Variable Density Sampling Research Articles

An Algorithm for Variable Density Sampling with Block-Constrained Acquisition

Reducing acquisition time is of fundamental importance in various imaging modalities. The concept of variable density sampling provides a nice framework to achieve this. It was justified recently from a theoretical point of view in the compressed sensing (CS) literature. Unfortunately, the sampling schemes suggested by current CS theories may not be relevant since they do not take the acquisition constraints into account (for example, continuity of the acquisition trajectory in Magnetic Resonance Imaging - MRI). In this paper, we propose a numerical method to perform variable density sampling with block constraints. Our main contribution is to propose a new way to draw the blocks in order to mimic CS strategies based on isolated measurements. The basic idea is to minimize a tailored dissimilarity measure between a probability distribution defined on the set of isolated measurements and a probability distribution defined on a set of blocks of measurements. This problem turns out to be convex and solvable in high dimension. Our second contribution is to define an efficient minimization algorithm based on Nesterov's accelerated gradient descent in metric spaces. We study carefully the choice of the metrics and of the prox function. We show that the optimal choice may depend on the type of blocks under consideration. Finally, we show that we can obtain better MRI reconstruction results using our sampling schemes than standard strategies such as equiangularly distributed radial lines.

Variable Density Sampling with Continuous Trajectories

Reducing acquisition time is a crucial challenge for many imaging techniques. Compressed Sensing (CS) theory offers an appealing framework to address this issue since it provides theoretical guarantees on the reconstruction of sparse signals by projection on a low dimensional linear subspace. In this paper, we focus on a setting where the imaging device allows to sense a fixed set of measurements. We first discuss the choice of an optimal sampling subspace (smallest subset) allowing perfect reconstruction of sparse signals. Its standard design relies on the random drawing of independent measurements. We discuss how to select the drawing distribution and show that a mixed strategy involving partial deterministic sampling and independent drawings can help breaking the so-called coherence barrier. Unfortunately, independent random sampling is irrelevant for many acquisition devices owing to acquisition constraints. To overcome this limitation, the notion of Variable Density Samplers (VDS) is introduced and defined as a stochastic process with a prescribed limit empirical measure. It encompasses samplers based on independent measurements or continuous curves. The latter are crucial to extend CS results to actual applications. Our main contribution lies in two original continuous VDS. The first one relies on random walks over the acquisition space whereas the second one is heuristically driven and rests on the approximate solution of a Traveling Salesman Problem. Theoretical analysis and retrospective CS simulations in magnetic resonance imaging highlight that the TSP-based solution provides improved reconstructed images in terms of signal-to-noise ratio compared to standard sampling schemes (spiral, radial, 3D iid...).

Stable and Robust Sampling Strategies for Compressive Imaging

In many signal processing applications, one wishes to acquire images that are sparse in transform domains such as spatial finite differences or wavelets using frequency domain samples. For such applications, overwhelming empirical evidence suggests that superior image reconstruction can be obtained through variable density sampling strategies that concentrate on lower frequencies. The wavelet and Fourier transform domains are not incoherent because low-order wavelets and low-order frequencies are correlated, so compressive sensing theory does not immediately imply sampling strategies and reconstruction guarantees. In this paper, we turn to a more refined notion of coherence-the so-called local coherence-measuring for each sensing vector separately how correlated it is to the sparsity basis. For Fourier measurements and Haar wavelet sparsity, the local coherence can be controlled and bounded explicitly, so for matrices comprised of frequencies sampled from a suitable inverse square power-law density, we can prove the restricted isometry property with near-optimal embedding dimensions. Consequently, the variable-density sampling strategy we provide allows for image reconstructions that are stable to sparsity defects and robust to measurement noise. Our results cover both reconstruction by ℓ1-minimization and total variation minimization. The local coherence framework developed in this paper should be of independent interest, as it implies that for optimal sparse recovery results, it suffices to have bounded average coherence from sensing basis to sparsity basis-as opposed to bounded maximal coherence-as long as the sampling strategy is adapted accordingly.

