New Strategy for Reconstructing Partial-Fourier ImagingData in Functional MRI

Xiaodong Zhang, Essa Yacoub, and Xiaoping Hu*

Most partial Fourier (PF) approaches use a low-resolutionphase estimate in the reconstruction to account for non-zerophases of the image. These methods may fail when there arelarge phase errors, a situation commonly encountered in T *2-weighted functional MRI (fMRI). To mitigate this problem, amethod was developed based on the inversion of a matrixformulated according to a phase map derived from iterativereconstruction. To make this method computationally practicalfor fMRI, a strategy was introduced such that the matrix inver-sion is performed only once for each slice in the time series,assuming that the phase map remains constant in the timeseries. To ensure the temporal phase invariance, physiologicalnoise correction and global phase correction were applied tothe data before the reconstruction. This method was demon-strated to be robust and efficient for fMRI. Magn Reson Med46:1045–1048, 2001. © 2001 Wiley-Liss, Inc.

Key words: partial Fourier imaging; fMRI; physiological noise;singular value decomposition; phase correction

In functional magnetic resonance imaging (fMRI), partialFourier (PF) or half Fourier (HF) acquisition can be usedfor 3D imaging (1) and high-resolution imaging (2). Jes-manowicz et al. (2) showed that PF acquisition applied inthe phase-encoding direction of an EPI sequence allowed ashort TE while achieving high spatial resolution, and theshort TE provided an improved signal-to-noise ratio(SNR). In fMRI, a short TE is desirable because the opti-mum TE is the tissue T*2 and may be impossible to achievewith full Fourier EPI. In addition, a PF acquisition in EPIalso reduces the TR, thereby increasing the imaging speed.Because of these advantages, PF acquisition in the phase-encoding direction in an EPI sequence was further devel-oped for fMRI in this work.

A major difficulty of PF imaging arises from non-zerophases in the image being acquired, which leads to arti-facts in the reconstructed image. To alleviate this problem,in practice a small portion of the low k-space is sampledsymmetrically around the origin and used to provide alow-resolution phase estimate, which is used in the sub-sequent reconstruction. Several methods (3–8) are avail-able for reconstruction from this type of PF data. Theseinclude the Margosian approach (4), the homodyne detec-tion method (5), and the iterative approach (6,7). Whilethese methods work reasonably well under most circum-stances, they may be inadequate for situations in which

phase errors are large, such as those in fMRI using T*2-weighted imaging. Thus, improved methods are needed.

In an earlier work, Haacke et al. (8) mentioned a methodbased on matrix inversion using singular value decompo-sition (SVD) for PF reconstruction. In that paper, the SVDmethod was implemented using a phase map from the lowk-space data. In the present work, the SVD approach isimplemented using a phase map obtained with the itera-tive reconstruction (6,7). The present approach is shown tobe less sensitive to a large phase offset in the data thanother PF methods. Furthermore, for fMRI, in which a hugeamount of data is acquired, the application of matrix in-version for each image becomes impractical. Thus, a strat-egy is developed such that the SVD reconstruction is per-formed only once, assuming that the phase-map obtainedfrom the iterative reconstruction at one time point can beused for all images of the slice in the entire time series.This assumption, however, is violated in practice becausephysiological activities, especially respiration, lead tophase fluctuations, and system hardware drift leads tophase drifts with time. Therefore, physiological noise cor-rection and phase drift removal were applied to the rawdata before the single inversion method was applied. Thisstrategy was shown to be robust and practical for fMRI.

METHODS

Mathematical Formalism

For simplicity, and without loss of generality, we considera 1D formalism in the spatial encoding direction alongwhich PF acquisition is applied. Ignoring relaxation ef-fects and approximating the integration in the image spacewith summation, the measured k-space data can be writtenas

Sk 5 Oj50

N21

rj exp~2i2pjk/N!exp~iDj! [1]

where Sk represents the measured signal at the kth spatialfrequency, and rj and Dj are the discretized spin densityand phase factor, respectively, at location j. Eq. [1] can alsobe written in a matrix form:

S 5 Ar [2]

where S represents the measured k-space data, A is acoefficient matrix determined by the Fourier phase factorand the spatially varying phase factor, and r is a matrixdenoting the magnitude of the image. Equation [2] can besolved with a least-squares approach to obtain r. To ensuresolution stability, a singular value decomposition (SVD)

Center for Magnetic Resonance Research, Department of Radiology, Univer-sity of Minnesota Medical School, Minneapolis, Minnesota.Grant sponsor: National Institutes of Health; Grant numbers: RO1NS34756;RO1MH55346; RR07809.*Correspondence to: Xiaoping Hu, Ph.D., Center for Magnetic ResonanceResearch, University of Minnesota Medical School, 2021 6th St. SE, Minne-apolis, MN 55455. E-mail: xiaoping@cmrr.umn.eduReceived 11 May 2001; revised 11 June 2001; accepted 12 June 2001.

