Self-Calibration Method for Radial GRAPPA/k-t GRAPPA

Feng Huang,* Sathya Vijayakumar, Yu Li, Sarah Hertel, Shahed Reza, andGeorge R. Duensing

Generalized autocalibrating partially parallel acquisitions(GRAPPA), an important parallel imaging technique, can beeasily applied to radial k-space data by segmenting the k-space. The previously reported radial GRAPPA method requiresextra calibration data to determine the relative shift operators.In this work it is shown that pseudo-full k-space data can begenerated from the partially acquired radial data by filtering inimage space followed by inverse gridding. The relative shiftoperators can then be approximated from the pseudo-full k-space data. The self-calibration method using pseudo-full k-space data can be applied in both k and k-t space. This tech-nique avoids the prescans and hence improves the applicabilityof radial GRAPPA to image static tissue, and makes k-tGRAPPA applicable to radial trajectory. Experiments show thatradial GRAPPA calibrated with pseudo-full calibration data gen-erates results similar to radial GRAPPA calibrated with the truefull k-space data for that image. If motion occurs during acqui-sition, self-calibrated radial GRAPPA protects structural infor-mation better than externally calibrated GRAPPA. However,radial GRAPPA calibrated with pseudo-full calibration data suf-fers from residual streaking artifacts when the reduction factoris high. Radial k-t GRAPPA calibrated with pseudo-full calibra-tion data generates reduced errors compared to the sliding-window method and temporal GRAPPA (TGRAPPA). Magn Re-son Med 57:1075–1085, 2007. © 2007 Wiley-Liss, Inc.

Key words: radial; GRAPPA; k-t GRAPPA; parallel imaging; dy-namic imaging; pseudo-full k-space

To increase the spatiotemporal resolution and reduce mo-tion artifacts, data acquisition speed is a crucial factor inmany applications, such as functional magnetic resonanceimaging (fMRI) and cardiac MRI (CMRI). One approach toreduce acquisition time is to use non-Cartesian acquisitiontrajectories, such as radial (1,2) or spiral (3,4) trajectories.Another approach is to acquire partial data and then makeuse of the correlation among receive channels and/or timeframes to interpolate the missing data. For example, par-tially parallel imaging (PPI) techniques (5–7) use the cor-relations among channels to reconstruct alias-free imageswith partially acquired data. Unaliasing by Fourier-encod-ing the overlaps using the temporal dimension (UNFOLD)(8), keyhole (9,10), and view-sharing strategies (11,12) usethe correlations among time frames to reconstruct finalimages. k-t SENSE (13) and k-t GRAPPA (14) exploit thecorrelations among both channels and time frames. Thesetwo approaches can be combined together to further re-duce the acquisition time (15–24) (i.e., the partially non-

Cartesian acquisition imaging technique). The iterativeconjugate gradient SENSE (CG-SENSE) method (15,24)and parallel MRI with adaptive radius in k-space (PARS)(17,25) provide general methods for PPI with a non-Carte-sian trajectory. Unfortunately, the application of thesetechniques is limited by long reconstruction times and adependence on sensitivity-map measurements. In 2003 therelative shift operator concept (16) was proposed forGRAPPA with partially acquired radial k-space data. Theradial k-space data are divided into many small segments,and each segment shares one relative shift operator. Therelative shift operator is actually a convolution operatorwith a small convolution kernel. This concept extends theGRAPPA technique to a non-Cartesian trajectory. The re-ported reconstruction times for non-Cartesian GRAPPA(19,22,23,26) techniques are much shorter than those forCG-SENSE and PARS because no iteration or calculationof large systems of linear equations is necessary. Further-more, the calculation of sensitivity maps is not necessaryfor GRAPPA-based methods. Hence, non-CartesianGRAPPA techniques are preferred in terms of reconstruc-tion time. However, early radial GRAPPA techniques re-quire extra full k-space calibration data to generate relativeshift operators. The purpose of this work was to find aself-calibration technique for convolution-based recon-struction methods, such as GRAPPA and k-t GRAPPA,with radial trajectory data. Self-calibration eliminates theneed for an external sensitivity reference, making parallelimaging reconstruction less susceptible to miscalibrationand image degradation resulting from changes in the coilsensitivity due to motion between the prescan and theactual accelerated scan.

