression with $$$\hat{c}=4$$$ coils, (C) our proposed Coil Sketching with $$$\hat{c}=4$$$ coils. Coil compression with $$$\hat{c}=4$$$ presents significantly lower image quality and shading artifacts, specially towards the lower borders of the volume. Coil Sketching yields virtually the same image as reconstruction with all $$$c=20$$$ coils.

Figure 1. The workflow of proposed method. 8-channel 4-contrast k-space data has been structured into a low-rank tensor. Subsequent higher-order singular value decomposition identifies both signal- and null-subspace bases. Different from low-rank reconstruction, all bases have been transformed to image space to estimate coil sensitivity and spatial support maps shared among contrasts by summation. The efficient image-space reconstruction was performed iteratively with the aforementioned procedures until convergence.

Figure 2. Estimated coil sensitivity and spatial support maps from fully sampled reference, uniformly undersampled, and reconstructed data (30th iteration) using the proposed method. Coil sensitivity and spatial support maps estimated from fully sampled reference data are nearly orthogonal complements within the brain region. The proposed iterative reconstruction can effectively and efficiently correct the estimation errors (locations indicated by red arrows) of each channel due to uniform undersampling (R = 4).

Figure 1: Relative B0 field changes $$$\omega(\textbf{r})$$$ in the head frame between two poses, obtained after registration, can be decomposed in lower-degree solid harmonics (n=2) and localised fields that are proportional to rotation angles $$$\theta_{pitch}$$$ and $$$\theta_{roll}$$$ (12 and 10 degrees for this example). The linear model $$$d(\textbf{r})$$$ (not shown here) was fitted on a total of 8 motion-free scans. Data shown here is not used in further reconstructions and only served to validate our model.

Figure 2: Sagittal (left), transversal (middle) and coronal (right) view of the (A) uncorrected (B) motion-corrected and proposed motion + B0 reconstruction (C) without and (D) with considering the solid harmonics (SH). Reduced motion corruption is observed when including the pose-dependent B0 fields with the biggest contribution coming from the linear B0 model, resulting in recovered signal near air-tissue interfaces (white arrows) and improved contrast, e.g. around the ventricles. Residual improvements are obtained when including the solid harmonics (red arrows).

Figure 3: Comparison of the proposed method with Low-Rank (LR) reconstructions. The first row shows representative images obtained by the LR reconstruction with rank 20 using k-space data of 3-min acquisition (reference). The rows 2 to 4 show reconstructions using only 45 sec of data. The 2^nd row shows images reconstructed using a global LR model of rank 20. The 3^rd row shows images obtained by performing a rank-10 local LR reconstruction (i.e. at each neighborhood). The 4^th row shows the images using the proposed method where the dimension of each tangent space is 10.

Figure 2: Global coordinates (spatial coefficients) and affine transform matrices of the proposed method. The 10×10 affine transform matrix $$${L}_c^{-1}$$$ is represented using grey levels. The 1st component of $$$T$$$ is a temporal average of all the neighborhoods. Our proposed model successfully describes the local coordinates by applying an affine transform $$${L}_c^{-1}$$$ to the global coordinates $$$T$$$ at each neighborhood $$$c$$$. The green dashed lines indicate the position of the liver for the 1^st component of the 1^st neighborhood.

Fig.4 Animated CINE for a third representative subject reconstructed with iterative SENSE (column 1), XD-GRASP (column 2) and the proposed MC-CINE (column 3) using 20 heartbeats (row 1), 2 heartbeats (row 2) and 1 heartbeat (row 3). The dynamics of the cardiac contraction are captured in every case, however considerable streaking and noise amplification are visible for XD-GRASP and especially iterative SENSE at higher accelerations. MC-CINE with 448 spokes (1 heartbeat) achieves comparable image quality to iterative SENSE and XD-GRASP with 8960 spokes (20 heartbeats).

Fig.1 Diagram of the proposed approach (considering here only three cardiac phases). 1) Acquired data is retrospectively binned into multiple cardiac phases. 2) Preliminary cardiac resolved images are obtained via XD-GRASP. 3) Each cardiac phase is registered to every other cardiac phase to estimate the cardiac motion between all phases. 4) The estimated motion is incorporated into the proposed MC-CINE to reconstruct each cardiac phase as a motion corrected image from all the data.

