Self-calibrated interpolation of non-Cartesian data with GRAPPA in parallel imaging

Abstract

Purpose:

To develop a non-Cartesian k-space reconstruction method using self-calibrated region-specific interpolation kernels (RIK) for highly accelerated acquisitions.

Methods:

In conventional non-Cartesian GRAPPA with through-time GRAPPA (TT-GRAPPA), the use of RIK has demonstrated improved reconstruction quality in dynamic imaging for highly accelerated acquisitions. However, TT-GRAPPA requires the acquisition of a large number of separate calibration scans. In order to reduce the overall imaging time, we propose Self-calibrated Interpolation of Non-Cartesian data with GRAPPA (SING) to self-calibrate RIK from dynamic undersampled measurements. SING synthesizes calibration data to adapt to the distinct shape of each RIK geometry, and additionally uses a novel local k-space regularization through an extension of TT-GRAPPA. This calibration approach is employed to reconstruct non-Cartesian images at high acceleration rates while mitigating noise amplification. The reconstruction quality of SING is compared with CG-SENSE and TT-GRAPPA in numerical phantoms and in vivo cine data sets.

Results:

In both numerical phantom and in vivo cine data sets, SING offers visually and quantitatively similar reconstruction quality to TT-GRAPPA, and provides improved reconstruction quality over CG-SENSE. Furthermore, temporal fidelity in SING and TT-GRAPPA is similar for the same acceleration rates. G-factor evaluation over the heart shows that SING and TT-GRAPPA provide similar noise amplification at moderate and high rates.

Conclusion:

The proposed SING reconstruction enables significant improvement of acquisition efficiency for calibration data, while matching the reconstruction performance of TT-GRAPPA.

Introduction

Lengthy scan times in MRI compared to other imaging modalities remain a limiting factor, necessitating accelerated data acquisitions. In most clinical settings, the scan time is reduced by undersampling data in k-space. Additional information is then used to reconstruct images from undersampled acquisitions. Parallel imaging techniques utilize properties of multi-coil receiver arrays to separate aliased pixels in image space or recover missing k-space data points using nearby acquired k-space samples. These methods are the most clinically used approaches in fast imaging.

By using information from coil sensitivity profiles, Sensitivity Encoding (SENSE) resolves superimposed pixels in a reduced field-of-view (FOV) image which is reconstructed from undersampled k-space.¹ While SENSE unfolds aliasing signals in the image domain, Generalized Partially Parallel Acquisitions (GRAPPA) recovers missing data in undersampled k-space acquisitions.² Each coils’ sensitivity profile spatially weights the signals and the spatial weighting generates correlations among local data points in k-space. GRAPPA utilizes the signal correlations across coils to interpolate missing k-space data points using neighboring acquired data points. A single missing k-space point in a single channel, called a target point, is reconstructed as a linear combination of neighboring acquired k-space points from all channels called source points. The mapping between source and target points for all channels is called a GRAPPA kernel. Autocalibration signal (ACS) data is separately acquired to calibrate the GRAPPA kernel in the conventional GRAPPA method.

While SENSE and GRAPPA are originally developed for uniform Cartesian undersampling, they both have been extended to non-Cartesian undersampling. Owing to advantages of non-Cartesian sampling, such as incoherent undersampling artifacts, non-Cartesian imaging has been an active area of research.^3–12 While the non-Cartesian extension is natural for SENSE, it is nontrivial for GRAPPA. On non-uniform non-Cartesian undersampling, each region of undersampled k-space has a distinct sampling pattern with varying distribution of acquired k-space points, leading to requirering distinct kernel geometry in each local region. The initial version of non-Cartesian radial GRAPPA used an angular-invariant kernel geometry to estimate non-Cartesian radial GRAPPA kernels using a single calibration scan.¹³ However, the use of angular-invariant kernels results in blurred artifacts in image reconstruction.¹⁴ Alternatively, GRAPPA Operator Gridding (GROG) self-calibrates region-specific extrapolation kernels by factorizing each kernel into bases powered by the relative distance between a target and a source point, and using these kernels to extrapolate the nearest Cartesian point of each acquired non-Cartesian data point.¹⁵ These bases are self-calibrated from the undersampled non-Cartesian data points. Although this self-calibrated extrapolation approach adapts to the distinct kernel geometry in non-Cartesian GRAPPA, the extrapolated Cartesian data points are limited within half of Nyquist distance from each acquired non-Cartesian data point, limiting GROG reconstruction to low acceleration rates.¹⁶ Seiberlich et al proposed the consecutive application of GROG and pseudo-Cartesian GRAPPA.¹⁷ The non-Cartesian measurements are extrapolated to nearest Cartesian points with GROG, and the extrapolated Cartesian points are then used to reconstruct missing Cartesian points with a Cartesian GRAPPA reconstruction.¹⁷ However, this approach suffered from additional g-factor loss due to the consecutive application of two parallel imaging techniques.

