Constrained Reconstruction Method Research Articles

Highly accelerated non-contrast-enhanced time-resolved 4D MRA using stack-of-stars golden-angle radial acquisition with a self-calibrated low-rank subspace reconstruction.

To develop a highly accelerated non-contrast-enhanced 4D-MRA technique by combining stack-of-stars golden-angle radial acquisition with a modified self-calibrated low-rank subspace reconstruction. A low-rank subspace reconstruction framework was introduced in radial 4D MRA (SUPER 4D MRA) by combining stack-of-stars golden-angle radial acquisition with control-label k-space subtraction-based low-rank subspace modeling. Radial 4D MRA data were acquired and reconstructed using the proposed technique on 12 healthy volunteers and 1 patient with steno-occlusive disease. The performance of SUPER 4D MRA was compared with two temporally constrained reconstruction methods (golden-angle radial sparse parallel [GRASP] and GRASP-Pro) at different acceleration rates in terms of image quality and delineation of blood dynamics. SUPER 4D MRA outperformed the other two reconstruction methods, offering superior image quality with a clear background and detailed delineation of cerebrovascular structures as well as great temporal fidelity in blood flow dynamics. SUPER 4D MRA maintained excellent performance even at higher acceleration rates. SUPER 4D MRA is a promising technique for highly accelerating 4D MRA acquisition without comprising both temporal fidelity and image quality.

J-LOUCR: Joint learned optimized undersampling and constrained reconstruction for accelerated MRI by reference-driven deep image prior

Undersampled magnetic resonance (MR) image reconstruction performance is related to two fundamental problems: where to sample and how to reconstruct the image. Previously developed deep learning-based magnetic resonance imaging (MRI) methods, including undersampled mask optimization and reconstruction, require large-scale, high-quality, and patient-based datasets for training, which presents a challenge in practical clinical scenarios. To reduce the dependence on datasets and perform undersampling and reconstruction simultaneously, we propose a novel joint learned optimized undersampling and constrained reconstruction method for accelerated MRI by a reference-driven deep image prior (J-LOUCR). Inspired by the deep image prior (DIP), J-LOUCR involves no pretraining procedure, resulting in no dependence on large-scale training datasets. The implementation of J-LOUCR includes learned optimization of the undersampled mask and constrained reconstruction with the reference-driven DIP. In the first step, based on the traditional DIP, we introduce a reference image as the network input and propose a joint optimization problem of the undersampled mask and network parameters. Then, the joint optimization problem is deconstructed into subproblems and solved by using the alternative iteration strategy to obtain the optimal undersampled mask guided by the reference image. In the second step, for the optimal undersampled mask, we consider and recommend that the structural similarity between the reference and target MR images be used as prior information. Under the dual drivers of the reference image as the network input and prior constraint, J-LOUCR optimizes the network parameters and then completes reconstruction. Experiments on invivo MR scans demonstrate that J-LOUCR can achieve more accurate reconstruction with more details and features.

A Physics-Based Algorithm to Universally Standardize Routinely Obtained Clinical T2-Weighted Images

MR images can be challenging for machine learning and other large-scale analyses because most clinical images, for example, T2-weighted (T2w) images, reflect not only the biologically relevant T2 of tissue but also hardware and acquisition parameters that vary from site to site. Quantitative T2 mapping avoids these confounds because it quantitatively isolates the biological parameter of interest, thus representing a universal standardization across sites. However, efforts to incorporate quantitative mapping sequences into routine clinical practice have seen slow adoption. Here we show, for the first time, that the routine T2w complex raw dataset can be successfully regarded as a quantitative mapping sequence that can be reconstructed with classical optimization methods and physics-based constraints. While previous constrained reconstruction methods are unable to reconstruct a T2 map based on this data, the expanding-constrained alternating minimization for parameter mapping (e-CAMP), which employs stepwise initialization, a linearized version of the exponential model and a phase conjugacy constraint, is demonstrated to provide useful quantitative maps directly from a vendor T2w single image data. This paper introduces the method and demonstrates its performance using simulations, retrospectively undersampled brain images, and prospectively acquired T2w images taken on both phantom and brain. Because T2w scans are included in nearly every protocol, this approach could open the door to creating large, standardized datasets without requiring widespread changes in clinical protocols.

Implicit Solutions of the Electrical Impedance Tomography Inverse Problem in the Continuous Domain with Deep Neural Networks.