Pulmonary perfusion MRI using interleaved variable density sampling and HighlY constrained cartesian reconstruction (HYCR)

To demonstrate the feasibility of performing single breathhold, noncardiac gated, ultrafast, high spatial-temporal resolution whole chest MR pulmonary perfusion imaging in humans. Eight subjects (five male, three female) were scanned with the proposed method on a 3 Tesla clinical scanner using a 32-channel phased-array coil. Seven (88%) were healthy volunteers, and one was a patient volunteer with sarcoidosis. The peak lung enhancement phase for each subject was scored for gravitational effect, peak parenchymal enhancement and severity of artifacts by three cardiothoracic radiologists independently. All studies were successfully performed by MR technologists without any additional training. Mean parenchymal signal was very good, measuring 0.78 ± 0.13 (continuous scale, 0 = "none" → 1 = "excellent"). Mean level of motion artifacts was low, measuring 0.13 ± 0.08 (continuous scale, 0 = "none" → 1 = "severe"). It is feasible to perform single breathhold, noncardiac gated, ultrafast, high spatial-temporal resolution whole chest MR pulmonary perfusion imaging in humans.

On Variable Density Compressive Sampling

We advocate an optimization procedure for variable density sampling in the context of compressed sensing. In this perspective, we introduce a minimization problem for the coherence between the sparsity and sensing bases, whose solution provides an optimized sampling profile. This minimization problem is solved with the use of convex optimization algorithms. We also propose a refinement of our technique when prior information is available on the signal support in the sparsity basis. The effectiveness of the method is confirmed by numerical experiments. Our results also provide a theoretical underpinning to state-of-the-art variable density Fourier sampling procedures used in magnetic resonance imaging.

Interleaved variable density sampling with a constrained parallel imaging reconstruction for dynamic contrast-enhanced MR angiography

For MR applications such as contrast-enhanced MR angiography, it is desirable to achieve simultaneously high spatial and temporal resolution. The current clinical standard uses view-sharing methods combined with parallel imaging; however, this approach still provides limited spatial and temporal resolution. To improve on the clinical standard, we present an interleaved variable density (IVD) sampling method that pseudorandomly undersamples each individual frame of a 3D Cartesian ky-kz plane combined with parallel imaging acceleration. From this dataset, time-resolved images are reconstructed with a method that combines parallel imaging with a multiplicative constraint. Total acceleration factors on the order of 20 are achieved for contrast-enhanced MR angiography of the lower extremities, and improvements in temporal fidelity of the depiction of the contrast bolus passage are demonstrated relative to the clinical standard.

Contrast-enhanced whole-heart coronary magnetic resonance angiography at 3 T with radial EPI

Whole-heart coronary magnetic resonance angiography is a promising method for detecting coronary artery disease. However, the imaging time is relatively long (typically 10-15 min). The goal of this study was to implement a radial echo planar imaging sequence for contrast-enhanced whole-heart coronary magnetic resonance angiography, with the aim of combining the scan efficiency of echo planar imaging with the motion insensitivity of radial k-space sampling. A self-calibrating phase correction technique was used to correct for off-resonance effects, trajectory measurement was used to correct for k-space trajectory errors, and variable density sampling was used in the partition direction to reduce streaking artifacts. Seven healthy volunteers and two patients were scanned with the proposed radial echo planar imaging sequence, and the images were compared with a traditional gradient echo and X-ray angiography techniques, respectively. Whole-heart images with the radial EPI technique were acquired with a resolution of 1.0 × 1.0 × 2.0 mm(3) in a scan time of 5 min. In healthy volunteers, the average image quality scores and visualized vessel lengths of the RCA and LAD were similar for the radial EPI and gradient echo techniques (P value > 0.05 for all). Anecdotal patient studies showed excellent agreement of the radial EPI technique with X-ray angiography.

Variable Density Compressed Image Sampling

Compressed sensing (CS) provides an efficient way to acquire and reconstruct natural images from a limited number of linear projection measurements leading to sub-Nyquist sampling rates. A key to the success of CS is the design of the measurement ensemble. This correspondence focuses on the design of a novel variable density sampling strategy, where the a priori information of the statistical distributions that natural images exhibit in the wavelet domain is exploited. The proposed variable density sampling has the following advantages: 1) the generation of the measurement ensemble is computationally efficient and requires less memory; 2) the necessary number of measurements for image reconstruction is reduced; 3) the proposed sampling method can be applied to several transform domains and leads to simple implementations. Extensive simulations show the effectiveness of the proposed sampling method.