Non-Cartesian GRAPPA reconstruction of highly accelerated data is significantly improved by region-specific interpolation kernels, which we call RIK for the rest of the paper. RIK was firstly developed in through-time GRAPPA (TT-GRAPPA).¹⁸ A RIK has a distinct distribution of sampling points with a distinct set of interpolation coefficients. Two examples of RIK sampling patterns are depicted in Figure 1 for the illustration of radial k-space reconstruction. Missing data points in each local region of undersampled k-space are interpolated from the linear RIK-weighted combination of neighboring acquired data. The use of RIK has enabled successful reconstruction of highly accelerated data for non-Cartesian dynamic imaging in clinical studies.^19–25 TT-GRAPPA calibrates each RIK by using multiple fully-sampled non-Cartesian calibration scans from which sufficient numbers of repeated sampling patterns for each RIK with the distinct geometry are obtained. For 3D stack of stars, using a consistent sampling pattern across partitions, the adaptation of TT-GRAPPA has been demonstrated using a single TT-GRAPPA calibration for all partitions.²⁶ The reconstruction quality of TT-GRAPPA was found to exceed CG-SENSE for acquisitions with higher acceleration rates.^18,26 For kooshball type 3D acquisition, the scan time may scale up to hours with such calibration, and to the authors knowledge have not been demonstrated yet.

Figure 1:

Two different region-specific interpolation kernel (RIK) are depicted to illustrate the k-space reconstruction using RIK. Each missing sample is reconstructed from the linear combination of a RIK weighting of acquired samples. For the radial trajectory, the shapes of sampling patterns in different regions are different.

Recently, self-calibrated non-Cartesian GRAPPA methods without the need for special sampling considerations in calibration data have been proposed.^27–29 Specifically Luo et al has shown that each RIK can be calibrated from a Cartesian ACS region, which is either acquired or reconstructed from a non-Cartesian dataset, by applying selected linear phase modulations in image space and then transforming to k-space via FFT to obtain repeated non-Cartesian sampling patterns necessary for calibration of each RIK.^28,29 The performance of the self-calibrated RIK in Luo et al’s work was found to match the reconstruction quality of CG-SENSE, while being computationally five times faster than CG-SENSE.

In this paper, we propose an alternative technique to obtain an overdetermined calibration system and a different estimation for each RIK to match the reconstruction quality of TT-GRAPPA. The proposed method called SING (Self-calibrated Interpolation of Non-Cartesian data with GRAPPA) synthesizes target and source points on the repeated sampling patterns with the distinct shape of each RIK geometry to eliminate the acquisitions of additional calibration frames. Then, the through-time calibration of TT-GRAPPA is extended to yield a novel local k-space regularization for RIK calibration in SING. To the best of our knowledge, the analysis of TT-GRAPPA for this extension has not been performed. The efficacy of SING is demonstrated on numerical data and in vivo cardiac imaging.

Theory

Synthesize autocalibration signal (ACS) data

SING synthesizes target and source points on shift-invariant sampling patterns with a distinct shape for each RIK geometry to establish a linear overdetermined system based on the shift-invariant property of GRAPPA interpolation kernels. The synthesis is illustrated in Figure 2. Dynamic non-Cartesian measurements in each channel are combined, independently of other channels, into a composite dataset with lower temporal resolution as shown in Figure 2A. The composite non-Cartesian measurements in each channel are resampled, independently of other channels, with density compensation, convolution with a Kaiser-Bessel function³⁰ of width L = Δk (Δk is Nyquist sampling interval in k-space), and resampled on Cartesian points, similar to a gridding reconstruction³¹. The region with the resampled Cartesian points from Nyquist sampled non-Cartesian data, shown as a circular region in Figure 2B, is referred to as the auto-calibration signal (ACS) region.