Electrical impedance tomography (EIT) is a non-invasive imaging modality used for estimating the conductivity of an object Ω from boundary electrode measurements. In recent years, researchers achieved substantial progress in analytical and numerical methods for the EIT inverse problem. Despite the success, numerical instability is still a major hurdle due to many factors, including the discretization error of the problem. Furthermore, most algorithms with good performance are relatively time consuming and do not allow real-time applications. In our approach, the goal is to separate the unknown conductivity into two regions, namely the region of homogeneous background conductivity and the region of non-homogeneous conductivity. Therefore, we pose and solve the problem of shape reconstruction using machine learning. We propose a novel and simple jet intriguing neural network architecture capable of solving the EIT inverse problem. It addresses previous difficulties, including instability, and is easily adaptable to other ill-posed coefficient inverse problems. That is, the proposed model estimates the probability for a point of whether the conductivity belongs to the background region or to the non-homogeneous region on the continuous space Rd∩Ω with d∈{2,3}. The proposed model does not make assumptions about the forward model and allows for solving the inverse problem in real time. The proposed machine learning approach for shape reconstruction is also used to improve gradient-based methods for estimating the unknown conductivity. In this paper, we propose a piece-wise constant reconstruction method that is novel in the inverse problem setting but inspired by recent approaches from the 3D vision community. We also extend this method into a novel constrained reconstruction method. We present extensive numerical experiments to show the performance of the architecture and compare the proposed method with previous analytic algorithms, mainly the monotonicity-based shape reconstruction algorithm and iteratively regularized Gauss-Newton method.

A constrained boundary gradient reconstruction method for unstructured finite volume discretization of the Euler equations

We propose a novel boundary gradient and solution reconstruction approach for the second-order cell-centered unstructured finite volume method for solving the inviscid Euler equations. The proposed constrained reconstruction method involving ghost cells is shown to be more accurate than several conventional boundary reconstruction methods and linear-exact in the near-boundary areas. In the present procedure, the solutions at the ghost cell centroids are given by solving constraint equations, which are established based on boundary conditions. At the same time, the solution gradients in boundary cells and the solutions in boundary surfaces are obtained altogether, while guaranteeing the compatibility of the boundary conditions with the reconstructed results. A variety of numerical test cases show that the new method effectively reduces the errors of near-boundary approximations and improves the overall numerical accuracy. Furthermore, because the constrained reconstruction is only used for the boundary cells, the extra computational cost barely changes the overall computational efficiency.

Damage detection in cementitious materials with optimized absolute electrical resistance tomography

Electrical resistance tomography (ERT) serves as a non-invasive, non-destructive, non-radioactive imaging technique. It has potential applications in industrial and biological imaging. This paper presents an optimized inverse algorithm, named Newton’s Constrained Reconstruction Method (NCRM), to detect damage in cementitious materials. Several constraints were utilized in the proposed algorithm to optimize initial parameters. The range and spatial distribution of conductivities within the sample were chosen as two main constraints. Two sets of numerical and a set of experimental voltage data were used to reconstruct conductivity distribution images based on this algorithm. To evaluate the quality of reconstructed images, two image quality evaluation indicators, correlation coefficient and position error were used. Results show that the proposed algorithm NCRM has the ability to enhance the reconstructed image quality with fewer artifacts and has better positioning accuracy.

Deep learning for radial SMS myocardial perfusion reconstruction using the 3D residual booster U-net

PurposeTo develop an end-to-end deep learning solution for quickly reconstructing radial simultaneous multi-slice (SMS) myocardial perfusion datasets with comparable quality to the pixel tracking spatiotemporal constrained reconstruction (PT-STCR) method. MethodsDynamic contrast enhanced (DCE) radial SMS myocardial perfusion data were obtained from 20 subjects who were scanned at rest and/or stress with or without ECG gating using a saturation recovery radial CAIPI turboFLASH sequence. Input to the networks consisted of complex coil combined images reconstructed using the inverse Fourier transform of undersampled radial SMS k-space data. Ground truth images were reconstructed using the PT-STCR pipeline. The performance of the residual booster 3D U-Net was tested by comparing it to state-of-the-art network architectures including MoDL, CRNN-MRI, and other U-Net variants. ResultsResults demonstrate significant improvements in speed requiring approximately 8 seconds to reconstruct one radial SMS dataset which is approximately 200 times faster than the PT-STCR method. Images reconstructed with the residual booster 3D U-Net retain quality of ground truth PT-STCR images (0.963 SSIM/40.238 PSNR/0.147 NRMSE). The residual booster 3D U-Net has superior performance compared to existing network architectures in terms of image quality, temporal dynamics, and reconstruction time. ConclusionResidual and booster learning combined with the 3D U-Net architecture was shown to be an effective network for reconstructing high-quality images from undersampled radial SMS datasets while bypassing the reconstruction time of the PT-STCR method.