Fig.3: Diagram of both decompositions: a) vGRAPPA: Measured low SNR SVS data (yellow) is used to generate the kernel to decompose the acquired SMVS data (pink) to their respective voxel regions. The channel-wise ACS was obtained by 2D-stacking of the data points and averages. Then, this matrix was convolved with an 11x2 kernel to generate the pseudo inverse matrix following⁹. b) SENSE-based decomposition: channel-wise sensitivity maps and noise covariance matrix, derived from image- and spectral data, respectively, are used to decompose the SMVS data.

Fig.2: in vivo acquisition: a) voxel region of the left (blue, V1) and right motor cortex (orange, V2). b-c) Spectral shapes of the two SVS acquisitions (cyan/orange) and d-g) the SVS acquisition decomposed like the SMVS acquisitions with both the vGRAPPA (blue/red) and the SENSE-based (dark blue/dark red) algorithm. Both algorithms should decompose the SVS voxel to its respective region (d and g), while ideally resulting in equally distributed noise for the other voxel (e and f).

Figure 5. High-resolution and high-SNR metabolite and neurotransmitter maps estimated from the in vivo data (a 3.4×3.4×5.3mm³ nominal resolution). Anatomical images (T1w) across different slices from the 3D imaging volume are shown in the top row, and maps of different metabolites, as well as the Glx and GABA components, are shown in subsequent rows.

Figure 2. Monte-Carlo analysis showing the normalized standard deviation (std) of the coefficient estimates for different components from the UoSS fitting (Row 1: metabolite, Row 2: Glx, and Row 3: GABA). Several alternative TE choices with an equivalent acquisition time are considered for a 2-TE case. Specifically, columns 1-4 display the std maps for the case of single-TE (35ms, 2 average), first 2 TEs (35 and 50ms), random 2 TEs (50 and 110ms), and optimized 2 TEs (65 and 80ms), respectively, The best std maps were achieved by the optimized 2 TEs, consistent with the CRLB prediction.

Figure 4 Reconstruction for 30-channel abdominal MR data at MB=2/R=2. Compared to SPSG, the proposed CPI results in improved SNR in both background and abdominal regions. As shown in the zoomed view, SPSG suffered from slight residual artifact (vertical ringing indicated by red arrow), which was not visible in results from proposed CPI.

Figure 1 Diagram of the proposed calibrationless parallel imaging (CPI) reconstruction. It iteratively updates the estimated k-space by sequentially promoting structural low-rankness and enforcing data consistency. (A) Structural low-rankness is promoted for individual slices by constructing a block-wise Hankel matrix, performing singular value decomposition, and forcing low-rankness through rank truncation. (B) The data consistency is enforced for individual blades by minimizing the difference between synthesized/acquired SMS blade data.

Figure 1. Diagram of the proposed r2f-ESPIRiT reconstruction for reduced FOV parallel imaging with wave encoded k-space trajectory. Although accelerated reduced FOV imaging scan is performed, the full FOV image can be reconstructed.

Figure 5. Wave encoded 3D imaging with 1 (PE) × 2 (PAR) acceleration. In the conventional ESPIRiT reconstruction, 2 sets of maps were estimated from a reduced FOV ACS scan (a), and residual aliasing artifacts (yellow arrows) remain in the reconstructed image of RO-PAR plane (c). In the proposed r2f-ESPIRiT, an ACS scan with 40% slice-oversampling was used to estimate the sensitivity maps (b), but it does not add additional scan time, and an artifact-free and full FOV image can be obtained (d).

Marchenko-Pastur Virtual Coil Compression (MP-VCC) For the aliased region of MB=2 experiment, we display a (A) local spatial patch, $$$X_C$$$, (B) its VC basis $$$X_{VC}$$$, and (C) its eigenvalue spectrum with the MP distribution in black. (D) Spatially varying virtual coil maps, $$$P_C$$$, are shown for every experiment.

Statistics of Discarded VCs. Properties of the normalized residuals $$$r=(\text{noisy}-\text{denoised})/\sigma$$$, characterizing the discarded VCs, are evaluated via (A,B) histograms showing Gaussian distribution of $$$r$$$; and (C,D) power spectrum analysis, showing no memory along the measurement dimension (temporal power-spectrum $$$\Gamma(\omega)$$$ of residuals is flat) and marginal low-frequency bias along the spatial dimension (spatial power-spectrum $$$\Gamma(k)$$$ of residuals flat for almost all $k$)

Fig.3: Images reconstructed from different under-sampled datasets with Standard nuFFT reconstruction methods (A) (B) and the modified PICS reconstruction methods(C)(D), (G) is the fully sampled image with Standard nuFFT reconstruction.