Radial trajectories offer inherent self-calibrating charac-teristics because of the densely sampled center. The feasi-bility of self-calibrating non-Cartesian SENSE has beenshown in the literature (24,27). In this work a self-calibra-tion technique for GRAPPA/k-t GRAPPA with radial tra-jectory data is presented in the Theory section. The imple-mentation details are discussed in the Materials and Meth-ods section. In the Results section, in vivo data obtainedwith a radial trajectory are used to demonstrate the perfor-mance of this technique.

THEORY

In this section the basic idea of radial GRAPPA is re-viewed. The self-calibration scheme is then presented.Brain anatomy data are used to demonstrate the idea.

Review of Conventional Radial GRAPPA

In GRAPPA, individual images are generated for eachchannel in the array by forming linear combinations ofneighboring lines to reconstruct missing data. In the case

Advanced Concept Development, Invivo Corporation, Gainesville, Florida,USA.*Correspondence to: Feng Huang, Invivo Corporation, 3545 SW 47th Ave.,Gainesville, FL 32603. E-mail: fhuang@invivocorp.comReceived 6 March 2006; revised 19 January 2007; accepted 16 February2007.DOI 10.1002/mrm.21233Published online in Wiley InterScience (www.interscience.wiley.com).

Magnetic Resonance in Medicine 57:1075–1085 (2007)

© 2007 Wiley-Liss, Inc. 1075

of Cartesian GRAPPA, the weights are optimized to shiftdata in k-space by a fixed amount in a specific direction,because the lines used in the fitting process are all shiftedby the same specific amount in the same direction. In thecase of non-Cartesian GRAPPA, however, relative shiftoperators need to be derived to shift data by a specificamount in different locations. For some non-Cartesian tra-jectories, such as radial and spiral, a set of positions canshare the same relative shift operator because of the sym-metry of the trajectory. For example, in the case of a radialtrajectory all points at a particular radius have the samerelative displacement with respect to the neighboring pro-jections. Therefore, the basic idea of radial GRAPPA is todivide the non-Cartesian spaces into many small segmentssuch that the projections are nearly parallel inside eachsegment. With this property, the entire segment can use asingle relative shift operator for reconstruction. The rela-tive shift operators are derived from a full k-space calibra-tion data set, which is usually from prescan. RadialGRAPPA segments the k-space data, and then calculatesthe relative shift operator and linearly interpolates themissing projections in each segment with its specific op-erator (16).

Self-Calibrated Radial GRAPPA

Self-calibration is possible in radial GRAPPA because thecenter of k-space is oversampled in radial trajectories.Even for accelerated acquisitions with reduced samplingtrajectories, the center of the k-space data can providesufficient information about coil sensitivity maps, whichare explicitly or implicitly needed in self-calibration(7,24,27). GRAPPA implicitly uses coil sensitivity infor-mation in the calculation of relative shift operators (28). InCartesian GRAPPA, the relative shift operators are thesame over the entire k-space and can be calculated fromthe center k-space data. In radial GRAPPA, however, atechnical challenge is that the relative shift operators aredifferent for different segments in k-space. Hence a set of

full k-space data is necessary for calibration. The mainpurpose of this paper is to provide an approach to generatea set of full k-space data for self-calibration from the par-tially acquired k-space data. For this purpose, the segmen-tation technique introduced in the Appendix can be ap-plied to the low-resolution image generated from the cen-ter k-space data. Zeros are then assigned to the pixels inthe background regions determined from the segmenta-tion. This truncation in image space is equivalent to thek-space convolution of the center k-space data with theinverse Fourier transform (FT) of the truncation windowfunction. Due to this convolution, the center k-space datais extrapolated to the entire k-space. Therefore, if the low-resolution image with the zeroed background region istransferred back to k-space, a full k-space data can begenerated. Apparently, this set of generated full k-spacedata is not exactly the true full k-space data, and hence itis termed “pseudo-full k-space data” in this paper. The useof background information can also be understood as theuse of a constraint in image space to improve the PPI(29,30) techniques. The flow chart of reconstruction withpseudo-full k-space data is shown in Fig. 1. Figure 2 showsadditional details for the procedure to calculate pseudo-full k-space data. The implementation of inverse griddingfor radial data can be simplified by using a Radon trans-form followed by a 1D fast FT (FFT). For simplicity, theprocess for one channel of the data is shown, but the sameprocess is applicable to all channels. In Fig. 2, one projec-tion out of every four projections is used to simulate theaccelerated acquisition. All of these projections areequally spaced. Six steps are employed to generate thepseudo-full k-space data:

Step 1. Generate Cartesian k-space data (k1,. . ., knc) withthe partially acquired non-Cartesian k-space data by grid-ding.

