concurrent correction of geometric distortion and motion...

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ng 26 (2008) 703–714
Magnetic Resonance Imagi
Concurrent correction of geometric distortion and motion using themap-slice-to-volume method in echo-planar imaging

Desmond T.B. Yeoa,b, Jeffrey A. Fesslera,b, Boklye Kima,⁎aDepartment of Radiology, University of Michigan Medical School, Ann Arbor, MI 48109, USA

bDepartment of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109, USA

Received 7 May 2007; revised 30 September 2007; accepted 24 November 2007

Abstract

The accuracy of measuring voxel intensity changes between stimulus and rest images in fMRI echo-planar imaging (EPI) data is severelydegraded in the presence of head motion. In addition, EPI is sensitive to susceptibility-induced geometric distortions. Head motion causesimage shifts and associated field map changes that induce different geometric distortion at different time points. Conventionally, geometricdistortion is “corrected” with a static field map independently of image registration. That approach ignores all field map changes induced byhead motion. This work evaluates the improved motion correction capability of mapping slice to volume with concurrent iterative fieldcorrected reconstruction using updated field maps derived from an initial static field map that has been spatially transformed and resampled.It accounts for motion-induced field map changes for translational and in-plane rotation motion. The results from simulated EPI time seriesdata, in which motion, image intensity and activation ground truths are available, show improved accuracy in image registration, fieldcorrected image reconstruction and activation detection.© 2008 Elsevier Inc. All rights reserved.

Keywords: Field inhomogeneity; fMRI; Motion correction; Geometric distortion; Activation detection

1. Introduction

In functional MRI (fMRI) studies, activation maps aregenerated by statistical analyses of voxel intensity changesbetween stimulus and rest images of echo-planar imaging(EPI) time series data. The accuracy of measuring theseintensity changes, which are typically in the range of 1% to4%, is severely degraded in the presence of head motion. Tocompensate for head motion, some studies employ in-planeand volumetric image registration techniques on EPI timeseries data [1–4]. Single-shot EPI data are also sensitive tomagnetic susceptibility(χ)-induced geometric distortions,especially for the mid to lower brain regions near air–tissueand bone–tissue interfaces. Subject motion causes imageshifts as well as field-map changes which result in time-varying local changes in geometric distortion [5,6]. Conse-

⁎ Corresponding author. Basic Radiological Sciences Division, Depart-ment of Radiology, University of Michigan Medical School, Ann Arbor, MI48109-2200, USA. Tel.: +1 734 763 5692; fax: +1 734 615 1471.

E-mail address: [email protected] (B. Kim).

0730-725X/$ – see front matter © 2008 Elsevier Inc. All rights reserved.doi:10.1016/j.mri.2007.11.001

quently, inconsistencies in voxel positions across the timeseries images degrade statistical testing of the signal changesin response to the given tasks in activation studies.

In fMRI, geometric distortion correction typically isperformed with a static field map independently of motioncorrection [6,7], thus ignoring field-map changes caused byhead motion. A prospective approach to this problem is toacquire field maps simultaneously with EPI data during anfMRI experiment to track temporal field-inhomogeneitychanges by collecting additional k-space data [8,9]. How-ever, acquisition of extra k-space data within reasonable timemay pose some limitations in field-map resolution since, toreduce scan time, the extra data acquired are typicallyconstrained to a low-pass filtered or truncated version of afull set of EPI image k-space data. Also, modifying pulsesequences may not be an available option in many clinicalscanners. In addition, some of the algorithms for imagereconstruction and post-processing are computationallyintensive [9]. A correction strategy [10,11] that does notrequire any a priori field maps is of notable interest. Instead,pairs of EPI images are acquired with opposing blipped

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http://dx.doi.org/10.1016/j.mri.2007.11.001

704 D.T.B. Yeo et al. / Magnetic Resonance Imaging 26 (2008) 703–714

phase encode gradient polarity, thus yielding image pairswith identical geometric distortion but in opposite directionsalong the phase encoding axis. The deformation fieldbetween each pair of images, from which the dynamic fieldmap is computed, is estimated from the images' intensityvalues. However, to acquire the pairs of EPI images, pulsesequence modification is required, and motion-inducedfield-map changes may occur between the acquisitions. Thismay lead to local differences in geometric distortion in eachpair of images, especially in regions with susceptibility-induced field inhomogeneity, which may yield inaccuratefield-map estimates. Another retrospective correctionmethod [5] adopts a least squares approach and modelsthe temporal change in B0 using a Taylor series expansionwith respect to motion parameters. Qualitative resultsindicate good correlation between estimated and measuredparameters. This model was designed for registration andgeometric distortion correction of mono-modality EPI timeseries images with a pre-selected EPI volume as a referencevolume. An anatomically correct structural volume was notused for registration.