Fig.2: Flowchart of PETRA data reconstruction with PICS algorithm

Fig 2. The aliasing of a zero-filled, density compensated estimate in the image and wavelet domains of a tenfold undersampled brain, and the wavelet-domain aliasing estimate $$$\boldsymbol{\tau}_0$$$. The histogram verifies that $$${\boldsymbol{r}}_0 \approx \boldsymbol{w}_0 + \mathcal{CN}(\boldsymbol{0}, \text{Diag}(\boldsymbol{\tau}_0))$$$ is an accurate model of the aliasing.

Fig. 4. The NMSE vs iteration of three example reconstructions, demonstrating the relative rapidity of convergence of P-VDAMP. The NMSE at the 0th iteration differs because the 0th estimate is defined to be after the first application of soft thresholding.

Fig4. In this single-coil, complex-valued, in vivo experiment with four contrasts, joint reconstruction with BaCS provides 2.0-fold to 1.2-fold RMSE reduction over sparseMRI for a wide range of 2D acceleration factors. It is 31x faster on the GPU, and 2.8x faster on the CPU compared to sparseMRI.

Fig5. Synergistic combination of BaCS with SENSE yields a significant, 1.4x improvement in RMSE over standard SENSE reconstruction at R=5-fold total acceleration using 32 channel data. This is made possible by the application uniform R₁=2-fold undersampling with SENSE, and an additional R₂=2.5-fold random undersampling on the reduced FOV coil images with joint BaCS reconstruction.

Figure 1. A step-by-step simulation experiment considering a single coil with homogeneous uniform sensitivity and 2-fold regular under-sampling to illustrate the priors of Wave in the proposed VCC-Wave model.

Figure 3. High resolution 2D imaging using a RF-spoiled GRE sequence and 24-channel head coils, with an isotropic resolution of 0.67 mm, under an acceleration factor of 6 (right to left).

Figure 1. Proposed reconstruction algorithm schematic.

Figure 4. MWF maps of the different method reconstructed images with difference maps (R=2*2).

Fig 2. Blurring due to T2 decay can be noticed in the imaging from a conventional scan (red arrow) and is mitigated in both SENSE and Temporal Harmonic Encoding reconstruction despite noise increment. In spite of contrast reduction around basal ganglia, Temporal Harmonic Encoding image retains the gray/white matter contrast in most of the brain. Both SENSE and Temporal Harmonic Encoding demonstrates similar SNR reduction compared to the conventional imaging. On the other hand, the regularization applied on Temporal Harmonic Encoding image restores SNR with improved image quality.

Fig 3. (a) The average of the other two harmonic components in Temporal Harmonic Encoding reconstruction indicates the signal changes with TE. Faster T2 decay region appears brighter in the harmonic image. (b) The inverse of (a) will convert the contrast to resemble a T2W image. (c) The T2 map is calculated from the multi-TE SENSE.

Figure 1: Diagram showing flow of stages of scan and image reconstruction.

Figure 5: Left: Subject. Center: Residual/error of MGRAPPA reconstruction of Subject. Right: Residual/error of GRAPPA reconstruction of Subject. The motion is rotation by 9 degrees and translation by 5 and 1 voxels/pixels in x and y directions respectively. GRAPPA reconstruction, but not MGRAPPA, showed a clear ghosting artifact. Accordingly, the L2-norm error decreased 41% with MGRAPPA.

Figure 2: One exemplary slice from the human scan is shown. Each row displays one b-value. In the first column a reconstruction without phase correction is shown, leading to strong ghosting in the diffusion weighted scans. The second column shows the GRAPPA reconstruction. The third column displays the ASeDiWA reconstruction. Both parallel imaging reconstruction methods correct the ghosting seen in the first column. However, ASeDiWA consistently provides higher SNR as the GRAPPA reconstruction.

Figure 3: Reconstruction results (b/w) and g-factor simulations (color) are shown. Each row displays data with a different number of simulated segments (2, 3, and 4). The first column shows the data reconstructed with GRAPPA. The following three columns show results from ASeDiWA without and with one and two iterations, respectively. In general, higher segmentation leads to higher g-factors. Independent of the segmentation, the comparison of ASeDiWA with GRAPPA shows a reduced g-factor. Further improvement can be achieved by iteration of the algorithm (ASeDiWA₁ and ASeDiWA₂).