High-resolution sodium imaging using anatomical and sparsity constraints for denoising and recovery of novel features.

To develop and evaluate a novel method for reconstruction of high-quality sodium MR images from noisy, limited k-space data. A novel reconstruction method was developed for reconstruction of high-quality sodium images from noisy, limited k-space data. This method is based on a novel image model that contains a motion-compensated generalized series model and a sparse model. The motion-compensated generalized series model enables effective use of anatomical information from a proton image for denoising and resolution enhancement of sodium data, whereas the sparse model enables high-resolution reconstruction of sodium-dependent novel features. The underlying model estimation problems were solved efficiently using convex optimization algorithms. The proposed method has been evaluated using both simulation and experimental data obtained from phantoms, healthy human volunteers, and tumor patients. Results showed a substantial improvement in spatial resolution and SNR over state-of-the-art reconstruction methods, including compressed sensing and anatomically constrained reconstruction methods. Quantitative tissue sodium concentration maps were obtained from both healthy volunteers and brain tumor patients. These tissue sodium concentration maps showed improved lesion fidelity and allowed accurate interrogation of small targets. A new method has been developed to obtain high-resolution sodium images with good SNR at 3 T. The proposed method makes effective use of anatomical prior information for denoising, while using a sparse model synergistically to recover sodium-dependent novel features. Experimental results have been obtained to demonstrate the feasibility of achieving high-quality tissue sodium concentration maps and their potential for improved detection of spatially heterogeneous responses of tumor to treatment.

Simultaneous recovery of both bright and dim structures from noisy fluorescence microscopy images using a modified TV constraint.

The quality and information content of biological images can be significantly enhanced by postacquisition processing using deconvolution and denoising. However, when imaging complex biological samples, such as neurons, stained with fluorescence labels, the signal level of different structures can differ by several orders of magnitude. This poses a challenge as current image reconstruction algorithms are focused on recovering low signals and generally have sample-dependent performance, requiring tedious manual tuning. This is one of the main hurdles for their wide adoption by nonspecialists. In this work, we modify the general constrained reconstruction method (in our case utilizing a total variation constraint) so that both bright and dim structures can drive the deconvolution with equal force. In this way, we can simultaneously obtain high-quality reconstruction across a wide range of signals within a single image or image sequence. The algorithm is tested on both simulated and experimental data. When compared with current state-of-art algorithms, our algorithm outperforms others in terms of maintaining the resolution in the high-signal areas and reducing artefacts in the low-signal areas. The algorithm was also tested on image sequences where one set of parameters are used to reconstruct all images, with blind evaluation by a group of biologists demonstrating a marked preference for the images produced by our method. This means that our method is suitable for batch processing of image sequences obtained from either spatial or temporalscanning. LAY DESCRIPTION: Fluorescence microscopy images of complex biological samples contain a wide range of signal levels. This signal variation leads current reconstruction algorithms, which aim to enhance the quality of the raw images, to have sample-dependent performance. In this work, we design a new optimization that allows the reconstruction to "pay equal eqattention to" both bright and dim structures. In this way, we can simultaneously recover both bright and dim structures within a single image or image sequence, as validated when the algorithm was quantitatively tested on both simulated and experimental data. When our method was evaluated alongside current state of art algorithms by a group of biologists, our algorithm was considered qualitativelysuperior.

Feasibility of multiple-view myocardial perfusion MRI using radial simultaneous multi-slice acquisitions.