Previously, in our group, a realistic motion-correctionapproach by mapping a slice-to-volume (MSV) for multi-slice EPI time series was developed [12]. This technique,which uses negated mutual information (MI) as the similaritycost metric, allows individual slices in the time series to bemapped to an anatomically correct reference volume and canaccurately correct image shifts due to 3D rigid head motion.Compared to the widely used volumetric registration of EPIvolumes, which ignore inter-slice motion, the MSVapproachimproved sensitivity and specificity in localizing activatedregions [12]. Moreover, receiver operating curve (ROC)analyses of detected activation from MSV-corrected simu-lated datasets and datasets processed by SPM suggest thatMSV improves activation detection results [13]. Althoughrigid-body transformations may suffice for activationlocalization in the sensorimotor cortex, geometric distortionsin conventionally reconstructed EPI slices acquired from themid to lower brain regions cause difficulty in localizingactivations, which makes language studies with fMRIdifficult. Consequently, MSV was expanded to include anonlinear warping function for studies involving mid-brainregions [14,15]. This improvement increases computationdue to the longer optimization process associated with higherdegrees of freedom in registration.

This work evaluates an extension of MSV withconcurrent field-inhomogeneity correction for EPI timeseries images and was introduced previously using a smalltest dataset with known motion parameters [16]. Theconcurrent field-map MSV (CFMMSV) method employsiterative field-corrected quadratic penalized least squares(QPLS) image reconstruction [17] followed by a field-mapupdate to enhance the MSV rigid body motion-correctionscheme, therefore accounting for field-inhomogeneitychanges with inter-slice head motion. The proposedmethod consists of iterative correction cycles, each with

a pair of QPLS image reconstruction and MSV motioncorrection stages. In each cycle, dynamic field-map slicesare resampled from a high-resolution 3D static field mapthat has been spatially transformed by a rigid bodytransformation function determined by MSV for therespective EPI slices. Since geometric distortion iscorrected incrementally in the QPLS stage after eachfield-map update, a rigid body MSV motion model shouldsuffice and requires less computation than nonlinear MSV.For two sets of realistically simulated EPI time series withdifferent ground truths for rigid body motion, imageintensities and activation regions, results show that theCFMMSV method improves the accuracy of the estimatedmotion parameters and reconstructed images when com-pared to a strategy that performs geometric distortion andmotion correction independently using just a static fieldmap. Nonparametric random permutation tests were alsoperformed on all datasets at various stages in theCFMMSV correction process to compute activationdetection (ROC) curves. The areas under these ROCcurves show that the CFMMSV method improves theactivation detection accuracy.

2. Background

2.1. EPI geometric distortion

In brain EPI data that are reconstructed without field-inhomogeneity correction, geometric distortion is observedin regions where the local magnetic field is inhomoge-neous, especially at the boundaries of tissues withsignificant magnetic susceptibility differences. Head motionthat changes the orientation of the inter-tissue boundarywith B0 (out-of-plane rotations) may induce significantfield-inhomogeneity changes in the region around theboundary. The effects of this type of motion, which includetwo out-of-plane rotations, are not modeled in this work.Translations and in-plane rotation are less likely to causesuch local changes in the susceptibility-induced componentof the field-map and mainly induce shifts and in-planerotation of the entire field map. In EPI, due to the longreadout time, field inhomogeneity causes pixel shifts,mainly in the phase encoding direction [18,19]. Theseshifts depend on the EPI slice readout time Treadout and thepoint field inhomogeneity ΔB(xi,yj). The space-variantpixel shift in the phase encoding direction causes geometricdistortion and intensity accumulation or spread, whichadversely affect fMRI activation detection.

2.2. Iterative field-corrected reconstruction

To perform geometric distortion correction, we use aniterative field-corrected reconstruction method that, unlikemany noniterative image reconstruction methods, does notassume a smooth field map [17]. The continuous object f andfield-map Δω are parameterized into a sum of weighted rectfunctions ϕ(r−rn) where r is the vector of spatial coordinates.