Magnetic Resonance in Medicine 46:1045–1048 (2001)

© 2001 Wiley-Liss, Inc. 1045

(9) is used. In SVD, the matrix A is decomposed into theform:

A 5 V¥UH [3]

where U and V are unitary matrices, the superscript Hindicates the Hermitian operator, and S is a diagonal ma-trix containing the singular values of A. In truncated SVD,the solution is regularized by truncating the smallest sin-gle values in S, leading to a pseudo-inverse of A:

A† 5 US¥̃21 00 0DVH. [4]

In the present implementation, singular values below 17%of the maximum singular value are truncated. This thresh-old was chosen empirically and used for all results shownin this paper. Reconstruction with a wide range of thresh-olds showed that the results were not very sensitive to itsvalue. With the pseudo-inverse, the reconstruction be-comes a simple matrix multiplication:

r 5 A†S. [5]

Estimation of the Phase Map

As can be seen in Eq. [2], the construction of the coefficientmatrix A requires knowledge of the phase map, Dj. In theSVD approach described previously (8), the phase mapwas obtained from the low-resolution image derived by theFourier transform of the low k-space data. In the presentapproach, an iterative reconstruction based on the methoddescribed by Haacke et al. (7,8,10) is first performed toobtain an initial estimate of the phase. The phase of theiterative reconstruction is then used to construct A.

Application to fMRI Time Series

Because the SVD procedure is very time consuming(1 min/image on an SGI Origin 2000 Workstation), it isimpractical to apply the SVD method image by image in anfMRI time series. To make the method practical for fMRI,the SVD operation is performed once, on a matrix derivedfrom one of the images or the average of several images inthe time series, and the resultant pseudo-inverse A† isapplied to the data at all time points in the series.

A problem arises when using a single A† for the entiretime series because of the existence of temporal phasevariations in the fMRI time series. Due to this phase vari-ation, the phase map derived from one of the images or theaverage of several images cannot be applied to every imagein the time series. Fortunately, this phase variation can bereadily corrected. The main source of the temporal phasevariation is the subject’s respiration. To remove this phasevariation, a retrospective method (11) is first applied toremove the physiological fluctuation in the k-space data.The second source of phase variation is a B0 field drift,which leads to a slow variation of phase with time (12).This drift, as it turns out, is approximately global in space.To remove this phase variation, after the physiologicalfluctuation correction, the phase of the zero k-space pointin the time series is calculated and smoothed by a low-pass

filter over time to generate a global phase time course. Thephase of the k-space data at each time point is then ad-justed by the difference between the global phase at agiven time point and the time point at which A† is calcu-lated before applying A† to the data to reconstruct theimage.

Data Acquisition

Experimental data were acquired on a 4T whole-body sys-tem (Varian, Palo Alto, CA; Siemens, Erlangen, Germany).Two data sets were acquired to demonstrate the method.The first was a series of 340 images of 10 coronal slicescovering the occipital cortex of a normal subject obtainedwith a single-shot T*2-weighted EPI sequence (TR 5 1 s,TE 5 30 ms, flip angle 5 60°, FOV 5 20 3 20 cm, matrix 564 3 64, slice thickness 5 6 mm). The second data set wasa time series obtained with a visually-guided motor para-digm on an axial slice across the motor area, acquired witha single-shot EPI sequence (TR 5 0.3 s, TE 5 30 ms, flipangle 5 40°, FOV 5 20 3 20 cm, matrix 5 64 3 64, slicethickness 5 7 mm). A total of 975 images were acquired,with five epochs, each consisting of a 45-s control periodand a 4.5-s stimulation period, followed by a 45-s baselineat the end.

Data Analysis

The algorithm was implemented in MATLAB 5.2 (Math-works, Inc., Boston, MA) and is available upon request. Todemonstrate the robustness of the SVD with an iteratively-derived phase map, the k-space data for the third slice indata set 1 was reconstructed with various approaches andthe resultant images were compared. In this case, the orig-inal full 64 3 64 k-space data matrix was cropped in thephase-encoding direction to a matrix corresponding to apartial k-space matrix of 64 3 40. Four different approach-es—1) the Margosian approach (4); 2) the iterative ap-proach (7); 3) SVD with phase map determined from thelow-resolution image (8); and 4) SVD with iterative phaseestimate—were used to reconstruct images from the partialk-space data. The reconstructed images were comparedwith the image derived from the full k-space data. The firstdata set was also processed to demonstrate the effective-ness of using a single pseudo-inverse for the entire timeseries.

The second data set was used to illustrate the perfor-mance of the reconstruction strategy for fMRI. In this case,the time series (minus the first epoch to account for theapproach to steady state) was reconstructed with 1) fullFourier reconstruction, 2) image-by-image inverse, 3) sin-gle inverse with both drift and physiological noise correc-tion, and 4) single pseudo-inverse with drift correctiononly. The resultant time courses in the activated regionswere compared.

