GRAPPA Operator for Wider Radial Bands (GROWL)with Optimally Regularized Self-Calibration

Wei Lin,* Feng Huang, Yu Li, and Arne Reykowski

A self-calibrated parallel imaging reconstruction method is

proposed for azimuthally undersampled radial dataset.

A generalized auto-calibrating partially parallel acquisition

(GRAPPA) operator is used to widen each radial view into a

band consisting of several parallel lines, followed by a stand-

ard regridding procedure. Self-calibration is achieved by

regridding the central k-space region, where Nyquist criterion

is satisfied, to a rotated Cartesian grid. During the calibration

process, an optimal Tikhonov regularization factor is intro-

duced to reduce the error caused by the small k-space area

of the self-calibration region. The method was applied to

phantom and in vivo datasets acquired with an eight-element

coil array, using 32-64 radial views with 256 readout samples.

When compared with previous radial parallel imaging techni-

ques, GRAPPA operator for wider radial bands (GROWL) pro-

vides a significant speed advantage since calibration is

carried out using the fully sampled k-space center. A further

advantage of GROWL is its applicability to arbitrary-view

angle ordering schemes. Magn Reson Med 64:757–766, 2010.VC 2010 Wiley-Liss, Inc.

Key words: projection reconstruction; partial parallel imaging;GRAPPA; self-calibration; regularization

In recent years, there has been an increased interest in

radial MRI due to its potential for highly accelerated

dynamic imaging (1,2), advantages in motion artifacts

reduction (3,4), and ability to achieve ultrashort echo

times (5). In many applications, radial images are

acquired using a phased-array coil, providing the oppor-

tunity for further acceleration using partial parallel imag-

ing techniques. General-purpose parallel imaging recon-

struction methods for non-Cartesian datasets have been

previously proposed and applied to undersampled radial

MRI, including the iterative conjugate gradient sensitiv-

ity encoding (SENSE) (6,7) and parallel MRI with adapt-

ive radius in k-space (PARS) (8). Streaking artifacts

resulting from azimuthal undersampling can be signifi-

cantly reduced with these parallel imaging reconstruc-

tion methods. However, the application of these techni-

ques is limited by long reconstruction time and a

dependence on sensitivity map measurements.

Several parallel imaging techniques have been devel-

oped for radial MRI, exploiting the k-space locality prin-

ciple that was first presented in the PARS method. In ra-

dial generalized auto-calibrating partially parallel

acquisition (GRAPPA) (9), k-space is divided into many

concentric rings (segments), and missing data within

each segment are estimated using a relative shift opera-

tor, i.e., a GRAPPA convolution kernel. The advantage of

radial GRAPPA over conjugate gradient SENSE is a

shorter reconstruction time since there is no need to

solve a large system of linear equations. When compared

with PARS, radial GRAPPA requires solving a smaller

number of linear equations. One limitation of the radial

GRAPPA technique when compared with its Cartesian

counterpart is the need for a separate full k-space for the

calibration of convolution kernels. This limits the useful-

ness of radial GRAPPA in some static imaging applica-

tions, such as ultrashort echo time imaging (5). The need

for an extra sensitivity reference scan also makes the

technique more susceptible to motion-induced coil sensi-

tivity discrepancy between the reference and the acceler-

ated scan. This limitation has been partially addressed

by later work generating pseudo full k-space using

image-support constraints (10,11). A second drawback

for radial GRAPPA that is more difficult to overcome, as

will be demonstrated later in this work, is some residual

blurring artifact at higher acceleration factors. A k-space

implementation of PARS reconstruction method was also

introduced that is self-calibrated and uses interpolated

weights to reduce the computation cost (12). More

recently, GRAPPA operator gridding (GROG) has been

proposed to overcome some difficulties associated with

radial GRAPPA (13–15). The data on the Cartesian grid,

near the acquired radial trajectory, are first estimated

using GROG, followed by conventional Cartesian

GRAPPA with various kernel shapes or conjugate-gradi-

ent-based iterative reconstruction.

In this work, an alternative self-calibrated parallelimaging technique, GRAPPA operator for wider radialband (GROWL), is proposed for azimuthally under-sampled radial datasets. This technique is based onexpanding each radial readout line into a wider bandusing GRAPPA relative shift operators. The resultingk-space sampling pattern is similar to that of periodicallyrotated overlapping parallel lines with enhanced recon-struction technique originally proposed for motion com-pensation (16). As will be demonstrated, image recon-structed from such a k-space sampling pattern is lesssusceptible to either streaking or blurring artifacts. Thecentral fully sampled k-space region is used for the self-calibration of GRAPPA operators. An optimal Tikhonovfactor is introduced to further reduce the GRAPPA opera-tor error when the calibration region is small. Whencompared with previous radial parallel imaging method,GROWL provides a significant speed advantage due to asmall GRAPPA operator kernel size and the small

Invivo Corporation, Philips Healthcare, Gainesville, Florida, USA.