Figure 2:

The schematic of SING synthesizing ACS data points. Firstly, dynamic accelerated data is combined into a multichannel composite data (A). The composite data in each channel is resampled with density compensation, convolution with a Kaiser-Bessel function, and resampled on Cartesian points, similar to a gridding reconstruction (B). The shape of the sampling pattern for each RIK is determined (C). The gridded Cartesian data is then convolved with a Kaiser-Bessel function and sampled on the repeated sampling patterns with the determined shape of each RIK to synthesize target and source points (D). The synthesized target and source points are utilized to establish a linear system for each RIK (E). The system is then regularized using the proposed local k-space regularization and solved via least-squares to estimate the RIK coefficients.

To extract one calibration equation target and source points are synthesized by convolving the resampled Cartesian points with a Kaiser-Bessel function with width L = Δk and then resampling at the locations which match the sampling pattern with a distinct shape for a RIK. The same sampling pattern is shifted vertically and horizontally relative to the first sampling pattern by a distance β ≥ L over the ACS region until all of the gridded Cartesian points are used to extract additional calibration equations, as illustrated in Figure 2D. Using all extracted equations, a linear overdetermined system with the GRAPPA source and target formulation for calculating a RIK can be established and solved to determine the signal correlations for GRAPPA interpolations in the local k-space region corresponding to a RIK, as specified in Figure 2E. The synthesis of target and source points is repeated to calibrate other RIK’s until all local k-space regions with missing points are interpolated. If β is smaller than L, the Kaiser-Bessel convolution results in correlations among adjacent patterns and negatively affects the conditioning of the calibration system.

Each RIK is the least-squares solution of a linear calibration system

$$\text{arg }\underset{\mathbf{x}}{\text{min}}{\Vert b-A\cdot \mathbf{x}\Vert }_2^2,$$ [1]

where b is a M×1 vector of synthesized target points, A is a M×N matrix of synthesized source points, and x is a N×1 vector of the coefficients in the RIK.

Local k-space regularization

Noise amplification in the estimation of a RIK via least-squares, which is determined by the conditioning of the linear system, can lead to noise like errors in the image reconstruction.³² Tikhonov regularization has been utilized to mitigate noise amplification in kernel calibrations,^33,34 but it often results in blurring artifacts in image reconstruction.³⁵ Furthermore, tuning a single regularization parameter for each RIK using the discrepancy principle or the L-curve method³³ requires multiple calculations of a matrix inversion and as the parameter tuning is repeated for different RIK, such a tuning process requires significant computations.

In order to develop a strategy with improved sharpness in image reconstruction that is comparable to TT-GRAPPA, we investigate the effect of signal-to-noise ratio (SNR) in calibration data on k-space reconstruction. Several studies have demonstrated that lower SNR ACS data can be advantageous for GRAPPA reconstruction.^36–37 Furthermore, it was shown that adding noise onto the source matrix (matrix A in Eq. [1]) for determining interpolation kernels has the same effect as Tikhonov regularization.³⁸ For the RIK calibrations in TT-GRAPPA, where local k-space is used for calibrating each RIK, the variable SNR in k-space as such implicitly regularizes each RIK in TT-GRAPPA differently.

Motivated by the regularization effect from noise inherent to TT-GRAPPA, we propose a local k-space regularization by employing local SNR level in k-space. The definition of local signal level is the l₂ norm of acquired k-space points across all channels in the local region where a RIK is applied for k-space interpolation. By proposing the use of local k-space SNR level for regularization, the computationally intense parameter tuning with discrepancy principle or L-curve can be eliminated. The proposed regularization adds noise to the calibration system of each RIK, matching the SNR effect of through-time calibration data on RIK calibrations in TT-GRAPPA. Before adding noise, the system is as specified in Eq. [1]. After adding noise, similar to what is done implicitly in TT-GRAPPA, the system is represented as:

$$\underset{x}{\text{min}}{\Vert \left(b+\Delta b\right)-\left(A+\Delta A\right)·x\Vert }_2^2.$$ [2]