Figure 1. Multiple maps modelling the aliasing of reduced field-of-view imaging could be created from full-FOV ESPIRiT CSM estimated from full-FOV calibration scan, and used in soft-SENSE reconstruction to reconstruct full-FOV image without fold-over artifacts.

Figure 3. Full-FOV reconstruction of reduced FOV parallel imaging along two phase encoding directions with 2x2 downsampling by proposed ESPIRiT 4 maps created from full-FOV calibration scan

Figure 2. Quantitative SNR maps derived from (a) the pseudo multiple-replica method (Mean SNR: 19.34), (b) directly from the proposed method (Mean SNR: 19.75), and (c) neglecting kernel phase correction (Mean SNR: 20.13). The difference map (d) is from (a) and (b), (e) is from (a) and (c). The deviation of the reconstructed synthetic noise and the calculated noise map is shown in (f) and (g), with similar RMS (~1.02). Both (d) and (f) show that the noise map with phase correction is more consistent with the reference method.

Figure 1. (a) schematic of the SG-DK method to obtain the k-space data for a unaliased slice. Two kernel sets are calculated for odd and even lines in each unaliased slice respectively; (b) schematic of the kernel rearranging. By exchanging some kernel values, the new kernel sets can be applied on disjunct sets of k-space in collapsed data respectively. The final unaliased slice is then obtained by adding two reconstructed data from these two kernels. It notes that the full k-space of collapsed data has been divided into two sets with zero-padded lines (indicated with dot lines).

DS-CSME: a deep learning solution for coils sensitivity estimation, allowing end-to-end learning framework for accelerated parallel magnetic resonance imaging reconstruction.

An example comparing the fully sampled target to predictions obtained using precomputed CSMs and those obtained by the end-to-end system with DC-CSME. At ~5× acceleration, predictions with precomputed CSMs show more artifacts.

Figure 1. Illustration of SB-EPI correction. Figure 1(a): SB-EPI with Nyquist artifact. Figure 1(b): Regions marked as green used for CSP comparison. Figure 1(c): A series of corrected complex images of one coil after applying different possible phase value. Figure 1(d): Reference Nyquist artifact-free CSP and CSPs generated from corrected complex images in Figure 1(c). Figure 1(e): Variations of CSP difference compared to reference Nyquist artifact-free CSP with different phase value.

Figure 2. Workflow of correction for SENSE MB-EPI ($$$Acc_{in}=2,Acc_{MB}=2$$$). (A): SENSE MB-EPI; (B): Possible phase value along frequency-encoding direction; (C): Performed correction; (D): Corrected complex images with different phase values; (E): Corresponding CSPs; (F): Nyquist artifact-free CSP; (G): Determined phase $$$C_{0}$$$; (H): Possible phase value along phase-encoding direction; (I): Cycled 2D phase map; (J): Unaliased CSP from scanner; (K): Performed correction with 2D phase map and unaliased CSP; (L) Final corrected images.

Magnitude reconstruction results for a brain acquired at acceleration factor 2, contrast T2, and field strength of 7T. The top row represents the reconstruction using the different methods, while the bottom one represents a zoom in the cerebellum region, an anatomical feature that was not present in the XPDNet training set.

Magnitude reconstruction results for a specific fastMRI slice with T2 contrast, at acceleration factor 8. The top row represents the reconstruction using the different methods, while the bottom row represents the error when compared to the reference.

Figure 4: A low-resolution head scan acquired with a fast bandwidth and angular under-sampling. Despite the increased dead-time gap and paucity of calibration data, ZINFANDEL provides an artefact free reconstruction.

Figure 2: Diagram of the ZINFANDEL algorithm. In step 1, a kernel matrix W is calculated from the calibration region. In step 2, a point in the dead-time gap is filled. By looping first over spokes and then missed samples, the matrix W can be progressively updated towards the center of k-space.

The workflow of the HyKE image reconstruction pipeline implemented with the BART phantom. The first row displays the k-space. The second row shows the corresponding images. The numerical phantom was retrospectively subsampled with the Hybrid k-space EPI sampling pattern (column 1) for each of the coil images (N = 8). SPIRiT was run for 20 iterations (column 2) followed by coil combination (column 3). The phantom then was split into two separate k-spaces (column 4). A POCS Partial Fourier image reconstruction (column 5) ran for 10 iterations resulting in the two reconstructed images.