Step 2. Generate images L1,. . ., Lnc by FFT with k1,. . .,knc.

FIG. 1. Flow chart of the proposed self-calibration method.

1076 Huang et al.

Step 3. Generate images fL1,. . ., fLnc by FFT with filteredk1,. . ., knc by FK, where FK is defined as

FK � � 1 � e��kx

2�ky2�c��w��1

� � 1 � e��kx

2�ky2�c��w��1

[1]

where kx and ky are the count of k-space lines along x andy directions, and c and w are two parameters to adjust thefilter (c decides the cutoff frequency, and w decides thesmoothness of filter boundary). The purpose of this filterFK is to generate low-resolution images with fewer arti-facts (ideally artifact free).

Step 4. Generate an image support mask with the squareroot of sum-of-squares (SSoS) of fL1,. . ., fLnc (the details ofthis technique are provided in the Appendix).

Step 5. Generate a new set of images L�1,. . ., L�nc withabsolute value of L1,. . ., Lnc and phase of fL1,. . ., fLnc, andthe image support mask through pointwise multiplication.

Step 6. Generate pseudo-full k-space data with L�1,. . .,L�nc by inverse gridding.

The relative shift operator Wp_full of each segment is thencalculated based on the pseudo-full k-space data (16). Themissing k-space data points are then interpolated usingWp_full. The final reconstructed images are the SSoS of theFT of the interpolated full k-space data in Cartesian space.The relative shift operator concept can be adopted by k-tGRAPPA, as well for reconstruction with radial data sets.Hence the proposed self-calibration scheme, which usespseudo-full k-space concept, can also be used for k-tGRAPPA to generate calibration data.

MATERIALS AND METHODS

Data Acquisition

To demonstrate the performance of the proposed method,one set of brain anatomy data and one set of dynamiccardiac function images were acquired. The brain data setwas acquired on a 1.5T Siemens Symphony system (Sie-mens Medical Solutions, Erlangen, Germany) with aneight-channel head coil (Invivo Corp., Gainesville, FL,USA). The acquisition parameters were FOV � 220 �220 mm2, matrix � 256 � 256 � 8, slice thickness � 5 mm,TR � 53.5 ms, TE � 3.4 ms, and flip angle � 70°. Thecardiac function data sets were acquired on a 3T SiemensTrio system using a cine True-FISP sequence with aneight-channel cardiac coil (Siemens Medical Solutions,Erlangen, Germany). The acquisition parameters wereFOV � 300 � 300 mm2, matrix � 256 � 256 � 17 � 8,TR � 36.64 ms, TE � 2.29 ms, and flip angle � 65°. For thebrain data, all 256 projections were acquired. For the car-diac data, all 256 projections were acquired for the firsttime frame, and for all other time frames 128 time-inter-leaved projections were acquired.

Evaluation Criteria

To evaluate the image quality of the reconstructed images,the difference map and relative error are used. The differ-ence map depicts the difference in magnitudes betweenthe reconstructed and reference images at each pixel andshows the distribution of error. The relative error or rela-tive energy difference is defined as the ratio of the SSoS ofall pixels of the difference map to the SSoS of all pixels ofthe reference image.

FIG. 2. Flow chart for calculationof the pseudo-full calibration datafor one channel. All channels usethe same scheme. The first chan-nel of the brain anatomy data isused in this example, and 64 outof 256 projections are used to cal-culate the pseudo-full k-spacedata. The abscissa and ordinatefor radial k-space are for radialand azimuthal direction.

Self-Calibration for Radial GRAPPA/k-t GRAPPA 1077

To evaluate the performance of radial GRAPPA cali-brated with pseudo-full k-space data, the results of theproposed method were compared with the results ob-tained by radial GRAPPA calibrated with ideal full k-spacedata. When full k-space data were acquired, only partialk-space data from the full k-space data were used forreconstruction to simulate the partially parallel acquisi-tion. The method that uses the same set of full k-space datafor calibration, besides generating the partial k-space data,is called calibration with ideal full k-space data. The cal-ibration projections are used only to derive the radialGRAPPA convolution kernels, and are not included in thefinal reconstruction. Notice that both the ideal full k-spacedata calibration method and the pseudo-full k-space datacalibration method are self-calibration methods. Ideal cal-ibration data should provide the best calibration informa-tion since the calibration information is derived from thetrue full k-space data.