We propose to determine the variance of Δb and ΔA to match the SNR between the synthesized ACS points and the measured k-space points in the local region of k-space corresponding to each calibrated RIK. For the source and target points of a m^th sampling pattern in the ACS, the variance of Δb and ΔA is calculated with the SNR ratio of these points to the undersampled signals in the local region corresponding to each RIK, scaled by the standard deviation of the thermal noise σ as

$${\text{w}}_m=\text{σ}\cdot \frac{{\text{a}}_m}{u},$$ [3]

where a_m is the SNR level of synthesized source points of the m^th pattern in the ACS (m^th row in matrix A in Eq. [1]), calculated with the l₂ norm of these points across all channels, and u the SNR level in the local region of acquired k-space where applying each calibrated RIK for k-space interpolation is sought, calculated as the l₂ norm of neighboring acquired signals across all channels. Thus, Δb and ΔA are generated as

$$\begin{gathered} \Delta b=W\cdot \chi \\ \Delta A=W\cdot X, \end{gathered}$$ [4]

where the entries of χ and X are drawn from a standard normal distribution. W is a M×M diagonal weighting matrix with the diagonal entry w_m as specified in Eq.[3].

The coefficients of each RIK are obtained by solving Eq. [2] via least-squares and then used to recover the missing data points in the local region of undersampled k-space corresponding to each RIK. The reconstructed fully-sampled multichannel non-Cartesian data is transformed to image space via gridding reconstruction channel-by-channel and a reconstructed image is obtained with the root-sum-of-square channel combination.

Methods

Numerical Simulations

The MRXCAT³⁹ cardiac numerical phantom was used for the simulations. 30 synthetic coil sensitivity profiles were simulated from an in vivo cine imaging dataset and added to the MRXCAT simulation. For each cardiac phase, a multichannel radially fully-sampled k-space with 216 radial projections was generated using inverse gridding on the individual coil sensitivity weighted phantom image under simulated breath-hold. Complex white Gaussian noise was added to yield an SNR of 40 calculated over the image object. An angularly uniform radial pattern was used to undersample the k-space in each cardiac phase and the bit-reversed ordering was utilized as the interleaved undersampling ordering throughout cardiac phases to suppress simulated motion effects. 20 free-breathing fully-sampled noisy radial k-space datasets, added with noise of the same distribution in the numerical undersampled data specified above, were generated using the cardiac phantom images under simulated free-breathing provided by the MRXCAT simulated datasets for TT-GRAPPA calibration purposes.

In vivo imaging

The imaging protocols were locally approved by the local institutional review board and the written informed consent was obtained from all subjects before each scan of this HIPAA-compliant study.

Cine cardiac MRI was acquired at 3T (Siemens Prisma) on two healthy subjects using a 30-channel body array with a GRE sequence with the imaging parameters: TE/TR/α=2.3ms/3.9ms/12°, bandwidth=440Hz/pixel, in-plane resolution=2×2mm², and FOV 300×300mm². The readout sampling rate was 2 times the acquisition bandwidth, and the number of total sampled readouts in each radial projection was 288. Fully-sampled radial dataset in each cardiac phase contains 216 radial projections using a linear view order in a single breath-hold acquisition. During each cardiac phase of a heartbeat, 9 radial projections were acquired in a linear order to yield a temporal resolution of 35ms for the fully-sampled cardiac images. Data was retrospectively undersampled in a uniform angular pattern for each cardiac phase with bit-reversed ordering for interleaved undersampling to suppress motion effect throughout cardiac phases, and the undersampled data are used for reconstruction. For TT-GRAPPA calibrations, additional 20 free-breathing fully-sampled acquisitions were obtained shortly after the breath-hold acquisition.

Effect of inclusion of central ACS region on calibration

The effect of using the central ACS, which yield residuals of high magnitudes in the calibration system due to high signal variations⁴⁰ on SING was investigated by performing two SING calibrations that included and excluded these central ACS points and then reconstructing two sets of images using two sets of RIK corresponding to these two calibrations. The central ACS region was defined as a circular region with a radius of γ · Δk and different values of γ ∈ {2, 4, 8, 12, 16} were evaluated. The proposed local k-space regularization was employed on all SING calibrations.