A schematic of the image reconstruction pipeline for the HyKE approach. (A) The hybrid k-space EPI incorporates several reconstructions techniques, including (B) SPIRiT, (C) conjugate coil combination from the coil sensitivity maps, (D) separating the different sampling k-space trajectories, and a (E) POCS partial Fourier reconstruction, in order to acquire (F) two separate images from one hybrid EPI acquisition. The resultant output images have multiple advantages; such as accelerated imaging and the possibility to characterize EPI distortion in a single scan.

Figure 1: Block diagram of the proposed CS-GROG-BPE scheme. Undersampled radial signal is used to generate the Bunch Phase Encoded k-space at the specified k_max and NBP, using GROG. Non-Cartesian BPE k-space is gridded to Cartesian k-space using a defined oversampling factor and step-size using GROG. Gridded k-space is then reconstructed coil-by-coil by CS algorithm with specified regularization parameters. Coil-by-coil reconstructed images are then combined using sum-of-square reconstruction to yield the final output image.

Figure 2: Reconstruction Results (a) shows fully sampled ground truth image. (b) shows results of CS-GROG-BPE method proposed in this work, at acceleration at AF = 8, 10, 12. (c) shows results of GROG-BPE CG-INNG method² with the same BPE generation and gridding parameters. Quantifying parameters including AF, RMSE and SNR have been provided underneath each reconstructed image.

Field-map-combined images for selected cardiac phases. The TV reconstructions have near-band-flow-related streaking artifacts (red arrows) and undersampling artifacts (peach arrow). In addition, some hyperintensities (yellow arrows) persist, possibly from using noisier field maps for phase-cycle combination. The MSLR reconstructions have smoother blood signal since flow artifacts were localized to near the bands, which were then removed by field-map combination. In addition, both blood-myocardium boundaries (cyan arrows) and background structures appear sharper.

Reconstructions from the inverse-gridded and retrospectively undersampled dataset. Six cardiac phases evenly spaced through the cardiac cycle are shown. The phase cycles were root-sum-of-squares combined. Multi-scale low rank results in sharper images and better removal of undersampling artifacts (e.g., in the background) than total variation.

Figure 2. Images for detecting myocardial infarction by (a) MDI PSIR and (b) the conventional PSIR reconstruction methods. (c) signal intensity analysis for the five representative regions (three in myocardium and two in the background) circled in (a) and correspondingly in (b).

Figure 1. Illustration of MDI PSIR reconstruction method

Figure 1: The SPIDER framework.

Figure 3: Comparison of images from Multitasking reconstruction (reference) and the proposed SPIDER method. From left to right: (a) Reference from Multitasking reconstruction; (b) Contrast-variated SPIDER real-time images; (c) Difference between (a) and (b); (d-f) Contrast-frozen T1w, T2w, and PDw SPIDER real-time images.

Figure 1. Illustrations of the data acquisition and reconstruction scheme. (a) A continuous golden-angle radial sampling method was used for i-MRI in this study (golden angle = 111.25^°). (b) Conventional dynamic image reconstruction based on a retrospective scheme. (c) The proposed group-based reconstruction method for real-time i-MRI reconstruction.

Figure 2. A comparison of different algorithms. The ground truth is the 150th simulated brain intervention image. A group-based reconstruction strategy (10 spokes per frame, 5 frames per group, a total of 200 frames for 2000 spokes) was adopted for reconstruction using NUFFT, GRASP, and LSFP. The acceleration factor was about 40.

FIG.2. Reconstruction results comparison for cardiac cine dataset with radial trajectory and r=2. From top to bottom: time-frame magnitude images, error map, x-t temporal profiles in white dotted line. The proposed method had lower artifacts for reconstruction of cardiac blood pools than the other methods by the arrows in the error map. By the arrows in the temporal profile, the methods to be compared present temporal blurring artifacts, which are effectively removed by the proposed mothed.

Table.1. PSNR/RMSE comparison of various reconstruction methods on PINCAT dataset, Cine dataset and Perfusion dataset.

Figure 2: Flow chart of the acceleration method. All calculations are done in the k-space domain.

Figure 3: A time evolution comparison of for two patients over 2.5 minutes using the spatial PCA method (red) and our proposed time-domain PCA method (blue) using normalised mean square error for an acceleration factor of 3.