*Correspondence to: Wei Lin, Ph.D., Invivo Corporation, Philips Healthcare,3545 SW 47th Ave, Gainesville, FL 32608. E-mail: wei.lin2@philips.com

Received 25 June 2009; revised 18 February 2010; accepted 19 February2010.

DOI 10.1002/mrm.22462Published online 25 May 2010 in Wiley Online Library (wileyonlinelibrary.com).

Magnetic Resonance in Medicine 64:757–766 (2010)

VC 2010 Wiley-Liss, Inc. 757

number of operators required. A further advantage ofGROWL over radial GRAPPA is its applicability to anyradial view-angle ordering such as the golden-anglescheme (17), which is beneficial in dynamic and motioncompensation applications. A detailed description of themethod is first provided, followed by evaluation in bothphantom and in vivo experiments.

MATERIALS AND METHODS

In an azimuthally undersampled radial dataset, the

Nyquist criterion is only satisfied within a central

k-space circle, which is denoted as the ‘‘Nyquist circle’’

in this study. The initial Nyquist circle has a radius of r0¼ N/(p field of view [FOV]), where N is the number of

radial readout lines (Fig. 1a). In the proposed GROWL

method, GRAPPA relative shift operator is used to

expand each radial line into a k-space segment consist-

ing of m parallel k-space lines (Fig. 1b). As a result, the

radius of the Nyquist circle increases to r ¼ mr0, and

therefore the streaking artifacts caused by azimuthal

undersampling will be reduced. As the data within the

initial Nyquist circle are fully sampled, they can be used

for self-calibration. The data within this region are first

regridded from the acquired radial lines onto the Carte-

sian grid. For each radial readout line, a shearing method

(18) is used to rotate the regridded Cartesian data set to

align with the readout line. The GRAPPA relative shift

operator weights can then be computed from this calibra-

tion region and used to expand each readout line into a

wider band (Fig. 1c).

There are two sources of reconstruction error in the

GROWL method: those introduced by the GRAPPA rela-

tive shift operator and those introduced by the regrid-

ding of non-Cartesian (periodically rotated overlapping

parallel lines with enhanced reconstruction like) k-space

dataset. The error caused by the GRAPPA operator can

be further divided into the approximation error, which

depends on the kernel shape and size, and noise amplifi-

cation error, which depends on the data noise level (19).

In this section, an optimal Tikhonov regularization factor

is first introduced for the calibration of the GRAPPA rel-

ative shift operator. Then the error caused by the

GRAPPA operator is examined on a Cartesian grid, fol-

lowed by the comparison of the GROWL reconstruction

results with the convolution regridding, GROG, and ra-

dial GRAPPA for undersampled radial datasets.

Optimal Regularization for GRAPPA Calibration

In this work, an optimal Tikhonov regularization factoris introduced to minimize the error introduced byGRAPPA relative shift operator since the available auto-calibration signal (ACS) only occupies a small portion ofthe k-space for highly undersampled radial datasets.When compared with previous methods (20,21) to dealwith a small calibration region, the proposed method ismuch less computationally intensive. Let t and S be thevector of target and the matrix of source data points(open and gray solid circles shown in Fig. 1c) from mul-tiple k-space locations and coil channels. Let w be the

weight vector for the GRAPPA relative shift operator.During the calibration process, weight vector w is deter-mined by solving the overdetermined linear equation

tACS ¼ SACSw: ½1�

Here the subscript ACS indicates that both target andsource data points are collected in the ACS region. Thestandard method to solve Eq. 1 is the linear least squareapproach, which seeks to minimize the residual error:

w0 ¼ argminwð tACS � SACSwk k2Þ: ½2�

Here �k k is the L2 norm. The optimal weight vector,however, should minimize the error over the entirek-space:

wopt ¼ argminwð tE � SEwk k2Þ: ½3�

Here the subscript E indicates that both target andsource data points are collected in the entire k-space. Inother words, the ideal weight vector is the solution tofollowing equation:

tE ¼ SEw: ½4�

In reality, only SE is known and tE is unknown. There-fore, Eq. 4 cannot be solved.