Effect of regularization on SING calibration

The regularization effect on RIK was evaluated by performing SING calibration with no regularization, Tikhonov regularization, and the proposed signal-based regularization. Tikhonov regularization parameters were chosen in a readout-dependent manner based on the decay rate of Fourier coefficients along the readout dimension. For each RIK, the Tikhonov parameter was chosen as

$$\text{σ}\cdot {\left(\frac{{\text{r}}_{\text{miss}}}{{\text{r}}_{\text{ACS}}}\right)}^{\text{q}},$$ [5]

with σ the standard deviation of thermal noise. q is a positive scalar and optimized empirically to q=2 after visually inspecting reconstructed images for q values from 1 to 3. r_miss denotes the readout coordinate of the missing data point reconstructed using the RIK. r_ACS denotes the averaged readout coordinate of sampling patterns in ACS region.

Reconstruction Evaluation

CG-SENSE,⁴¹ TT-GRAPPA, and SING were compared in numerical simulations and in vivo imaging. Two acceleration rates R∈{6,12} were included. CG-SENSE was implemented with Tikhonov regularization, where the regularization parameter was empirically set to λ=0.1 for both R=6 and 12 after visually inspecting the reconstructed images for λ values from 0.001 to 1. The coil sensitivities were calculated from gridding reconstructed fully-sampled coil images using ESPIRiT.⁴² In TT-GRAPPA, each RIK with size 2×5 (angle × readout) was estimated using the local through-time acquired calibration data points in the neighboring region spanning ±4 readouts samples, spanning 2Δk (Δk: Nyquist sampling interval) in the readout dimension, and ±4 projections similar to the original proposed approach.¹⁸ The size of this neighboring region was empirically validated after testing TT-GRAPPA reconstructed images corresponding to different sizes with ±i × ±j (readouts × projections) for i, j ∈{2,4,6,8}. In SING, the gridded Cartesian points (gridded with oversampling ratio of 2 from fully-sampled non-Cartesian acquisitions) in the central region of size 60Δk×60Δk and 72Δk×72Δk are used for R=6 and R=12, respectively, to obtain an 8 times overdetermined linear system for calculation of the RIKs. For k-space reconstruction each RIK was used to reconstruct two samples angularly positioned between the two acquired projections, and four samples radially, as employed in TT-GRAPPA.¹⁸ Specifically for the data used, the corresponding number of RIKs is 180*288/8=6480 at R=6 and 198*288/8=7128 at R=12 (R: acceleration rate) respectively where 180 and 198 denote the number of missing radial projections at R=6 and R=12, respectively. For each reconstructed image, both the difference image and the normalized root mean squared error (NRMSE) were computed with respect to the gridding reconstruction of the fully-sampled acquisition (the reference image). G-factor maps were calculated using the pseudo-replica method⁴³ with 5000 repetitions for CG-SENSE, TT-GRAPPA, and SING.

Results

Effect of inclusion of central ACS region on calibration

Figure 3 shows two sets of reconstructed images corresponding to two SING calibrations that included and excluded central ACS region in the numerical simulation and an in vivo dataset. The radius γ of the excluded central circular ACS region was empirically set to γ=4 after validating the reconstructed images corresponding to different values of γ ∈ {2, 4, 8, 12, 16}. With the inclusion of these central ACS data points in SING calibration, the reconstructed images showed blurring and streaking artifacts at R=6 and 12. By excluding these central ACS data in SING calibration, the reconstructed images demonstrated no visually noticeable blurring or streaking artifacts. For the rest of SING reconstructions, the central ACS points were excluded from the calibration of each RIK.

Figure 3:

The reconstructed images from SING that includes and excludes central ACS data points in numerical simulations (top-half) and a cine dataset (bottom-half) at R=6 and 12. The difference image corresponding to each reconstructed image is shown in the right side of the reconstructed images in the same row.