PurposeDynamic contrast enhanced MRI of the heart typically acquires 2–4 short-axis (SA) slices to detect and characterize coronary artery disease. This acquisition scheme is limited by incomplete coverage of the left ventricle. We studied the feasibility of using radial simultaneous multi-slice (SMS) technique to achieve SA, 2-chamber and/or 4-chamber long-axis (2CH LA and/or 4CH LA) coverage with and without electrocardiography (ECG) gating using a motion-robust reconstruction framework.Methods12 subjects were scanned at rest and/or stress, free breathing, with or without ECG gating. Multiple sets of radial SMS k-space were acquired within each cardiac cycle, and each SMS set sampled 3 parallel slices that were either SA, 2CH LA, or 4CH LA slices. The radial data was interpolated onto Cartesian space using an SMS GRAPPA operator gridding method. Self-gating and respiratory states binning of the data were done. The binning information as well as a pixel tracking spatiotemporal constrained reconstruction method were applied to obtain motion-robust image reconstructions. Reconstructions with and without the pixel tracking method were compared for signal-to-noise ratio and contrast-to-noise ratio.ResultsFull coverage of the heart (at least 3 SA and 3 LA slices) during the first pass of contrast at every heartbeat was achieved by using the radial SMS acquisition. The proposed pixel tracking reconstruction improves the average SNR and CNR by 21% and 30% respectively, and reduces temporal blurring for both gated and ungated acquisitions.ConclusionAcquiring simultaneous multi-slice SA, 2CH LA and/or 4CH LA myocardial perfusion images in every heartbeat is feasible in both gated and ungated acquisitions. This can add confidence when detecting and characterizing coronary artery disease by revealing ischemia in different views, and by providing apical coverage that is improved relative to SA slices alone. The proposed pixel tracking framework improves the reconstruction while adding little computational cost.

Radial simultaneous multi-slice CAIPI for ungated myocardial perfusion

ObjectiveSimultaneous multi-slice (SMS) imaging is a slice acceleration technique that acquires multiple slices in the same time as a single slice. Radial controlled aliasing in parallel imaging results in higher acceleration (radial CAIPIRINHA or CAIPI) is a promising SMS method with less severe slice aliasing artifacts as compared to its Cartesian counterpart. Here we use radial CAIPI with data undersampling and constrained reconstruction to improve the utility of ungated cardiac perfusion acquisitions. We test the proposed framework with a traditional saturation recovery fast low-angle shot (turboFLASH) sequence and also without saturation recovery as a steady-state spoiled gradient echo (SPGR) sequence on animal and human studies. MethodsSimulations and phantom studies were performed for both the turboFLASH and the SPGR radial CAIPI methods. Ungated undersampled golden ratio radial CAIPI data with saturation recovery were acquired in 8 dogs and 2 human subjects. The CAIPI data without saturation pulses were acquired in 4 human subjects. For both methods, slice acceleration factors of two and three were used. A new spatio-temporal reconstruction using total variation and patch-based low rank constraints was used to jointly reconstruct the multi-slice multi-coil images. ResultsPhantom scans and computer simulations showed that ungated SPGR generally provides better contrast to noise ratio (CNR) than the saturation recovery sequence if the saturation recovery time is less than 100ms. Both of the ungated radial CAIPI methods demonstrated promising image quality in terms of preserving dynamics of the contrast agent and maintaining anatomical structures, even with three slices acquired simultaneously. ConclusionUngated simultaneous multi-slice acquisitions with either a saturation recovery turboFLASH sequence or a steady-state gradient echo SPGR sequence are feasible and provide increased slice coverage without loss of temporal resolution. Compared with a sensitivity encoding (SENSE) SMS reconstruction, the constrained reconstruction method provides better image quality for undersampled radial CAIPI data.

ACCELERATING MAGNETIC RESONANCE IMAGING VIA DEEP LEARNING.

This paper proposes a deep learning approach for accelerating magnetic resonance imaging (MRI) using a large number of existing high quality MR images as the training datasets. An off-line convolutional neural network is designed and trained to identify the mapping relationship between the MR images obtained from zero-filled and fully-sampled k-space data. The network is not only capable of restoring fine structures and details but is also compatible with online constrained reconstruction methods. Experimental results on real MR data have shown encouraging performance of the proposed method for efficient and effective imaging.

Split Bregman multicoil accelerated reconstruction technique: A new framework for rapid reconstruction of cardiac perfusion MRI.

Rapid reconstruction of undersampled multicoil MRI data with iterative constrained reconstruction method is a challenge. The authors sought to develop a new substitution based variable splitting algorithm for faster reconstruction of multicoil cardiac perfusion MRI data. The new method, split Bregman multicoil accelerated reconstruction technique (SMART), uses a combination of split Bregman based variable splitting and iterative reweighting techniques to achieve fast convergence. Total variation constraints are used along the spatial and temporal dimensions. The method is tested on nine ECG-gated dog perfusion datasets, acquired with a 30-ray golden ratio radial sampling pattern and ten ungated human perfusion datasets, acquired with a 24-ray golden ratio radial sampling pattern. Image quality and reconstruction speed are evaluated and compared to a gradient descent (GD) implementation and to multicoil k-t SLR, a reconstruction technique that uses a combination of sparsity and low rank constraints. Comparisons based on blur metric and visual inspection showed that SMART images had lower blur and better texture as compared to the GD implementation. On average, the GD based images had an ∼18% higher blur metric as compared to SMART images. Reconstruction of dynamic contrast enhanced (DCE) cardiac perfusion images using the SMART method was ∼6 times faster than standard gradient descent methods. k-t SLR and SMART produced images with comparable image quality, though SMART was ∼6.8 times faster than k-t SLR. The SMART method is a promising approach to reconstruct good quality multicoil images from undersampled DCE cardiac perfusion data rapidly.