Choice of Parameters for Self-Calibrated Radial GRAPPA

The performance of radial GRAPPA is dependent on thechoice of several parameters. Among these parameters, thechoice of the convolution kernel and the number of itera-tions are not trivial and hence are described in more detail.All other related parameters are briefly described.

Parameters for k-Space Segmentation, Low-Pass Filter,and Gridding

In our implementation, the following definitions are used:k-space segmentation scheme (32 segments along radialdirection, no segmentation along azimuthal direction); thechoice of low-pass filter (c � #PR/6, where #PR is thenumber of readouts on each projection); for gridding, boththe width � of Kaiser-Bessel kernel (31) and over gridfactor � are 4, the Bessel function parameter � is calculatedfrom � � ��2/�2 � � � 0.5�2 � 0.8, and the samplingdensity function is calculated from the Voronoi diagrams(32); and inverse gridding is implemented with the near-est-neighbor regridding method (33). Based on our exper-iments, changing these predefined parameters will not im-prove the result significantly.

Choice of the Convolution Kernel

In Cartesian GRAPPA, a slightly larger size of the convo-lution kernel (the relative shift operator) provides moreaccurate results (34,35) at the expense of a longer recon-struction time. Our experiments show that radial GRAPPAalso prefers a slightly larger size of the convolution kernel.Figure 3 shows some examples of convolution kernels. Forradial GRAPPA calibrated with full k-space data, kernel

FIG. 3. Examples of convolution kernels.One sector of the radial trajectory is used fordemonstration. The black dots show the lo-cations of acquired data. The circles showthe locations of missing data. The solid boxshows the location for the data point beinginterpolated. The dotted box shows theshape and size of the convolution kernel.

1078 Huang et al.

4 generates better results than other kernels. However,kernel 4 is not the best choice for radial GRAPPA cali-brated with pseudo-full k-space data. Table 1 shows therelative errors of the results calibrated with pseudo-fullk-space data. Brain anatomy data and the first time frameof the cardiac function data are used in this table. Fromthis table, it can be seen that kernels 2 and 3 alwaysprovide better results than the others. To balance thechoice of neighbors along radial and azimuthal directions,kernel 3 is recommended. Kernel 3 is used in all of theexperiments presented in the following sections.

Number of Iterations

To generate the pseudo-full k-space calibration data, threeconstraints from the reconstructed image are used: theimage support mask, the image phase, and the magnitudeof the result of direct gridding. These constraints originatefrom the partially acquired k-space data and have errors.These constraints can be updated with the reconstructedfull k-space data to generate better calibration data. There-fore, iteration methods potentially generate better results.To show the advantages of iteration and to find the termi-nation condition for the iteration, the two data sets de-scribed in the previous section are used again as test sets.

Figure 4 shows the results of the brain data set. Figure 4ashows the relative error from different numbers of itera-tions. From top to bottom, the plots are for reductionfactors 8 (32 projections), 6 (43 projections), 4 (64 projec-tions), and 2 (128 projections), respectively. It can be seenthat iterative reconstruction can reduce relative errors forradial GRAPPA with pseudo-full k-space calibration data,especially when the reduction factor is high. Figure 4b–fshow the images at reduction factor 6. Clearly, iterationsimprove spatial resolution. From Fig. 4a, it can be seenthat the change in relative error after three iterations istrivial. Figure 4e shows the difference of Fig. 4c (threeiterations) and d (six iterations). Figure 4e is brightenedfive times. The difference is insignificant, but reconstruc-tion time is doubled because of the six iterations. Hencethree iterations are sufficient. Similar observations can bemade from Fig. 5 regarding the cardiac images.