Effect of regularization on calibration

Figure 4 and 5 demonstrate the effect of the regularized SING calibration on image reconstructions in numerical simulations and cine imaging, respectively. The signal-based regularization offered visually improved noise performance over the un-regularized calibration at R∈{6,12}. At R=6, the signal-based and Tikhonov regularization provided visually similar image sharpness and noise performance. At R=12, the signal-based regularization offered visually sharper images over Tikhonov regularization.

Figure 4:

SING reconstructed images in numerical simulations using un-regularized (top-row), Tikhonov regularized (middle-row), and local k-space regularized (bottom-row) RIK calibrations at R∈{6,12}. The difference image corresponding to each reconstructed image is shown at the right side of the reconstructed images in the same row, together with the NRMSE value in the bottom-right side of each difference image.

Figure 5:

SING reconstructed images in cine imaging using un-regularized (top-row), Tikhonov regularized (middle-row), and local k-space regularized (bottom-row) RIK calibrations at R∈{6,12}. The difference image corresponding to each reconstructed image is shown at the right side of the reconstructed images in the same row, together with the NRMSE value in the bottom-right side of each difference image.

Reconstruction of undersampled acquisitions

Figure 6 shows the results of the numerical simulations with the reconstructions using CG-SENSE, TT-GRAPPA and SING at R=6 and 12. For R=6, no visual difference was observed between the TT-GRAPPA and SING reconstructed images, while blurring artifacts were observed in the CG-SENSE image. For R=12, TT-GRAPPA and SING images were visually similar, while CG-SENSE image showed more blurring and undersampling artifacts.

Figure 6:

Image reconstructions of radially undersampled data using Tikhonov regularized CG-SENSE (top row), TT-GRAPPA (middle row), and SING (bottom row) at R∈{6,12}. The difference image corresponding to each reconstructed image is shown at the right side of the reconstructed images in the same row, together with the NRMSE value in the bottom-right side of each difference image.

Figure 7 shows the reconstruction results from a cine data set using CG-SENSE, TT-GRAPPA, and SING at R∈{6,12}. There was no visual difference between TT-GRAPPA, SING and the reference image at R=6, while CG-SENSE resulted in streaking artifacts. At R=12, there were no visually noticeable blurred or streaking artifacts in TT-GRAPPA and SING images, while CG-SENSE yielded to blurred and streaking artifacts. For the other cine data set in which similar results were obtained, NRMSE values at R∈{6,12} were 8.9%, 15.3% for CG-SENSE; 8.2%, 10.5% for TT-GRAPPA; and 8.5%, 10.9% for SING.

Figure 7:

Image reconstructions of the accelerated data in a cine data set using CG-SENSE (top row), TT-GRAPPA (middle row), and SING (bottom row) at R=6 and 12. The difference image corresponding to each reconstructed image is shown at the right side of the reconstructed images in the same row, together with the NRMSE value in the bottom-right side of each difference image.

Figure 8 depicts the temporal intensity profile across the indicated line for CG-SENSE, TT-GRAPPA, and SING. The profiles showed no temporal blurring at R=6 for TT-GRAPPA and SING, consistent with the absence of any temporal constraints during reconstruction. For CG-SENSE, the profile at R=6 showed the undersampling artifacts. For R=12, noise amplifications were observed in the profiles of TT-GRAPPA and SING, while the profile of CG-SENSE showed more undersampling artifacts.

Figure 8:

Temporal intensity profiles from 20 cardiac frames across 70 voxels indicated by the yellow line using CG-SENSE (top row), TT-GRAPPA (middle row), and SING (bottom row) at R=6 and 12.

Figure 9 shows g-factor maps of CG-SENSE, TT-GRAPPA, and SING at R=6 and 12. For R=6 and 12, TT-GRAPPA and SING offered similar g-factor over the heart, while CG-SENSE yielded higher g-factor than either method. The g-factor maps of TT-GRAPPA and SING were in agreement with the reconstructed images shown in Figure 7, in which either method resulted in similar noisy appearances and NRMSE values at R∈{6,12}.

Figure 9:

The g-factor maps for CG-SENSE, TT-GRAPPA and SING reconstructions in the central dashed region at R=6 and 12. The average g-factor value in the same region of each reconstruction was calculated and shown in the bottom-right side of each corresponding g-factor map.