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

PurposeTo evaluate the potential value of combining multiple constraints for highly accelerated cardiac cine MRI. MethodsA locally low rank (LLR) constraint and a temporal finite difference (FD) constraint were combined to reconstruct cardiac cine data from highly undersampled measurements. Retrospectively undersampled 2D Cartesian reconstructions were quantitatively evaluated against fully-sampled data using normalized root mean square error, structural similarity index (SSIM) and high frequency error norm (HFEN). This method was also applied to 2D golden-angle radial real-time imaging to facilitate single breath-hold whole-heart cine (12 short-axis slices, 9–13s single breath hold). Reconstruction was compared against state-of-the-art constrained reconstruction methods: LLR, FD, and k-t SLR. ResultsAt 10 to 60 spokes/frame, LLR+FD better preserved fine structures and depicted myocardial motion with reduced spatio-temporal blurring in comparison to existing methods. LLR yielded higher SSIM ranking than FD; FD had higher HFEN ranking than LLR. LLR+FD combined the complimentary advantages of the two, and ranked the highest in all metrics for all retrospective undersampled cases. Single breath-hold multi-slice cardiac cine with prospective undersampling was enabled with in-plane spatio-temporal resolutions of 2×2mm2 and 40ms. ConclusionHighly accelerated cardiac cine is enabled by the combination of 2D undersampling and the synergistic use of LLR and FD constraints.

A novel shape-similarity-based elastography technique for prostate cancer assessment.

Association between tissue stiffness alteration and pathology is well known. This has formed the basis for prostate elastography imaging techniques where images of prostate tissue mechanical properties are reconstructed. In this paper, the authors present a novel prostate elastography technique which, unlike other techniques, relies on magnitude image data only. This proposed technique works in conjunction with ultrasound or magnetic resonance imaging (MRI) imaging modalities and it requires the prostate's pre- and postdeformation images as input. It uses a constrained reconstruction method where the elastic moduli of the prostate's normal and pathological tissues are determined based on an essential subset of the tissue deformation provided by the images data. The elasticity reconstruction technique uses optimization where similarity between calculated and observed shape features of the postcompression prostate image is maximized. The method was validated with an in silico phantom study followed by studies using ultrasound and MR with tissue-mimicking phantoms. Using the proposed methods, it was observed that the maximum uncertainties of the reconstructed Young's modulus ratios of tumor to normal tissue were 15.6% and 9.7%, which were obtained from the transrectal ultrasound (TRUS) and MR tissue-mimicking phantom studies, respectively. This novel prostate elastography technique relies on prostate TRUS or MRI images that can be routinely acquired without additional imaging hardware. The phantom studies provided evidence that the proposed technique has a good potential to reconstruct prostate stiffness maps noninvasively particularly when applied in conjunction with MRI. Further studies are necessary to evaluate the technique's merits for clinical use.

Accelerated MR parameter mapping with low-rank and sparsity constraints.

To enable accurate magnetic resonance (MR) parameter mapping with accelerated data acquisition, utilizing recent advances in constrained imaging with sparse sampling. A new constrained reconstruction method based on low-rank and sparsity constraints is proposed to accelerate MR parameter mapping. More specifically, the proposed method simultaneously imposes low-rank and joint sparse structures on contrast-weighted image sequences within a unified mathematical formulation. With a pre-estimated subspace, this formulation results in a convex optimization problem, which is solved using an efficient numerical algorithm based on the alternating direction method of multipliers. To evaluate the performance of the proposed method, two application examples were considered: (i) T2 mapping of the human brain and (ii) T1 mapping of the rat brain. For each application, the proposed method was evaluated at both moderate and high acceleration levels. Additionally, the proposed method was compared with two state-of-the-art methods that only use a single low-rank or joint sparsity constraint. The results demonstrate that the proposed method can achieve accurate parameter estimation with both moderately and highly undersampled data. Although all methods performed fairly well with moderately undersampled data, the proposed method achieved much better performance (e.g., more accurate parameter values) than the other two methods with highly undersampled data. Simultaneously imposing low-rank and sparsity constraints can effectively improve the accuracy of fast MR parameter mapping with sparse sampling.