One key observation is that Eqs. 1 and 4 will have dif-ferent condition numbers, which is defined as the ratiobetween the maximal and minimal singular values of thesource data matrix S using the singular value decomposi-tion. The condition number measures the stability in the

FIG. 1. The basic principle of GROWL. a: For an azimuthal under-sampled radial dataset, the Nyquist criteria are only satisfied inthe small circle (‘‘the Nyquist circle’’) near the center of the k-

space. b: After the GRAPPA relative shift operators are used toexpand each radial readout lines into a wider band, the Nyquist

circle is enlarged. c: For each radial line, calibration is first per-formed in the initial Nyquist circle (the large gray circle), followedby GRAPPA relative shift operations from source (gray) points to

target (white) points.

758 Lin et al.

GRAPPA operation by giving the maximum ratio of therelative error in weight w divided by the relative error inthe target data t. If each GRAPPA operator kernel con-tains NS source data points (from multiple k-space loca-tion and coil channels) and the ACS region containsNACS k-space points, then the size of the source data ma-trix S is NACS � NS. Typically, NACS � NS. The maximalsingular value of S is determined by coil sensitivity pro-files and therefore is essentially independent of NACS.Assuming that each channel of MR data contains inde-pendent gaussian noise with a standard deviation (SD) ofs, it can be shown that the minimal singular value smin

of a random NACS � NS (NACS � NS) rectangular matrixis approximately (22):

smin �ffiffiffiffiffiffiffiffiffiffiffiNACS

ps: ½5�

Therefore, a smaller ACS region results in a lower smin

value and a higher condition number. On the otherhand, a higher noise level in the data helps to stabilizethe system by lowering the condition number (23).

A well-established method to deal with ill-conditionedlinear systems is the Tikhonov regularization (24,25),which solves Eq. 1 by minimizing:

wopt ¼ argminw tACS � SACSwk k2þl2 wk k2� �: ½6�

Here l is known as the Tikhonov factor. The solutionto Eq. 6 is:

wopt ¼Xn

j¼1

sj

s2j þ l2uHj tvj : ½7�

Here uj, vj, and sj are the left singular vectors, rightsingular vectors, and singular values of S, respectively,generated by singular value decomposition, with singularvectors and singular values indexed by j.

The hypothesis used in this work is that optimalTikhonov factor for the GRAPPA operator calibration canbe determined from the full k-space calibration equation(Eq. 4) in order to minimize the residual error root meansquare error (RMSE) computed from all the k-space data.It should be noted that the error minimized here con-tains both contributions from the approximation error,which depends on the kernel shape and size, and thenoise amplification error. The goal here is to strike a bal-ance between these two error sources and to minimizethe overall error. If the entire k-space contains NE datapoints, we have determined empirically using experi-mental data that the optimal Tikhonov factor lopt isapproximately the minimal singular value of the fullk-space calibration equation (Eq. 4):

lopt �ffiffiffiffiffiffiffiNE

ps ½8�

Evaluation of GRAPPA Relative Shift Operator

Simulations were carried out to examine the error intro-duced by the GRAPPA relative shift operator and theeffectiveness of the proposed regularization scheme. Forthis purpose, a noise-free T1-weighted Cartesian brain MR

dataset was downloaded from a simulated brain database(http://www.bic.mni.mcgill.ca/brainweb/). The complexsensitivity profile of a head coil array with eight coil ele-ments equally spaced around a cylinder was computedusing an analytic Biot-Savart integration. The k-spacedata for each individual channel were then derived usingthe Fourier transform. To simulate different noise levels,gaussian distributed random noise was added to both realand imaginary components of each channel of k-spacedata, resulting in a noise SD in the range of 0.1%-5.0% ofsignal intensity of the white matter (the dominant tissue)in the final images reconstructed using the square-root-of-sum-of-square channel combination.

To evaluate the error introduced by GRAPPA relativeshift operators, the entire set of Cartesian k-space (sourcek-space) was shifted by operators to generate another setof Cartesian k-space (target k-space). The differencebetween two k-space datasets was evaluated by the RMSEof their corresponding images using the square-root-of-sum-of-square reconstruction. This process was repeatedfor operators with different kernel shapes by changingboth the number of source data points Nx along the read-out direction and the distance shifted perpendicular tothe readout direction Dky. GROWL bears some similarityto a k-space implementation of PARS method (12). InGROWL, GRAPPA extrapolation operator use sourcepoints from a single radial line. In the modified PARSmethod, sources points from multiple radial lines wereused in the GRAPPA interpolation operator. The errors ofthese two types of GRAPPA operators were compared atdifferent noise levels and kernel sizes. To evaluate theeffectiveness of the proposed regularization method, ACSregions with sizes in the range of 20 � 20 to 160 � 160were used for the calibration of the GRAPPA weights,which corresponds to the available fully sampled centralk-space region when 32-256 radial lines are acquired.