RESULTS

In Fig. 1, the results of the PF reconstructions are pre-sented along with the full-Fourier reconstruction. The up-per row shows the reconstructed images and the lower rowshows the difference (scaled up by a factor of 2) between

1046 Zhang et al.

the PF images and the full Fourier image. The averageabsolute error (in arbitrary units) for each PF reconstruc-tion is indicated in the difference image. The SVD withiterative phase estimation generated the best reconstruc-tion, while the Margosian reconstruction led to the worstimage. In the Margosian reconstruction, there are wide-spread regions with artifactual signal loss arising from theincorrect phase information in the low-resolution phaseestimate. These signal losses are recovered in the other PFreconstruction approaches. According to the average error,the performance of these techniques can be ordered in thefollowing ascending order: Margosian, SVD with low-res-olution phase, iterative reconstruction, and SVD with iter-ative phase. In Fig. 1, it should also be noted that there islocalized signal loss in all PF images on the right side. Thisis due to a substantial phase variation in that area, whichnot only led to more T*2 decay but also to more phaseerrors degrading all PF reconstructions.

From data set 1, image 100 was reconstructed with A†

derived from image 170 with (Fig. 1a) and without (Fig. 1b)correction for phase variations in the time series as well aswith A† derived from the data at time point 100 itself (Fig.1c). The average difference between Fig. 1b and c was 2.71,indicating that without phase correction, the pseudo-in-verse derived from time point 170 cannot be applied totime point 100. In contrast, the average difference betweenFig. 1a and c was very small (0.01), indicating that afterremoving the temporal phase variations, A† derived fromone point in the time series can be applied to other timepoints in the series.

Time courses of an activated ROI from the visual stim-ulation study, reconstructed by different versions of SVDwith an iterative phase estimate, are shown in Fig. 2. Withdrift correction alone, the single inverse approach workedreasonably well. However, fluctuation due to phase errorsarising from physiological noise is evident. Phase correc-tion for both drift and physiology essentially replicated theresult of image-by-image inversion, which is virtually

identical to the full Fourier reconstruction except for theexpected reduction in SNR.

DISCUSSION

As shown in Fig. 1, the Margosian approach, a widely usedapproach in practice, is inadequate for T*2-weighted imag-ing in fMRI because of the presence of large phase offsetsin the image. The iterative method seems to provide abetter phase estimate and thereby a better image by itself.The reconstruction method introduced in this paper com-bines SVD with the phase estimation from iterative recon-struction. This method is demonstrated to have better per-formance than other methods. In addition, because a trun-cated SVD is used, the noise in the reconstructed image isalso reduced.

The use of a single A† for a time series is needed to makethe method more efficient for fMRI. However, a straight-forward implementation of such a strategy does not workdue to temporal phase variations in the image. Two types

FIG. 1. a: Full Fourier image and PF images reconstructed with (b) the Margosian method, (c) iterative reconstruction, (d) SVD withlow-resolution phase map, and (e) SVD with iterative phase map. The difference between each PF reconstruction and the full Fourier imageis shown below each image along with the average absolute error (in arbitrary unit). Note that the difference images were scaled by a factorof 2.

FIG. 2. Time course of an activated area reconstructed from thetime series reconstructed with the (a) full Fourier data, (b) PF datawith direct reconstruction, (c) PF data with single inverse and phasecorrection for both respiration and drift, and (d) single inverse withdrift correction alone.

Partial Fourier Imaging for fMRI 1047

of phase variation were identified and removed. Our re-sults showed that with the removal of these phase varia-tions, the use of a single pseudo-inverse for the entire timeseries is virtually equivalent to the use of a time-point-specific A† in the reconstruction, providing substantialsavings in the reconstruction time (by a factor of 35).

In applying PF imaging to fMRI, Stenger et al. (1) uti-lized a strategy similar to that described above for obtain-ing the phase map. Their approach was to acquire a sepa-rate full Fourier image to produce the phase map, andapply the PF acquisition along the third direction in 3Dspiral imaging. Such an approach may be problematic for2D EPI because the difference in the applied gradientsbetween a full Fourier acquisition and PF acquisition mayrender the phase map from the former inappropriate forthe latter. On the other hand, the corrections of temporalphase variation used in the present work can be incorpo-rated with that strategy as well to improve the robustnessof the method.

CONCLUSIONS

This work introduced a new method for reconstructing PFdata in the presence of fairly large phase offsets in theimage, such as those encountered in T*2-weighted images.The method, based on using SVD along with a phaseestimate derived from an iterative reconstruction, reducesthe reconstruction error substantially. Furthermore, astrategy was developed for fMRI such that only a singleSVD operation per slice is used for the entire time series,substantially reducing reconstruction time. To ensure therobustness of this strategy, temporal phase variations inthe data are removed before the reconstruction is per-

formed. This strategy was found to be effective for recon-struction of fMRI time series, and is expected to be appli-cable for routine fMRI applications in which PF acquisi-tion is used.

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