Discussion

We propose a k-space reconstruction technique called SING for accelerated radial imaging. This method self-calibrates each RIK without additional acquisitions of multiple calibration scans for shortened total imaging time and uses these for k-space interpolation. SING and TT-GRAPPA have been compared under conditions where it is possible to acquire the additional calibration data required for TT-GRAPPA. SING enables successful reconstructions of highly accelerated radial acquisitions and the reconstruction quality is similar to TT-GRAPPA which has been clinically utilized in 2D cine imaging. When using SING, it is sufficient to have a central Nyquist-sampled k-space region to subsequently determine RIKs for undersampled regions of k-space where calibration data have not been obtained – similarly to how Cartesian GRAPPA is typically employed with variable density sampling, and benefits dynamic and volumetric imaging. SING requires an additional resampling relative to TT-GRAPPA, but both the gridding and the resampling to the patterns for each RIK can be performed efficiently, with the same computational complexity as TT-GRAPPA.

For GRAPPA calibration it is important to obtain unique representations of the correlations between channels for an unbiased estimation of each interpolation kernel. In hybrid radial GRAPPA/TT-GRAPPA the unique representations are obtained through enough motion/signal variability in the calibration data of TT-GRAPPA which is challenging to control for. SING avoids this by using different parts of k-space for RIK estimation.

The ACS data in SING can be obtained with gridding and inverse gridding operations or as proposed here with convolution-based resampling. In gridding reconstructions, the effect of the convolution function is corrected in image space and has the same effect across all channels. For determining GRAPPA interpolation functions, consistent intensity variations across do not have an impact for determining the correlation between channels.

The exclusion of ACS data at k-space center was found to improve SING reconstructions and was one way to match the quality of the TT-GRAPPA reconstruction. Previous work on Cartesian GRAPPA^44,45 found that “outliers,” which have large residuals in a linear calibration system, were from ACS signals at the central k-space region, and the exclusion of central ACS data could improve GRAPPA reconstructions. For SING the k-space signal intensity is scaled in the proposed regularization approach such that it should be matched in an l2 sense. The l2 estimate is a patch measure, and there can still be large variations of the calibration signal in a patch.

Apart from the self-calibration aspect of SING, the proposed local k-space regularization, which adapts to local signal information of the neighboring k-space undersampled data points in the local region of each RIK, is also an important component for improved reconstruction quality. This regularization outperforms Tikhonov regularization with improved sharpness in image reconstruction. By employing local SNR levels, the regularization parameters corresponding to each RIK is directly calculated with a scalar division and a scalar multiplication to eliminate alternative computationally intense parameter tuning processes for each RIK. As discussed earlier, such ideas are inherent to TT-GRAPPA, where the local SNR for the calibration data is the same as the local SNR for the dynamic series. In SING, similar regularization can be maintained for different dynamics, and across different relaxations. The k-space shifting from Luo et al²⁹ applies equivalent weighting on the signal and noise portion in the measured ACS data, offering an equivalent form of un-regularized SING RIK calibration, which was shown to suffer from noise amplification at R=6 and 12.

The computational complexity of SING for resampling target and source points on repeated sampling patterns with the distinct shape of each RIK geometry can be compared to the approach by Luo et al,²⁹ which has similar quality to the un-regularized SING. For a 2D Cartesian ACS region of size N×N, the computational complexity of resampling via Kaiser-Bessel convolution is O(N). Alternatively, the resampling can be computed by applying linear phase modulations in image space and then transforming to k-space via inverse Fourier transform. The corresponding computational complexity is then dominated by the fast Fourier transform from image-space to k-space, and is O (N log N). For SING the inversion of the GRAPPA type equation (A′A)⁻¹ A′ is the most time consuming, and is similar to the approach by Luo et al.²⁹ Once performed, the RIK for all missing points within a patch of acquired data can efficiently be computed.