GROWL Reconstruction

The performance of GROWL reconstruction and its de-pendence on parameter selections at different noise lev-els were first evaluated using simulated radial datasets.The noise-free brain dataset previously used for the eval-uation of the GRAPPA operator was used for this pur-pose. The Cartesian dataset was first inverse regridded togenerate a 384-view radial dataset, which approximatelysatisfies the Nyquist criteria in the entire k-space for a256-point readout matrix size. Gaussian-distributed ran-dom noise with different SD in the range of 0.1%-5.0%was then added to both real and imaginary componentsof each channel and every radial data point. Under-sampled 64- and 32-view data sets were subsequentlygenerated by extracting every 6th and 12th radial k-spaceline from the full dataset. To examine the effect of widthof radial band (Ny), GROWL reconstructions were carriedout for Ny ¼ 3, 5, 7, and 9. To examine the effect of regu-larization, Tikhonov factors in the range of 0.1-5.0 timesof the proposed optimal value (Eq. 8) were used forGRAPPA operator calibration.

The performance of the optimal GROWL algorithm wascompared with convolution gridding, GROGþSENSE, andradial GRAPPA using both 64- and 32-view undersampled

GRAPPA Operator for Wider Radial Bands 759

radial datasets, with noise standard deviations rangingfrom 0.1-5.0%. In our implementation of GROGþSENSE(26), after radial data are used to estimate data on thenearest Cartesian grid point, total variation is used as thesparsity constraint term for the subsequent iterativeSENSE reconstruction. For radial GRAPPA, k-space wasdivided into 16 concentric rings, with separate calibrationand GRAPPA interpolations. The size of the convolutionkernel is 9 (readout points) � 4 (radial views). The fullysampled, 384-view k-space was used for the calibration ofradial GRAPPA kernels. The iterative method first pro-posed by Pipe and Menon (27) and Pipe (28) was used toweight data prior to regridding using a Kaiser-Bessel ker-nel with a width of 4. The RMSE of the square-root-of-sum-of-square reconstruction for all algorithms was com-puted for comparison.

The performance of GROWL with actual radial data-sets was examined and compared with radial GRAPPA.In vivo brain datasets were acquired from a healthy vol-unteer on a 3.0-T Achieva scanner (Philips, Best, TheNetherlands) with an eight-channel head coil array (Invivo, Gainesville, FL), using a multislice two-dimen-sional radial gradient echo sequence. Scan parameterswere FOV 230 � 230mm2, slice thickness 5mm, matrixsize 256 (readout) � 256 (view no.), pulse repetitiontime/echo time ¼ 250/4.2 ms, flip angle ¼ 80�. Twoview-angle ordering schemes were used: a linear schemewith uniform view-angle distribution and a golden-anglescheme where adjacent views differs by 111.15�. Thegolden-angle view-ordering scheme results in a nearlyuniform view-angle distribution for arbitrary view num-bers. Every 4th and 8th radial readout line was extractedfrom the first dataset to generate undersampled 64- and32-view data, while the first 55 views from the seconddataset were used for the reconstruction of under-sampled golden-angle images.

To further demonstrate the application of GROWL, athree-dimensional (3D) radial dataset was generated fromthe T1-weighted brain images used in the previous simu-lation. The sensitivity profile of an eight-channel coilwas simulated using one-sided gaussian functions. Foreach of the eight channels, a total of 2800 radial projec-tions were generated from the 256 � 256 � 256 imagevolume, using the inverse regridding. The tips of the ra-dial projections form a two-dimensional spiral on thespherical surface, from the pole to the equator. Theamount of available data represents a reduction factor ofR ¼ 23.4 from a fully sampled 3D Cartesian trajectory.GROWL was then applied to expand each radial lineinto a 3D rod. Each 3D rod consists of the originalacquired radial line, two additional parallel lines alongazimuthal direction, and two additional parallel linesalong zenithal direction. The self-calibration of theGRAPPA operator is performed using the central fullysampled k-space region, following the inverse regriddingonto a rotated Cartesian grid.