705D.T.B. Yeo et al. / Magnetic Resonance Imaging 26 (2008) 703–714

Ignoring spin relaxation and assuming uniform receiver coilsensitivity, the observed MR raw data of slice frame l are

ul tmð Þ ¼ sl tmð Þ þ el tmð Þ; l ¼ 0 N L� 1; ð1Þand the parameterized MR signal equation of slice l is

sl tmð Þ ¼ U k tmð Þð ÞXN�1

n¼0

f ln e�jDxl

ntme�j2p k tmð Þ�rnð Þ; l ¼ 0 N L� 1;

ð2Þwhere sl(tm) is the baseband MR signal sample at time tmduring readout, el(tm) denotes white Gaussian noise [20],Φ(k(tm)) denotes the Fourier transform of ϕ(r), N is thenumber of pixels in a slice, L is the total number of sliceframes in the EPI time series, and fn

l and Δωnl are the object

intensity and field-inhomogeneity values, respectively, at rn.The matrix–vector form of Eq. (1) can be written as follows:

ul ¼ Alf l þ el ð3Þwhere f l=( f0

l,…, fN-1l ) and elements of the M×N system-

object matrix Al are

alm;n ¼ U k tmð Þð Þe�jDxlntme�j2p k tmð Þ�rnð Þ: ð4Þ

To estimate the unknown object slice f l from the observedk-space data, the iterative conjugate gradient algorithm isused to minimize the QPLS cost function

W1 f l� � ¼ jjul � Alf ljj 2 þ b jjCf l jj 2 ð5Þ

where C is a first-order difference matrix, and β is aparameter that controls the tradeoff between obtaining adata-consistent estimate and a smoothed, regularized esti-mate. The QPLS estimate of f l is

flQPLS ¼ argmin

fW1 fð Þ ¼ Al4Al � bC4C

� ��1Al4ul; ð6Þ

where an asterisk denotes a matrix complex conjugatetranspose operation. However, we minimize Eq. (5) usingthe conjugate gradient algorithm instead of evaluatingEq. (6) directly.

To accurately perform field-corrected reconstruction inthe presence of head motion, every slice of observed data ul

should be paired with a dynamic field-map slice Δωl thatdescribes the field inhomogeneity at frame l of the fMRIexperiment. Typically, however, only a static field mapΔωstatic is available. This field map is usually acquiredbefore or after the experiment and does not track field-inhomogeneity changes during the acquisition of the fMRItime series images. Each volume in the EPI time seriestypically has a lower spatial resolution and larger slicethickness than Δωstatic. A simple approach to obtain field-corrected fMRI images is to register each time series volumeto the image intensity volume acquired in the samecoordinate space as the static field-map volume and thenuse resampled slices of Δωstatic in place of Δωl inminimizing Eq. (5). This correction method ignores inter-

slice motion and field-inhomogeneity changes due to headmotion and thus yields potentially significant imagereconstruction errors.

2.3. Slice-to-volume motion correction

The map-slice-to-volume (MSV) motion correctionscheme [12] models the 3D motion of multislice EPI databy allowing each slice to have its own six degrees of freedommotion. In MSV, each reconstructed EPI slice f l is registeredwith a 3D reference volume gref using the six degrees offreedom rigid body transform denoted by Tαl. The vector αl

consists of the six MSV motion parameters tx, ty, tz, θx, θy,θz for slice l. This is performed by minimizing a function Ψ2

(αl) that measures the dissimilarity between f l and gref. Inthe implementation of MSV, the negated MI is used, whichperforms well for multi-modality datasets, i.e., T2⁎-weightedEPI slices registered with a T1-weighted reference volume.The motion parameters αl are estimated by minimizing thefollowing cost function over αl using the Nelder–Meaddownhill simplex optimization algorithm

W2 αl� � ¼ �MI gref ; f

l Tαl rð Þð Þ� � ð7Þ

Each set of optimized motion parameters is then used totransform and interpolate (trilinear) its respective slice f l intoa volume with the same spatial coordinates as the referencevolume. In the original MSV method, the motion of eachslice is computed independent of other slices and allows sixdegrees of freedom between each slice acquisition. Forsingle-shot acquisition, intra-slice motion is negligible.Given that head motion is typically correlated in time andthat MSV may generate outlier estimates, especially for topslices where the information content is reduced, we applytemporal median filtering on the recovered MSV motionparameters before use. A median filter was chosen becausethe MSV motion estimates obtained from data withsimulated smooth motion were observed to track the groundtruth except for intermittent outlier estimates. It is noted thatmedian filtered MSV motion parameters are used to computefield map updates in the CFMMSV method and not forrepositioning EPI slices into the anatomical referencevolume, i.e., EPI motion correction. The latter is done withraw unfiltered MSV motion parameters.

3. Materials and methods

All experiments were performed on Intel Pentium 4 Xeon3.0-GHz CPUs using MATLAB (The Mathworks, Inc.,Natick, MA, USA) and Advanced Visual Systems(Advanced Visual Systems, Inc., Waltham, MA, USA).