Spectrum reconstruction based on the constrained optimal linear inverse methods

The dispersion effect of birefringent material results in spectrally varying Nyquist frequency for the Fourier transform spectrometer based on birefringent prism. Correct spectral information cannot be retrieved from the observed interferogram if the dispersion effect is not appropriately compensated. Some methods, such as nonuniform fast Fourier transforms and compensation method, were proposed to reconstruct the spectrum. In this Letter, an alternative constrained spectrum reconstruction method is suggested for the stationary polarization interference imaging spectrometer (SPIIS) based on the Savart polariscope. In the theoretical model of the interferogram, the noise and the total measurement error are included, and the spectrum reconstruction is performed by using the constrained optimal linear inverse methods. From numerical simulation, it is found that the proposed method is much more effective and robust than the nonconstrained spectrum reconstruction method proposed by Jian, and provides a useful spectrum reconstruction approach for the SPIIS.

Edge-enhanced spatiotemporal constrained reconstruction of undersampled dynamic contrast-enhanced radial MRI

Dynamic contrast-enhanced magnetic resonance imaging (MRI) is a technique used to study and track contrast kinetics in an area of interest in the body over time. Reconstruction of images with high contrast and sharp edges from undersampled data is a challenge. While good results have been reported using a radial acquisition and a spatiotemporal constrained reconstruction (STCR) method, we propose improvements from using spatially adaptive weighting and an additional edge-based constraint. The new method uses intensity gradients from a sliding window reference image to improve the sharpness of edges in the reconstructed image. The method was tested on eight radial cardiac perfusion data sets with 24 rays and compared to the STCR method. The reconstructions showed that the new method, termed edge-enhanced spatiotemporal constrained reconstruction, was able to reconstruct images with sharper edges, and there were a 36%±13.7% increase in contrast-to-noise ratio and a 24%±11% increase in contrast near the edges when compared to STCR. The novelty of this paper is the combination of spatially adaptive weighting for spatial total variation (TV) constraint along with a gradient matching term to improve the sharpness of edges. The edge map from a reference image allows the reconstruction to trade-off between TV and edge enhancement, depending on the spatially varying weighting provided by the edge map.

Accurate and robust CFD algorithms applied to 3D arbitrary polyhedral grids

A parallel, accurate, robust and grid-transparent CFD solver was developed to solve compressible flow on 3D arbitrary polyhedral grids. To improve spacial accurate, a constrained least-squares reconstruction method is developed. To accelerate convergence, a matrix free implicit method GMRES+LU-SGS, and a parallel method using OpenMP are presented on shared-memory parallel systems with help of a special grid reordering technique. Several typical test cases, including subsonic, transonic and hypersonic flows, prove that the resulting solver performs fast, accurate and robust.

Sub-Nyquist acquisition and constrained reconstruction in time resolved angiography.

In 1980 DSA provided a real time series of digitally processed angiographic images that facilitated and reduced the risk of angiographic procedures. This technique has become an enabling technology for interventional radiology. Initially it was hoped that intravenous DSA could eliminate the need for arterial injections. However the 2D nature of the images resulted in overlap of vessels and repeat injections were often required. Ultimately the use of smaller arterial catheters and reduced iodine injections resulted in significant reduction in complications. During the next two decades time resolved MR DSA angiographic methods were developed that produced time series of 3D images. These 4D displays were initially limited by tradeoffs in temporal and spatial resolution when acquisitions obeying the Nyquist criteria were employed. Then substantial progress was made in the implementation of undersampled non-Cartesian acquisitions such as VIPR and constrained reconstruction methods such as HYPR, which removed this tradeoff and restored SNR usually lost by accelerated techniques. Recently, undersampled acquisition and constrained reconstruction have been applied to generate time series of 3D x-ray DSA volumes reconstructed using rotational C-arm acquisition completing a 30 year evolution from DSA to 4D DSA. These 4D DSA volumes provide a flexible series of roadmaps for interventional procedures and solve the problem of vessel overlap for intravenous angiography. Full time-dependent behavior can be visualized in three dimensions. When a biplane system is used, 4D fluoroscopy is also possible, enabling the interventionalist to track devices in vascular structures from any angle without moving the C-arm gantrys. Constrained reconstruction methods have proved useful in a broad range of medical imaging applications, where substantial acquisition accelerations and dose reductions have been reported.