In conclusion, for radial GRAPPA calibrated with pseu-do-full k-space data, 32 segments along the radial direc-tion, convolution kernel 3, and three iterations are used inour implementation. For radial GRAPPA calibrated withideal or external full k-space data, except for the convolu-tion kernel 4 being used, all other parameters are same.The proposed method, self-calibrated GRAPPA/k-t

GRAPPA, was implemented in the MATLAB programmingenvironment (MathWorks Inc., Natick, MA, USA). In all ofour experiments, MATLAB codes were run on anxw4100 hp workstation (Hewlett-Packard Co., Palo Alto,CA, USA) with 3.2 GHz CPU and 2 GB RAM.

RESULTS

In this section the results of self-calibrated GRAPPA/k-tGRAPPA as applied to brain MRI and CMRI with radialtrajectory are presented.

Comparison of Radial GRAPPA Calibrated With Ideal andPseudo-Full k-Space Data

To show the performance of the proposed method, theresults of radial GRAPPA calibrated with pseudo-full k-space data were compared with the ideal results fromradial GRAPPA, which is calibrated with ideal full k-spacedata. Figure 6 shows the comparison with brain anatomydata, and Fig. 7 shows cardiac function data. From theplots of relative errors (Figs. 6b and 7b), it can be seen thatthe relative errors of both methods are almost the samewhen the reduction factor is �4; the difference of relativeerror becomes larger with the increase in reduction factor.At reduction factor 8 (32 projections), the relative errors ofthe ideal results are 14.7% (brain) and 11% (cardiac), andthose of the proposed method are 16.3% (brain) and 12.5%(cardiac). From the images (Figs. 6c–h and 7c–h), it can beobserved that 1) images reconstructed by both methods arealmost identical (Figs. 6c–f and 7c–f) when the reductionfactor is �4, 2) image spatial resolution is reduced whenthe reduction factor is higher for both methods, 3) thespatial resolutions of both methods are comparable, and 4)the results of the proposed method have more residualartifacts than the ideal results.

In summary, radial GRAPPA with pseudo-full calibra-tion data can generate results with comparable spatialresolution to the ideal result, but with more visible resid-ual artifacts when the reduction factor is higher than 4.Hence, pseudo-full k-space calibration data can be usedwhen there is no external full k-space calibration dataavailable.

Comparison of Externally Calibrated and Self-CalibratedGRAPPA

The results of radial GRAPPA calibrated with pseudo-fullk-space data were compared with those of externally cal-ibrated GRAPPA. One potential problem with externallycalibrated GRAPPA is that the coil sensitivities mightchange between the acquisition of the calibration data andthe accelerated imaging data due to motion. This problemcan be avoided by using self-calibrated GRAPPA. As anexample, the cardiac function data set was used to showthe advantages of self-calibrated GRAPPA. For externallycalibrated radial GRAPPA, the fully acquired k-space data(256 projections) for the first time frame was used forcalibration. The convolution kernels generated from thefirst time frame were applied to all other time frames.Figure 8 shows the results of time frame 10 at reductionfactors 2 (128 projections), 4 (64 projections), and 8 (32

Table 1Relative Errors (%) Radial GRAPPA With Pseudo-Full CalibrationData With Different Convolution Kernels

Kernel

R factor

Brain anatomy data Cardiac function data

2 4 6 8 2 4 6 8

1 6.4 12.4 16.2 19.3 5.0 9.4 13.1 16.02 5.6 11.6 15.7 19.3 4.6 8.3 11.1 13.63 5.5 11.8 15.9 19.3 4.6 8.3 11.0 13.64 6.2 13.4 16.7 19.6 4.6 8.3 11.0 13.7

Self-Calibration for Radial GRAPPA/k-t GRAPPA 1079

projections). At reduction factor 2, the images recon-structed by radial GRAPPA with two different calibrationdata are almost identical (Fig. 8a, b, and d). The imagereconstructed by externally calibrated radial GRAPPAwith 128 projections (Fig. 8a) is used as the referenceimage because no fully acquired k-space data (256 projec-tions) were available at time frame 10 for reference, andFig. 8a is one of the best reconstructions available. FromFig. 8e–l it can be seen that 1) when the reduction factorbecomes larger, the images reconstructed by radialGRAPPA lose more structural information and havemore residual streaking artifacts; 2) self-calibrated radialGRAPPA (Fig. 8f, h, j, and l) preserves the structural

information better than externally calibrated radialGRAPPA (Fig. 8e, g, i, and k); and 3) images recon-structed by self-calibrated radial GRAPPA have moreresidual streaking artifacts. The results demonstrate thatthe self-calibration technique can preserve the struc-tural information better when the sensitivity mapschange during acquisition.