Reconstructive compounding for IVUS palpography

This study proposes a novel algorithm for luminal strain reconstruction from sparse irregularly sampled strain measurements. It is based on the normalized convolution (NC) algorithm. The novel extension comprises the multilevel scheme, which takes into account the variable sampling density of the available strain measurements during the cardiac cycle. The proposed algorithm was applied to restore luminal strain values in intravascular ultrasound (IVUS) palpography. The procedure of reconstructing and averaging the strain values acquired during one cardiac cycle forms a technique, coined as reconstructive compounding. The accuracy of strain reconstruction was initially tested on the luminal strain map, computed from 3 in vivo IVUS pullbacks. The high quality of strain restoration was observed after systematically removing up to 90% of the initial elastographic measurements. The restored distributions accurately reproduced the original strain patterns and the error did not exceed 5%. The experimental validation of the reconstructed compounding technique was performed on 8 in vivo IVUS pullbacks. It demonstrated that the relative decrease in number of invalid strain estimates amounts to 92.05 +/- 6.03% and 99.17 +/- 0.92% for the traditional and reconstructive strain compounding schemes, respectively. In conclusion, implementation of the reconstructive compounding scheme boosts the diagnostic value of IVUS palpography.

Fabrication and characterization of a compressive-sampling multispectral imaging system

We describe the fabrication and characterization of a compressive-sampling multispectral imaging (CS-MSI) system that uses single-pixel detectors to capture multiple spectral images concurrently, without mechanical scanning. In particular, we pay special attention to the hardware/software characterization of the CS-MSI system, wherein we discuss the optical realization of the proposed system, measure the effective sensitivity range, and investigate the relationship between the digital micromirror device frame rate and the reconstruction quality. We also compare the imaging performance of the Hadamard-pace variable-density sampling method, which is the CS method implemented with our hardware architecture, with the conventional random sampling method. We propose a compressive-sampling modulation transfer function (CS-MTF) to measure the amplitude response of different spatial frequencies by using different CS methods.

Contrast-enhanced intracranial magnetic resonance angiography with a spherical shells trajectory and online gridding reconstruction

To evaluate the feasibility of applying the shells trajectory to single-phase contrast-enhanced magnetic resonance angiography. Several methods were developed to overcome the challenges of the clinical implementation of shells including off-resonance blurring (eg, from lipid signal), aliasing artifacts, and long reconstruction times. These methods included: 1) variable TR with variable readout length to reduce fat signal and off-resonance blurring; 2) variable sampling density to suppress aliasing artifacts while minimizing acquisition time penalty; and 3) an online 3D gridding algorithm that reconstructed an 8-channel, 240(3) image volume set. Both phantom and human studies were performed to establish the initial feasibility of the methods. Phantom and human study results demonstrated the effectiveness of the proposed methods. Shells with variable TR and readout length further suppressed the fat signal compared to the fixed-TR shells acquisition. Reduced image aliasing was achieved with minimal scan time penalty when a variable sampling density technique was used. The fast online reconstruction algorithm completed in 2 minutes at the scanner console, providing a timely image display in a clinical setting. It was demonstrated that the use of the shells trajectory is feasible in a clinical setting to acquire intracranial angiograms with high spatial resolution. Preliminary results demonstrate effective venous suppression in the cavernous sinuses and jugular vein region.

Whole-body MR angiography using variable density sampling and dual-injection bolus-chase acquisition

Conventional bolus-chase acquisition generates peripheral runoff images using a single injection of the contrast material. Low spatial resolution, small slice coverage and venous contamination are major problems especially in the distal stations. A technique is presented herein in which whole-body magnetic resonance angiography is performed using a dual-contrast-injection four-station acquisition protocol. Bolus sharing was performed between two stations: the abdomen and calf stations share the first bolus injection, while the thorax and thigh stations share the second bolus injection. The combination of variable density sampling and elliptical centric acquisition order was applied to the abdomen and thorax stations. The scan time was extended to generate high spatial resolution arterial phase images with broad slice coverage for the calf and thigh stations. The feasibility of this technique was demonstrated using phantom and in vivo human volunteer studies.