A visualization of the reconstruction concept: A moving object is measured in a series of undersampled k-spaces. In this abstract, we simulated a 4-times acceleration per k-space with the 8 center k-space lines being measured additionally. The OT reconstruction allows not only to recover the image-series, but also the momentum fields of the object. For illustration purposes, only every second frame is shown.

Zoom-in on the results from figure 2: The reconstructed image closely mirrors the ground truth. Despite every frame only having approximately one fourth of k-space sampled with a single coil, the brain could be reconstructed in detail with only the optimal transport regularization. The registered image is slightly blurry, showing potential problems with the motion field.

Figure 2. CRBs for resolving 5 thin slices as a function of the number of voxels sharing the same phase value, plotted for several different RF encoding strategies. Note that without phase constraints, the CRB blows up to infinity whenever the number of RF encodings is smaller than the number of thin slices (5 in this case).

Figure 3. Simulated reconstruction of 5 thin slices from the set of 4 optimized RF encodings, for both phase-constrained and minimum norm reconstruction. For ease of visualization, we only show two out of the five reconstructed slices for each case.

Figure 5: (A) T2-weighted b0 image. (B) Naive combination of 12-shot data has severe multi-shot ghosting resulting in signal dropout. (C) Navigator shot combination has some residual ghosting (green). (D) SLLR reconstruction of uniformly sampled b500 data has loss of detail (yellow, orange), and reduced organ contrast (red). (E) Random sampling improves detail (yellow, orange). (F) Center oversampling further enhances detail (yellow, orange), restores organ contrast, matching the navigator combination (red), and improves SNR (white).

Figure 1: Random sampling patterns were generated by dividing k-space into $$$ETL$$$ segments, represented by different colours. Each sample is randomly assigned to a shot. K-space is fully covered when shots are combined. For center oversampling, the center two ky lines were added to the beginning of a randomly generated shot. A low-resolution phase navigator was explicitly acquired with uniformly sampled data because the shot undersampling factor is too high for parallel imaging to recover phase maps.

Fig. 3. (a): Sum-of-squares image obtained from all individual blade images in 2a. (b): The image obtained by the proposed MBIR with virtual coil LLR. (c): B0 field map obtained from 6-echo fast gradient-echo images via fat-water separation. The red circles highlight severe geometric distortion (a) and its improvement with the proposed scheme (b) in the region with strong field inhomogeneity as shown in (c).

Fig.1. 1D phase differences between positive and negative readout echoes (2x-oversampling not removed). Their offsets and slopes appear slightly different among the eight different blade angles.

Figure 1: 15 interleaves of the generated trajectory within normalised k-space.

Figure 2: Simulated PSFs in the $$$xy$$$-plane with $$$z=0$$$ of the presented approach according to the undersampling factors $$$R_1$$$ in a), $$$R_2$$$ in b), $$$R_3$$$ in c) and $$$R_4$$$ in d).

Fig.2. Comparison of CG convergence with and without the learned and the circulant preconditioner. (a) Flair brain scan (128x128). The learned preconditioner reduces the number of iterations in CG by a factor 2.9. This is a slightly larger reduction compared to the factor of 2.7 obtained with the circulant preconditioner. (b) Similar results are obtained for a TSE scan with a twice as large matrix size compared to that in the training set. (c) The speed up is slightly lower in the knee (128x128), which is an anatomy that the network has not observed during training.

Fig.1. The network’s input and output for two training examples. Each 37-channel input contains a sampling mask (R), regularization masks (λ and γ), 16 complex coil sensitivity maps and a complex image (y). Note that the complex channels are first split into real and imaginary components. The network’s prediction My is close to the ground truth, which is confirmed by the small error values both for the brain case (a) and for the noise case (b) (normalized norm < 0.16).

Figure 1: Distorted FID, distorted Projection, corrected FID and corrected projection fro simulated hollow cube object.

Figure 2: Brain Images of Health Adult Subject.
Bottom Row Left to Right: Un-corrected ZTE intrinsic contrast, IR-ZTE and T₂-ZTE
Top Row Left to Right: Corrected (Nd = 3) ZTE intrinsic contrast, IR-ZTE and T₂-ZTE

Axial T2SE images of the prostate of one subject: (A) standard 3mm thick T2SE (acquired in 4:14) and (B-F) 1mm thick images. (B) was reconstructed using the linear KZM method[1], while (C-F) were generated with the new MBIR at the λ values shown. Note some improved sharpness of (B) vs. (A) owing to the thinner slice but an increased noise level. MBIR with increasing λ provides progressive reduction in noise in (C-F). However, use of λ=0.00075 caused undesirable blurring, e.g. between the peripheral and transition zones (F, arrow).