There has been other recent work using both k-space interpolation and extrapolation⁴⁴ for self-calibrated non-Cartesian GRAPPA reconstruction. More recently Luo et al have shown that self-calibrated non-Cartesian GRAPPA reconstruction can be 3–5 times faster than CG-SENSE, while matching the quality of CG-SENSE.²⁹ In our work, where SING resamples non-Cartesian patches without the need of FFT, SING should, in principle, be faster than Luo et al²⁹ and we can use that as an upper limit of computational speed. As such, an optimized implementation of SING can potentially be 3–5 times faster than CG-SENSE, while outperforming the reconstruction quality of CG-SENSE. Good sensitivity estimation is important for CG-SENSE reconstructions which requires either a separate acquisition or dynamic imaging and can then be determined using e.g. ESPIRIT. Since SING calibrates RIKs on a central Nyquist sampled k-space region, the undersampled acquisitions with an oversampled central region can, in principle, be used to calibrate RIKs in SING, enabling potential uses for the reconstruction of accelerated acquisitions in static imaging. For dynamic imaging iterative reconstructions are typically combined with wavelet, total variation, and low-rank⁴⁷ constraints for improved reconstruction quality and sensitivity estimation is easier to accommodate. Further investigation is also here warranted to determine how SING combined with a noise-variance reduction technique compares with a GROG type hybrid reconstruction or with conventional iterative approaches in terms of quality and computation.

SING may be extended to complicated 3D sampling trajectories such as kooshball⁴⁸ because of the use of self-calibration, whereas it is challenging to use TT-GRAPPA in such acquisitions due to the lengthy scanning of multiple fully-sampled calibration datasets. The kooshball trajectory has been actively utilized, owing to advantages such as high isotropic resolution, wide spatial coverage, mild undersampling artifacts, reduced sensitivity to motion and self-navigation properties.^49–56 Due to the prolonged scan time of kooshball acquisitions, undersampled acquisitions are often performed. When kooshball acquisitions are undersampled, the reconstruction methods such as compressed sensing (CS) model-based reconstruction with the sparsity or the low-rank constraint are often employed to mitigate undersampling artifacts in reconstructed images.^57–60 However, the computational time of iterative CS-based non-Cartesian reconstruction is prolonged due to gridding/inverse gridding computation in each iteration. Alternatively, direct k-space reconstruction is well-suited to the parallel computation due to independent interpolation between different missing data points. By accelerating the k-space reconstruction using a graphics processing unit (GPU), 2D non-Cartesian TT-GRAPPA has been utilized in real-time imaging.⁶¹ With more advanced GPU hardware, 3D non-Cartesian k-space reconstruction can potentially be further shortened. In terms of computational efficiency, the k-space reconstruction is more preferable than the iterative CS-based reconstruction, enabling SING as a potential candidate for the reconstruction of accelerated kooshball acquisitions. Furthermore, the missing data can also be reconstructed directly on a Cartesian grid - which can be undersampled and then subsequently reconstructed with a Cartesian parallel imaging technique, similar to the approach employed by Seiebrlich et al.¹⁷ But unlike GROG, SING is not restricted to resampling to a Cartesian undersampled grid in such a setting.

While the ACS data points in SING are generated from the combined dynamic measurements during the whole series, potential temporal variations of coil sensitivity may yield artifacts in SING reconstructions from highly accelerated acquisitions. The artifacts due to coil sensitivity mismatches in dynamic acquisitions were studied in TGRAPPA.⁶² In the temporal profile of SING at R=12 shown in Figure 8, the noisy appearance observed in the difference image could result from the coil sensitivity mismatches, together with the g-factor of the coils and undersampling. Furthermore, since the combined dynamic measurements are utilized to generate ACS data in SING, accelerated free-breathing acquisitions may cause the movement of coils due to respiratory motion. This coil movement may yield aliasing artifacts in k-space reconstruction which uses kernels calibrated from the combined dynamic measurements.⁶²

Future work will investigate how to extend SING to other 2D non-Cartesian trajectories such as spiral, rosette and BLADE/PROPELLER and 3D non-Cartesian trajectories such as kooshball and 3D stack-of-stars. Furthermore, SING may potentially interpolate onto neighboring Cartesian points to save the computational time of Kaiser-Bessel convolution in gridding reconstruction. Additionally, a natural extension with a receiver array with more than 32 channels, such as 64 channels,⁶³ or higher field strength, such as 7 Tesla,^64–65 may potentially lead to improved acceleration and SNR gain in accelerated imaging with SING.