Generation and visualization of four-dimensional MR angiography data using an undersampled 3-D projection trajectory

Time-resolved contrast-enhanced magnetic resonance (MR) angiography (CE-MRA) has gained in popularity relative to X-ray Digital Subtraction Angiography because it provides three-dimensional (3-D) spatial resolution and it is less invasive. We have previously presented methods that improve temporal resolution in CE-MRA while providing high spatial resolution by employing an undersampled 3-D projection (3D PR) trajectory. The increased coverage and isotropic resolution of the 3D PR acquisition simplify visualization of the vasculature from any perspective. We present a new algorithm to develop a set of time-resolved 3-D image volumes by preferentially weighting the 3D PR data according to its acquisition time. An iterative algorithm computes a series of density compensation functions for a regridding reconstruction, one for each time frame, that exploit the variable sampling density in 3D PR. The iterative weighting procedure simplifies the calculation of appropriate density compensation for arbitrary sampling patterns, which improve sampling efficiency and, thus, signal-to-noise ratio and contrast-to-noise ratio, since it is does not require a closed-form calculation based on geometry. Current medical workstations can display these large four-dimensional studies, however, interactive cine animation of the data is only possible at significantly degraded resolution. Therefore, we also present a method for interactive visualization using powerful graphics cards and distributed processing. Results from volunteer and patient studies demonstrate the advantages of dynamic imaging with high spatial resolution.

Variable‐density one‐shot fourier velocity encoding

In areas of highly pulsatile and turbulent flow, real-time imaging with high temporal, spatial, and velocity resolution is essential. The use of 1D Fourier velocity encoding (FVE) was previously demonstrated for velocity measurement in real time, with fewer effects resulting from off-resonance. The application of variable-density sampling is proposed to improve velocity measurement without a significant increase in readout time or the addition of aliasing artifacts. Two sequence comparisons are presented to improve velocity resolution or increase the velocity field of view (FOV) to unambiguously measure velocities up to 5 m/s without aliasing. The results from a tube flow phantom, a stenosis phantom, and healthy volunteers are presented, along with a comparison of measurements using Doppler ultrasound (US). The studies confirm that variable-density acquisition of kz-kv space improves the velocity resolution and FOV of such data, with the greatest impact on the improvement of FOV to include velocities in stenotic ranges.

Optimizing spherical navigator echoes for three‐dimensional rigid‐body motion detection

Spherical navigator (SNAV) echoes show promise in correcting for three-dimensional rigid-body motion. In this paper, several important parameters in the design and performance of the SNAV technique are discussed, including a novel sampling strategy, the optimal k-space radius and sampling density of the navigator, and the execution of the SNAV trajectory by the scanner hardware. A variable-sampling density (VSD) helical-spiral SNAV trajectory, which can acquire data on the entire spherical shell without exceeding the maximum slew rate of the scanner, is presented. To ensure that the VSD SNAV trajectory was properly executed by the scanner hardware, the gradient waveforms were verified using a self-encoding technique. The ability of the VSD SNAV to measure rotational and translational motion was studied with in vitro experiments at various k-space radii and sampling densities. The results of this study show that the best accuracy was attained at k-space radii of 1.4 and 1.6 cm(-1), with 2400 to 4000 samples acquired over the sphere.

Artifact and noise suppression in GRAPPA imaging using improved k‐space coil calibration and variable density sampling

A parallel imaging technique, GRAPPA (GeneRalized Auto-calibrating Partially Parallel Acquisitions), has been used to improve temporal or spatial resolution. Coil calibration in GRAPPA is performed in central k-space by fitting a target signal using its adjacent signals. Missing signals in outer k-space are reconstructed. However, coil calibration operates with signals that exhibit large amplitude variation while reconstruction is performed using signals with small amplitude variation. Different signal variations in coil calibration and reconstruction may result in residual image artifact and noise. The purpose of this work was to improve GRAPPA coil calibration and variable density (VD) sampling for suppressing residual artifact and noise. The proposed coil calibration was performed in local k-space along both the phase and frequency encoding directions. Outer k-space was acquired with two different reduction factors. Phantom data were reconstructed by both the conventional GRAPPA and the improved technique for comparison at an acceleration of two. Under the same acceleration, optimal sampling and calibration parameters were determined. An in vivo image was reconstructed in the same way using the predetermined optimal parameters. The performance of GRAPPA was improved by the localized coil calibration and VD sampling scheme.