RESULTS

GRAPPA Relative Shift Operator

Figure 2 shows the performance of GRAPPA relativeshift operators with various kernel shapes at different

signal-to-noise ratio (SNR) conditions. Figure 2a showsthe T1-weighted brain image used in this and all subse-quent simulation experiments. Figure 2b shows two fac-tors that determine the geometry of a GRAPPA relativeshift operator: the number of source points (Nx) alongthe readout direction and the distance of the shiftedpoints from the source readout lines (Dky). Figure 2c,dshows the dependence of RMSE on the kernel geometryfor both noise-free and noisy data. The approximationerror decreases when more source points are included inthe relative shift operator. This decrease is most signifi-cant when Nx increases from 1 (which is the case forGROG) to 3 but levels off at around Nx ¼ 9. Therefore Nx

¼ 9 is used in the current implementation of GROWLalgorithm to achieve a reasonable performance/speed bal-ance. The GRAPPA operator error increases with Dky,when the target point moves farther away from theacquired readout line. The comparison between noise-free (Fig. 2c) and noisy (Fig. 2d) data shows that the con-tribution of noise amplification becomes more significantfor large Dky values. Figure 2e shows the SNR perform-ance of the GRAPPA relative shift operator. For the lineclosest to the acquired readout line (Dky ¼ 0.5/FOV), op-erator errors remain low even for the noisy dataset. Forthe line far from the acquired readout line (e.g., Dky ¼2.0/FOV), however, operator error could become quitehigh when data contain a high level of noise, indicatinga high level of noise amplification. It should be pointedout that the curves shown in Fig. 2c-e are RMSEs intro-duced by individual GRAPPA relative shift operatorsinstead of the overall GROWL reconstruction algorithm.In the GROWL reconstruction, many GRAPPA operatorsare used to estimate several parallel lines from a singleacquired radial line. Therefore, GROWL reconstructionerror is a weighted sum of different GRAPPA relativeshift operators with different Dky values.

Figure 2f further compares the error for two differenttypes of GRAPPA operators, those using source pointsfrom a single radial readout (solid, GROWL) and thoseusing source points from multiple readouts (dashed,modified PARS). For the modified PARS method, thedistance between two parallel source lines is 6/FOV,which is approximately the gap between adjacent radiallines at the outermost k-space for a 64-view radial data-set with 256 readout points. Results show that theGRAPPA extrapolation operator yields error levels simi-lar to those of the interpolation operator for target datapoints at distances Dky ¼ 1-2/FOV. At Dky ¼ 3/FOV,extrapolation operators gives higher error than interpola-tion operators. For both types of GRAPPA operator, noiseamplification is significant for large Dky values.

Figure 3 shows the results regarding the proposed reg-ularization scheme. The condition number of theGRAPPA fitting equation increases dramatically when asmaller ACS region is used for calibration, as shown inFig. 3a. As a result, without regularization the GRAPPAoperator error become very high when a small ACSregion is used (Fig. 3b). With the proposed regularizationscheme, however, the GRAPPA operator errors arereduced and become independent of the area of the ACSregion (Fig. 3c). Figure 3d further shows that the mini-mal GRAPPA operator error is always achieved when the

760 Lin et al.

proposed optimal Tikhonov factor lopt (Eq. 8) is used forthis dataset. These results demonstrate the effectivenessof the proposed regularization method.

GROWL Reconstruction

Figure 4 shows the impact of two input parameters onthe GROWL performance. Figure 4a shows that there isan optimal number of parallel lines (Ny) within each ra-dial band that gives a minimal RMSE. While a higher Ny

improves the k-space coverage, the additional parallellines will also introduce errors that increase with thedistance from the acquired readout lines (see Fig. 2c,d).The optimal value for Ny is a function of the noise leveland is higher for high SNR data (N

opty ¼ 9 when s ¼

0.1%) than for noisy data (Nopty ¼ 5 when s ¼ 5.0%). Fig-

ure 4b demonstrates the effectiveness of the proposedregularization method. At all three noise levels, thereconstruction errors were minimized when the optimalTikhonov factor as proposed in Eq. 8 was used.