Self-Calibrated Radial k-t GRAPPA

The self-calibration scheme was applied to k-t GRAPPAand the results were compared with those obtained bythe sliding-window method and temporal GRAPPA

FIG. 4. Comparison of results of self-calibratedradial GRAPPA with different number of iterations.Brain anatomy data are used in this image. a: Plotof relative errors with different numbers of itera-tions. From top to bottom, the plots are for reduc-tion factors 8 (32 projections), 6 (43 projections), 4(64 projections), and 2 (128 projections); Reductionfactor 6 (43 out of 256 projections) is used for b–e.b–d: Reconstruction with one, three, and six iter-ations. e: The difference map between (c) and (d).Image (e) was brightened five times for visibility.

1080 Huang et al.

(TGRAPPA) with adaptive calibration data. In non-Carte-sian k-t GRAPPA, seven to eight neighbors from each chan-nel were used for interpolation (36). The neighbors wereselected from all the acquired data with a distance of �2 inthe azimuthal direction in the adjacent 2 � R –1 timeframes, where R is the acceleration factor. TGRAPPA (37)refers to radial GRAPPA with adaptive full k-space cali-bration data generated by the sliding-window method. Forthe implementation of TGRAPPA, all parameters were the

same as those for radial GRAPPA calibrated with ideal fullk-space data, as described in the Materials and Methodssection.

Cardiac cine function images were used in this experi-ment. The results with reduction factor 8 (32 projections)are shown in Fig. 9. By comparing the images in the firstrow of Fig. 9, it can be seen that the resulting images fromradial k-t GRAPPA (Fig. 9b) and the sliding-window

FIG. 6. Comparison of results from radial GRAPPA calibrated byideal full k-space and pseudo-full k-space data. Brain anatomy dataare used in this image. a: The reference image reconstructed with256 projections. b: Plot of relative errors of the results of radialGRAPPA with two different calibration data, ideal full k-space data(solid line), and pseudo-full k-space data (dotted line). c–h: Imagesreconstructed by radial GRAPPA calibrated by ideal full k-spacedata (c, e, and g) and pseudo-full k-space data (d, f, and h) with 128projections (c and d), 64 projections (e and f), and 32 projections (gand h).

FIG. 5. Comparison of results from self-calibrated radial GRAPPAwith different numbers of iterations. The first time frame of thecardiac function data is used in this image. Only the zoomed-in ROIis shown in (b–e). a: Plot of relative errors with different numbers ofiterations. From top to bottom, the plots are for reduction factors 8(32 projections), 6 (43 projections), 4 (64 projections), and 2 (128projections). Reduction factor 8 (32 of 256 projections) is used for(b–e). b–d: Reconstruction with one, three, and six iterations. e: Thedifference map between (c) and (d). Image e was brightened 10times for visibility.

Self-Calibration for Radial GRAPPA/k-t GRAPPA 1081

method (Fig. 9d) are sharper than that from TGRAPPA(Fig. 9c). From the zoomed images, it can be seen that theimage from radial k-t GRAPPA (Fig. 9f) has reduced stripeartifacts compared to the image obtained from the sliding-

window method (Fig. 9h). Figure 9a and the zoomed image(Fig. 9e), which are the results of reconstruction byTGRAPPA with 128 projections, are used as references.Relative errors (with respect to TGRAPPA with 128 pro-jections) of all time frames reconstructed by three differentapproaches are provided in Fig. 10. It demonstrates againthat the images reconstructed by radial k-t GRAPPA havethe least errors for all time frames.

In summary, this section demonstrates that images re-constructed by self-calibrated radial k-t GRAPPA couldhave higher spatial resolution than those reconstructed byradial TGRAPPA using adaptive calibration data, and re-duced artifacts compared to those reconstructed by thesliding-window method.