Results from a second case. Fig. 2B vs. A better portrays a lesion (B, arrow), but MBIR provides further improved depiction (D, yellow arrow) and definition of a second lesion (orange arrow). Higher λ values again cause blur. Across the patient studies the radiologist’s perceived preference was for λ=0.00025.

Visualization of local phase images from the different reconstruction procedures. Panels (A) and (B) show the local phase from Bilgic et al⁷, while panel (C) shows the local phase obtained from our automatic WaveCS reconstruction procedure. Panel (D) shows the reference. In parenthesis it is shown the under-sampling factor for each reconstruction.

Visualization of magnitude Images from the different reconstruction procedures. Panels (A) and (B) show the results from Bilgic et al⁷, while panel (C) shows the result from our automatic WaveCS reconstruction. Panel (D) shows the reference. In parenthesis it is shown the under-sampling factor for each reconstruction.

Figure 1. k-space data obtained from the scanner is undersampled using the cartesian sampling mask. The Joint Reconstruction Algorithm is used to reconstruct the series of T1w images and their parameter maps. The process is repeated to reconstruct T2w/T2*w images and their parameter maps. Validation of the results is carried out by identifying the tissue types like muscle, fluid, tumor and adipose and using the rules on T1 and T2 maps. The T2*w images are subjected [1] to estimate PDFF, PDwF, & in the muscle, fluid, tumor and adipose tissue type.

Figure 2. T1w reconstructed images for 2 mouse slices with mask of ~27%, ~36% k-space data are shown.The sampling masks are shown in col 1. The results for the Repetition times (TR) 0.4s and 5s are shown in col 2 & 3 for mouse 1 and col 4 & 5 for mouse 2. The 5th & 6th column shows the detailed version of the region highlighted in the red box.The MI value is computed for 4 undersampling masks 18%, 27%, 36%, & 52% shown on x-axis. The y-axis shows the MI value when the ground truth |u| are compared to the reconstructed |u|. The mean MI and standard error of the mean (S.E.M) is calculated over n = 30 mouse slices.

Knee images from the different strategies to define the regularization weighting. We compare (A) $$$ \lambda_{\textrm{Lcurve}} $$$ (14 iterations) and (B) $$$ \lambda_{\textrm{NMSE}} $$$ (14 iterations) strategies to (C) our approach, $$$ \lambda_{\textrm{auto}} $$$, using the 4th-level Daubechies mother wavelet (1 iteration). Panel (D) shows the reference and panel (E) shows quantitative results. Smaller values along each axis in (E) represent more accurate reconstructions.

Visualization of magnitude and phase images from simulations. Panels (B) and (E) show the images from the reconstruction process using our approach for USF=5 and $$$\sigma=5$$$. Panels (A) and (D) show the reference. Panels (C) and (F) show the difference between both columns. All images are scaled according to their respective reference.

Figure 1. The reconstruction results using all comparison methods at AF= 4, 5. The values at left-top of images are the corresponding RE($$$\%$$$) and PSNR(dB).

Figure 2. The RE (left) and PSNR (right) plots of all reconstructions at various AFs (AF=3, 4, 5).

Fig. 2: Variable density poisson disc sampling patterns for different values of $$$\gamma$$$ and different aspect ratios.

Table 1: Run times for generating variable density poisson-disc sampling patterns with the proposed algorithm and the Tulleken algorithm. Time is reported in milliseconds. In all cases, the proposed algorithm is faster by $$$30-50\%$$$.

Impact of the size of the dataset for automatic low rank reconstructions using SNR 5 and an under-sampling factor of 4x. The first row corresponds to the mean appearance of the reconstructions, $$$\hat{X}$$$, for 5,10,40,100, and 600 images (from left to right, respectively). The second row corresponds to the absolute difference with respect to the reference, $$$X$$$, as $$$5 \times |\hat{X}-X|$$$.

Impact of the size of the dataset for automatic low rank reconstructions using SNR 30 and an under-sampling factor of 14x. The first row corresponds to the mean appearance of the reconstructions, $$$\hat{X}$$$, for 5,10,40,100, and 600 images (from left to right, respectively). The second row corresponds to the absolute difference with respect to the reference, $$$X$$$, as $$$5 \times |\hat{X}-X|$$$.