Figure 5 shows simulation results comparing GROWLreconstruction with convolution regridding, GROG, andradial GRAPPA reconstructions, using 32 radial lines toreconstruct a 256 � 256 image. For the convolutionregridding (data weighted with a RamLak filter), signifi-cant streaking artifact is present in the reconstructedimage. However, the image resolution is preserved reason-ably well. Combining GROG and sparsity-constrainedSENSE reconstruction significantly improves the imagequality, but some residual streaks still exist. RadialGRAPPA generates a k-space pattern that looks quite simi-lar to the reference fully sampled k-space. However, someimage blurring can be observed, particularly on thezoomed image. GROWL reconstruction widens each ra-dial line into a wider band, therefore enlarging fullysampled k-space region. Data in some peripheral k-spaceregion are still not estimated due to the limited width ofthe expanded radial bands. From the zoomed image, itcan be clearly seen that a better image resolution is pro-vided by GROWL when compared with radial GRAPPA,while the noise level is also higher in the GROWL image.

FIG. 2. Results from the simulationexperiment evaluating the perform-

ance of GRAPPA relative shift op-erator. a: Noise-free digital brainphantom image used for the simu-

lation. b: The kernel shape of theGRAPPA relative shift operator,showing both the number of

source readout points (Nx) and thedistance Dky between the target

point and the source readout line.c,d: GRAPPA operator error versusNx for noise-free (c) and noisy data

(d). The four curves, from top to bot-tom, correspond to Dky ¼ 2/FOV,

1.5/FOV, 1.0/FOV and 0.5/FOV. In (d),the SD of gaussian noise is 1.0% ofthe signal intensity of white matter

tissue. e: GRAPPA operator error ver-sus noise SD. f: Error comparison of

GRAPPA extrapolation (solid) andinterpolation (dashed) operators. ForGRAPPA interpolation operators, the

distance between two parallel sourcelines is 6/FOV.

GRAPPA Operator for Wider Radial Bands 761

Figure 6 compares the RMSE of the four algorithms forboth 32- and 64-view datasets at various noise levels (SD0.1%-5.0%). RMSEs for GROWL reconstructions aremuch lower than those of regridding and GROGþSENSE.When compared with radial GRAPPA, RMSE for GROWLis essential identical at low noise levels (s �1.0%) butslightly higher when the data noise level becomes higher(s 2.5%). However, as shown in Fig. 5, GROWL imageprovides a higher image resolution. Note that GROWLdid not use any additional calibration data, while radialGRAPPA used a fully sampled k-space for the calibrationpurpose, which is typically not available from a singleaccelerated scan.

Figure 7 shows the results comparing GROWL withregridding and radial GRAPPA using actual in vivo ra-

dial data set. The top row shows images reconstructedwith conventional regridding using 256, 64, and 32 ra-dial views. GROWL and radial GRAPPA reconstruction,with 64 and 32 radial readout lines, is shown in middleand bottom rows. The SNR in the white matter region isaround 20 for the 256-view radial dataset, as determinedby the intensity ratio to a signal-free background region.Convolution regridding introduces significant streakingartifacts that are particularly severe when only 32 radiallines were used for image reconstruction. Both radialGRAPPA and GROWL algorithms were able to removemost streaking artifacts. The zoomed images of the 32-view reconstruction reveal the different characteristics ofthe two algorithms. The radial GRAPPA image has lessnoise, but some image details are lost due to image

FIG. 4. Results from the simulation experiment evaluating GROWL performance using 32-view radial datasets. a: RMSE of GROWLimages versus the number of parallel lines within each radial band (Ny), when different levels of noise were added to the data. b: RMSE

of GROWL images when different Tikhonov factors are used for regularized calibration. lopt is determined using Eq. 8. Optimal Ny val-ues as determined from the curves shown in (a) were used in all cases.

FIG. 3. Results from the simula-tion experiment evaluating theperformance of the proposed reg-

ularization scheme for GRAPPAoperator. In this experiment, theACS is a square centering on the

k-space origin with different side-lengths. The SD of gaussian noise

is 1.0% of the signal intensity ofwhite matter tissue. a: Conditionnumbers for the GRAPPA calibra-

tion equation (Eq. 1) versus ACSregion side-length. b: GRAPPA

operator error versus ACS regionside-length when no regularizationis applied. c: GRAPPA operator

error versus ACS region sidelength when the proposed regula-

rization method is applied. d:GRAPPA operator errors whendifferent Tikhonov factors are

used for the regularization duringthe calibration. lopt is determinedusing Eq. 8. The ACS region size

is 32 � 32.

762 Lin et al.

blurring. In contrast, image details are preserved betterin the GROWL image, but the noise level is higher. Forthese datasets, the SNR limit as defined by the SNR onthe fully sampled image divided by the square root ofthe acceleration factor becomes fairly low (about 5-7).