CONCLUSIONS

A technique is introduced for self-calibrated radialGRAPPA/k-t GRAPPA. This technique makes radialGRAPPA applicable without extra calibration data, and k-tGRAPPA applicable to radial trajectory. The self-cali-brated radial GRAPPA technique can generate results withcomparable spatial resolution to radial GRAPPA calibratedwith ideal full calibration data, with less loss of structuralinformation than externally calibrated radial GRAPPA, atreduction factors as high as 8. However, self-calibratedradial GRAPPA suffers from more residual streaking arti-facts and requires a longer reconstruction time than exter-nally calibrated radial GRAPPA. Compared to the sliding-window method, images generated by self-calibrated ra-dial k-t GRAPPA have less streak artifacts. Compared toTGRAPPA with adaptive calibration data, images recon-structed by k-t GRAPPA have higher spatial resolution. Onthe other hand, self-calibrated radial k-t GRAPPA requiresa longer time for reconstruction compared to the sliding-window and TGRAPPA methods because one additionalgridding and inverse gridding step is needed at each timeframe for pseudo-full k-space data. Iterative reconstructionimproves the image quality of self-calibrated k-t GRAPPA.However, the reconstruction time due to the iterations maybe too long for dynamic imaging. Therefore, iterative re-construction is not recommended for high-speed dynamicimaging. This pseudo-full k-space-based self-calibrationtechnique can be used for dynamic imaging, where thecalibration data may be time-dependent and time-consum-ing, as well as for imaging static objects without extracalibration data.

ACKNOWLEDGMENTS

The authors thank Dr. Eugene G. Kholmovski and Dr.Edward V.R. DiBella at the University of Utah for provid-ing the cardiac MRI data, and the anonymous reviewers fortheir invaluable feedback.

APPENDIX

Segmentation of Signal and Background

The segmentation technique is based on two natural as-sumptions. First, the averages of the intensity values inbackground and signal regions differ significantly. This

FIG. 7. Comparison of results from radial GRAPPA calibrated byideal full k-space and pseudo-full k-space data. The first time frameof the cardiac function data is used in this image. Only thezoomed-in ROI is shown. a: Reference image reconstructed with256 projections. b: Plot of relative errors of the results of radialGRAPPA with two different calibration data, ideal full k-space data(solid line), and pseudo-full k-space data (dotted line). c–h: Imagesreconstructed by radial GRAPPA calibrated by ideal full k-spacedata (c, e, and g) and pseudo-full k-space data (d, f, and h) with 128projections (c and d), 64 projections (e and f), and 32 projections (gand h).

1082 Huang et al.

assumption sets the basic criterion for segmentation to bethe intensity level. Second, the signal regions have atendency to be connected, i.e., there are not many iso-lated small segments of signal region. The purpose ofthis assumption is to remove some isolated high-inten-sity noise points from signal regions and to includesome low-intensity signal points in some connected sig-nal regions with a higher intensity scale. Based on theidea of the level-set technique (38) and the Mumford-Shah model (39), Chan et al. (40) introduced the famoussegmentation scheme known as “active contour withoutedges” in 2001 for piecewise constant images, and thenextended this technique for piecewise smooth/textureimages (41) in 2002. Unlike published applications in

the literature, in the current work this technique isapplied to distinguish signal and noise in the image.Since the image needs to be segmented into only twoparts, one level set is enough to describe the segmenta-tion. The level set is a continuous function. It is greaterthan zero at signal regions, less than zero at noise re-gions, and equal to zero at the boundaries between sig-nal and noise regions. If the segmentation is correct,then the intensity variation in each segment should beminimized because of the first assumption. Further-more, the length of the boundary should be minimizedwith the constraint of the minimization of intensityvariation because of the second assumption. Based onthe analysis above, the energy functional is defined by

FIG. 8. Comparison of externallyand self-calibrated radial GRAPPA.The results of the 10th time frameare shown. The fully acquired k-space data of the first time frameare used as external calibrationdata for externally calibrated ra-dial GRAPPA. The first two col-umns show the images recon-structed by externally calibratedradial GRAPPA and self-cali-brated radial GRAPPA, and thelast two columns show the differ-ence maps of the first two col-umns. Image a is used as the ref-erence. These difference mapsare in the same intensity scaleand are brightened five times forvisibility. These three rows are forreduction factors 2 (128 projec-tions), 4 (64 projections), and 8(32 projections), respectively.

FIG. 9. Reconstruction of timeframe 10 by (a) TGRAPPA with128 projections, (b) k-t GRAPPAwith 32 projections, (c) TGRAPPAwith 32 projections, and (d) thesliding-window method with 32projections. Images (e–h) are thezoomed versions of (a–d).