Bayesian Statistics-Based Procedure for Sampling of Contaminated Sites

A variable density sampling pattern based on Bayesian statistics is presented and compared to a uniform density statistical pattern and a judgmental approach in a real case study. The Bayesian statistics, supported by a software tool, supplied a soil sampling plan similar to the judgmental one, especially for the number of sampling points and their location. It allowed statistical goals to be set and expert judgment to be included in the sampling strategy in a transparent and systematic procedure. For these reasons, it appears quite suitable for inclusion into Quality Assurance Quality Control plans.

Improved depiction of small anatomic structures in MR images using Gaussian-weighted spirals and zero-filled interpolation

Partial-volume artifacts reduce the contrast and continuity of small structures in magnetic resonance images. Zero-filled interpolation (ZFI) has been known for some time as a useful technique to reduce partial-volume artifacts and improve the appearance of small structures and edges. However, its use is limited by the fact that ZFI can exacerbate image artifacts. For example, it can exacerbate Gibbs ringing, also known as the truncation artifact, which manifests itself as spurious ringing around sharp edges. Currently, the most common technique to address this problem is post-acquisition filtering, which causes blurring in the image. Using ZFI in conjunction with a variable-density sampling method designed to reduce ringing is proposed as a possible solution to this problem. This approach is demonstrated with a Gaussian-weighted spiral and is compared to conventional spiral sampling both with and without the application of a filter used to reduce ringing. The two spiral sampling techniques are compared using simulations, phantom images, and in vivo brain images. The Gaussian-weighted spiral demonstrates reduced ringing without the loss of spatial resolution commonly associated with post-acquisition filtering. Additionally, this sampling technique is shown to work well in conjunction with ZFI to reduce partial-volume artifacts without the apparent increase in Gibbs ringing usually associated with zero-filled reconstruction. This approach will be most useful for imaging techniques such as MR angiography which are known to be sensitive to partial-volume effects, as well as when imaging anatomic regions associated with more severe Gibbs ringing.

Reduced aliasing artifacts using variable-density k-space sampling trajectories.

A variable-density k-space sampling method is proposed to reduce aliasing artifacts in MR images. Because most of the energy of an image is concentrated around the k-space center, aliasing artifacts will contain mostly low-frequency components if the k-space is uniformly undersampled. On the other hand, because the outer k-space region contains little energy, undersampling that region will not contribute severe aliasing artifacts. Therefore, a variable-density trajectory may sufficiently sample the central k-space region to reduce low-frequency aliasing artifacts and may undersample the outer k-space region to reduce scan time and to increase resolution. In this paper, the variable-density sampling method was implemented for both spiral imaging and two-dimensional Fourier transform (2DFT) imaging. Simulations, phantom images and in vivo cardiac images show that this method can significantly reduce the total energy of aliasing artifacts. In general, this method can be applied to all types of k-space sampling trajectories.

Reduced spatial side lobes in chemical-shift imaging.

Density-weighted k-space sampling with spiral trajectories is used to reduce spatial side lobes in chemical-shift imaging (CSI). In this method, more time is spent collecting data at the center of k space and less time at the edges of k space in order to make the sampling density proportional to a given apodization function, subject to constraints imposed by gradient performance and Nyquist sampling. The efficient k-space coverage of spiral-based trajectories enables good control over the sampling density within practical in vivo scan times. The density-weighted acquisition is compared to a conventional, nonweighted spiral sampling without the application of a window function. For a fixed voxel size and imaging time, the noise variance is observed to be the same for both cases, while spatial side lobes are greatly reduced with the variable-density sampling. This method is demonstrated on a normal volunteer by imaging of brain metabolites at 1.5 T with both single slice CSI and volumetric CSI. Magn Reson Med 42:314-323, 1999.