Figure 8 shows in vivo brain images when radial viewswere collected using a golden-angle ordering scheme.When only 55 views are used for image reconstruction,conventional regridding generates an image with severestreaking artifacts, while the GROWL image is compara-ble with the 256-view reference image. Due to the vari-able angular spacing between adjacent views and pseu-dorandom ordering of successive views, the radialGRAPPA method cannot be used to reconstruct such adataset.

Figure 9 shows images from the 3D radial data-set reconstructed using two regridding methods and

GROWL. When the Voronoi diagram is used for densitycompensation prior to the regridding, significant streaksare present in the reconstructed images (first row) due todata undersampling. When a small kernel width is usedto compute the Pipe’s density compensation function(28), the streaks can be significantly reduced at a cost ofreduced image resolution (second row). In comparison,GROWL images (third row) reduce streaks while preserv-ing the image resolution. The entire GROWL reconstruc-tion time is 10 min for the entire 256 � 256 � 256 imagevolume.

DISCUSSION

In this work, a self-calibrated parallel imaging recon-struction method is proposed for azimuthally under-sampled radial datasets. Each radial view is widened

FIG. 5. Simulation results comparing four different reconstruction methods: convolution regridding, GROG, and SENSE reconstruction,

radial GRAPPA, and GROWL. The left column is the reference 256 � 256 Cartesian data, while other columns use 32-view radial data.The noise SD is 1.0% of the signal intensity of white matter tissue. Top row: k-space of the 1st channel signal shown in the logarithmic

scale. Middle row: Square-root-of-sum-of-square reconstruction (except GROGþSENSE). Bottom row: Zoomed-in images.

FIG. 6. Simulation results com-paring RMSE at different noise

levels for four different recon-struction methods: convolutionregridding, GROGþSENSE, radial

GRAPPA, and GROWL, usingundersampled radial datasets; (a)32-view datasets; (b) 64-viewdatasets.

GRAPPA Operator for Wider Radial Bands 763

into a band consisting of several parallel lines, usingGRAPPA relative shift operator and therefore enlargingthe fully sampled k-space region. There are two adjusta-ble parameters in GROWL: the number of source pointsNx for GRAPPA relative shift operator and the number ofparallel lines within each widened radial band Ny. Asshown in Fig. 2c, the GRAPPA operator error is reducedwith a larger Nx (no. of source points), but the rate oferror reduction gradually levels off. However, the com-putation cost increases with the kernel size. Therefore,five to nine sources points can be used, depending onthe tradeoff between accuracy and speed. A larger Ny

value (no. of parallel lines within each radial band) willresult in a larger Nyquist circle and therefore a furtherreduction in streaking artifacts. However, with the

increasing Ny values, the outer parallel lines within eachradial band will have larger Dky (distance between thetarget and source points) values. As shown in Fig. 2d,the error from the GRAPPA relative shift operator couldbecome very high for large Dky values, when the noiselevel is high. As a result, there is an optimal Ny valuethat gives the minimum reconstruction error, dependingon both the number of acquired radial views and theimage SNR (as demonstrated in Fig. 4a). For accelerationfactors of 4-8, five to nine lines can be included in eachradial band, corresponding to a Dky ¼ 1/FOV to 2/FOVfrom the acquired readout line and the outermost shiftedline.

A second contribution of this work is a methodto determine the optimal Tikhonov factor (Eq. 8) to

FIG. 7. Reconstruction results using an in vivo radial brain dataset with the uniform view angle spacing. Top row: regridding reconstruc-

tion. Middle row: GROWL images. Bottom row: Radial GRAPPA images. The number of radial views used for reconstruction is shown inthe right upper corner of each image.

FIG. 8. Reconstruction results

using an in vivo radial brain data-set with the golden-angle viewangle ordering. Left: Regridding

reconstruction with 256 views.Center: Regridding reconstruc-

tion with 55 views. Right:GROWL reconstruction with 55views.

764 Lin et al.

regularize the GRAPPA calibration process. For the data-sets examined in this work, it is shown that minimalerrors are achieved with the proposed optimal regulariza-tion factor for all ACS region sizes and noise levels(Figs. 3d, 4b). Future work will examine whether thismethod can be applied to other applications that requirethe calibration of GRAPPA weights.