Self-Calibration for Radial GRAPPA/k-t GRAPPA 1083

E� �u� � ��

�u � Ia�2 H � � ��

�u � Ib�21 � H ���

� � ��

��H �� [A1]

In Eq. [A1], u is the image that needs be segmented, is thelevel set described above, Ia is the mean of image intensity

at signal regions Ia ��

� uH �

�� H �

, Ib is the mean of image

intensity at noise regions Ib ��

� u1 � H ��

�� 1 � H ��

, and H is the

Heaviside function, H � � 1, � 00, � 0 . The first term cal-

culates the sum of variations in two segments. � is thegradient operator, the second term calculates the length ofthe boundaries between signal and noise regions, and � isthe parameter to balance these two terms. Given the imageu, the level set that minimizes the energy functional willbe the segmentation result. Details can be found in Ref. 40.

The solution to Eq. [A1] usually requires solving a Ham-ilton-Jacobi type equation. Song (42) proposed a fastsweeping algorithm in his dissertation that results in afaster solution to Eq. [A1]. The outline of the fast sweepingalgorithm and our implementation is described below:

Step 1. Initialization: Construct an initial partition (par-tition one for � 0, partition two for � 0) and computethe value of E according to . Because the length of theboundaries can be calculated numerically using

��

��H ��

� i,j

�H i�1,j� � H i,j��2 � H i,j�1� � H i,j��

2 [A2]

where i,j is the value of at the pixel (i, j), there is no needto compute derivatives. Hence a two-value function, in-stead of a continuous , can be used to describe the seg-mentation. In our implementation the biggest circle in theimage domain is found, and then � 1 inside or on thecircle, and � �1 outside of the circle are set.

Step 2. Sweeping: For each point x in the image, if theenergy E decreases when we change (x) to � (x), thispoint is then updated by (x) � � (x); otherwise, (x)remains unchanged. We sweep the pixels in some pre-scribed order. For example, either a Gauss-Seidel or aJacobi iteration can be used in each sweep. In our imple-mentation the Jacobi iteration is used and the value of themeans of intensity is updated after every 100 pixels areswept. Based on the Euler-Lagrange equation of Eq. [A1],for a pixel x, when i,j is originally 1 and is changed to –1,the difference of the first term of E can be calculated by

�E1 � ui,j� � Ib�2

nn � 1

� ui,j� � Ia�2

mm � 1

[A3]

where m and n are the number of pixels for � 1 and ��1. For the second term of E, it can be seen from Eq. [A2]that only three neighboring points will be affected whenthe value of i,j is changed. Hence the change of this termis

�E2 � k�i�1

i l�j�1

j

� �H k�1,l� � H k,l��2 � H k,l�1� � H k,l��

2 [A4]

If �E � �E1 � ��E2 is less than 0, it means the energy isreduced, and hence we keep the change. Otherwise, we donot change i,j. The calculation for the case where thevalue of i,j is originally �1 is similar.

Step 3. Repeat step 2 until the energy E remains un-changed or for a given number of iterations.

This algorithm converges in less than 10 sweeps. For thesegmentation of signal and noise, we routinely use onlyone sweep because the intensity difference between signaland noise is usually significant. For simplicity, the param-eter � could set to be zero when a high-SNR pulse se-quence is used, because there is no isolated high-intensitynoise point. Thus only the first term in Eq. [A1] is neededfor segmentation. The segmentation time for a 256 � 256image is less than 1 s in our implementation.

In some cases there are some very bright regions in theimage. For example, the regions that are very close to thesurface coil could be very bright. If no intensity correctionis processed, the segmentation with one level set may failin this case. This is because the difference between theaverage intensity of these bright regions and the averageintensity of other regions in the image support is largerthan the difference between the average intensity of pixelsin image support and the average intensity in the back-ground. To solve this problem, one additional level setmust be added. With two level sets, the image will bedivided into four pieces based on the intensity. The regionwith the smallest average intensity value is the back-

FIG. 10. Relative errors of images reconstructed by three differentalgorithms with 32 projections.

1084 Huang et al.

ground. In the two level sets case, the initial level sets instep 1 can be chosen as two sets of overlapped smallcircles. Based on our experiments, this algorithm will con-verge in five to 10 iterations with this kind of choice ofinitial level sets. More discussions on the technique withtwo or more level sets can be found in Section 2.5.2 of Ref.42.

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