GROG (13–15) provides another approach to recon-struct undersampled radial datasets. The key differencebetween GROG and GROWL is the kernel size of theGRAPPA relative shift operator. In the GROG-basedmethod, only one source point is used from each coilelement to estimate the target data point. In contrast,GROWL uses multiple (up to nine in this work) sourcepoints for the relative shift operator. Figure 2c showsthat GRAPPA operator errors are large when the targetdata point moves farther away from the acquired readoutline, and errors can be reduced dramatically when morethan one source point is used. Therefore, in GROG, theGRAPPA operator is only used to estimate data pointsthat are no more than Dky ¼ 0.5/FOV from the acquiredradial lines. Because of the use of multiple sourcepoints, however, GROWL can estimate data points far-ther away from the source points more accurately andtherefore can generate a larger coverage in k-space, asdemonstrated in Fig. 5.

When compared with radial GRAPPA where the newlyestimated data remain on radial trajectories, the k-spacesampling pattern achieved using GROWL is similar to theperiodically rotated overlapping parallel lines withenhanced reconstruction technique previously proposedfor motion compensation. Although the k-space coverageof GROWL may not be as large as radial GRAPPA (asshown in the top row of Fig. 5), images presented showthat the GROWL image has a higher spatial resolution (asshown in the bottom row of Fig. 5). However, the GROWLimage does also show a higher level of noise. The overallRMSE is similar at various noise levels for both 32-viewand 64-view dataset (Fig. 6). Compared with radialGRAPPA technique, GROWL has two other advantages:First, GROWL is self-calibrated. By rotating the centralfully sampled k-space region, calibration data are providedfor every readout line. The second advantage of GROWL isits ability to reconstruct data with arbitrary view-anglespacing and ordering schemes, which is useful for dynamicimaging and motion-compensation applications. Conven-tional radial GRAPPA requires equal data spacing alongthe azimuthal direction. Hence, it is difficult to use for ra-dial imaging with arbitrary view-angle spacing.

GROWL is also related to the modified PARS method(12). In GROWL, GRAPPA extrapolation operator usessource points from a single radial line. In the modified

FIG. 9. 3D reconstruction resultswith 2800 radial views (image

matrix 256 � 256 � 256). Toprow: Regridding using the Voro-noi diagram for density compen-

sation. Middle row: Regriddingusing the Pipe’s (28) density

compensation function with asmall kernel width. Bottom row:GROWL. From left to right: axial,

coronal, and sagittal images.

GRAPPA Operator for Wider Radial Bands 765

PARS method, source points from multiple radial lineswere used in the GRAPPA interpolation operator. Asshown in Fig. 2f, GRAPPA extrapolation operator yieldserror levels similar to those of the interpolation operator,except for target data points farther away (Dky 3/FOVfor the eight-channel coil) from the acquired radial lines.Therefore GROWL may generate higher errors then themodified PARS method at higher acceleration factors.However, GROWL provides a significant speed advant-age due to two factors. First, the number of weight vec-tors is limited to the number of acquired radial lines.Second, the smaller number of source points furtherreduces the time required for matrix inversion.

The acceleration factor achievable with the GROWLreconstruction depends on the number of coil elementsand noise levels in the data. Our results show that 256 �256 images with decent qualities can be reconstructedwith 32 radial views, using a commercial eight-channelhead coil at a low SNR condition (about 5-7). Pipe (28)previously showed that for an undersampled dataset, thetradeoff between the image resolution and reconstructionerror can be adjusted by the choice of the convolutionkernel used to compute the data-weighting factor. How-ever, our tests with GROWL datasets showed that thesize of the convolution kernel for regridding has very lit-tle effect on the final image quality and the total recon-struction error. This is most probably due to the fact thata large central portion of the k-space is already fullysampled after the application of the GRAPPA relativeshift operator. The computational cost of GROWL is verylow when compared with most existing parallel imagingmethods due to the small calibration region and thesmall number of GRAPPA weights required. The recon-struction time of current version of GROWL is about 10sec for an eight-channel 256 (readout) � 32 (view) two-dimensional dataset on a 2.2-GHz personal computer.The method to interpolate GRAPPA operator weights, aspresented in the modified PARS algorithm (12), can beused to further improve the speed for GROWL.

One possible way to further improve the performanceof GROWL at higher acceleration factors is to combineGROWL with radial GRAPPA or the modified PARSmethod. As shown in Fig. 2f, estimation errors becomehigh when Dky increases to 3/FOV or above and thenoise level is high, limiting the ability to estimate datapoints farther away from the acquired readout line, usingGRAPPA relative shift operators. Radial GRAPPA andthe modified PARS method, however, are able to esti-mate the radial line that lies midway between twoacquired radial lines. Therefore, it is expected that thecombination of these techniques will further improve thequality of image reconstruction.

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