3.1. Concurrent field-inhomogeneity correction with MSV

To design a concurrent correction technique that involvesEqs. (5) and (7) with a rigid body transformation function,the main challenge is the approximation of the dynamic field

Fig. 1. Median filtered MSV motion parameter ty recovered at variouscorrection cycles for Dataset A. Field inhomogeneity-induced geometricdistortion in the phase encoding direction y causes significant MSV errorsfor the distorted EPI data as well as the corrected data in cycle 0 (κ=0).Correction cycles 1 to 3 yield estimates of ty that are close to the ground truthas shown in Tables 1 and 2. The above plot shows 130 simulated volumesinstead of the 120 volumes used for subsequent activation analyses.


map from the initial static field map. To partially account forfield-map changes due to 3D head motion during the fMRIexperiment, we propose and evaluate the following “con-current” correction approach. The concurrent field-map andMSV motion correction framework (CFMMSV) loopsthrough several correction cycles, each of which consistsof a field-corrected reconstruction stage followed by MSV.The changing field map is approximated using the recoveredMSV motion estimates and the static field map. As thenumber of correction cycles increases, the geometricdistortion is incrementally corrected in the image reconstruc-tion stage. Thus, a rigid body transformation function inMSV is expected to be sufficient to correct for head motion.This leads to a faster image registration process compared tothe use of 3D nonlinear warping functions.

Since the true dynamic field map Δωl in Eq. (4) isunavailable, the EPI time series image reconstruction in theinitial cycle (κ=0) of the concurrent correction algorithm isperformed with the static field-map volume Δωstatic. Theelements of the system matrix Al,κ=0 can be written as

al;j¼0m;n ¼ U k tmð Þð Þe�jDxl

n;statictme�j2p k tmð Þ�rnð Þ ð8Þ

where κ denotes the correction cycle number. The firstfield-corrected estimate of slice frame l, f l;j¼0, is thenobtained by minimizing Eq. (5) with Al=Al ,κ=0. Thereconstructed slices are then registered via MSV to thereference volume gref by minimizing Eq. (7). Each set ofmedian filtered motion parameters αl;j

filt, l=0,…L−1, is appliedto the original static field-map volume and the respectiveslice within the transformed field-map volume is re-sampledand stacked into a new field-map volume

Δxl;jþ1 ¼ Hl Δxstatic Tα l;j rð Þ� �� ; l ¼ 0; N ; L� 1; ð9Þ

where Hl{U} denotes an operator which resamples slice lfrom a volume U. This updated field map is then used in thenext cycle to reconstruct field-corrected images again fromthe original k-space data.

Since geometric distortion in EPI is predominantly in thephase encoding direction, the recovered MSV motionparameters in the phase encoding direction are not used totransform the static field map in the initial field-map updatewhen κ=0. This is because the EPI image shifts in the phaseencoding direction may be largely influenced by fieldinhomogeneity-induced geometric distortion rather thanobject motion. This is illustrated in the MSV estimates oftranslations in the phase encoding direction (Fig. 1) for asimulated EPI time series (Dataset A) with applied motion inthree translational directions and in-plane rotation about thez axis. In Fig. 1, field inhomogeneity-induced geometricdistortion in the phase encoding direction causes significantMSV errors for the distorted EPI data as well as the field-corrected EPI data in cycle 0 (κ=0). Correction cycles 1 to3 yield estimates of ty that are close to the ground truth. Inaddition to translation in the phase encoding direction, the

field map may change significantly with out-of-planerotations which would render these motion parametersunreliable for the initial field-map update. Thus, for κ=0,the motion parameters ty (translation in phase encodingdirection), θx and θy (out-of-plane rotations) are omittedwhen applying the transformation Tαl;j¼0 to the static fieldmap. For the following cycles, κ≥1, all six degrees offreedom are used when applying Tαl;j to update the field map.The original raw data u and static field-mapΔωstatic are usedin each cycle to approximate the dynamic field-map andfield-corrected EPI images to avoid error propagation due tointermediate processing steps as the number of cyclesincreases. The CFMMSV method is summarized as follows:

Algorithm. Concurrent QPLS-MSV for EPI motion andfield-inhomogeneity correction

Initial data: Dxl;j¼0 (slice l of static field-mapΔωstatic), ul

(k-space data), l=0…L−1for κ=0…K (correction cycles)Step 1: f l;j ¼ argmin

fjjul � AðDxl;jÞf jj 2 þ b jjCf jj 2

Do for l=0…L−1 (QPLS)Step 2:αl;j ¼ argmin

α�MI gref rð Þ; f

l;jTα rð Þ

� � on

Do for l=0…L−1 (MSV)


Step 3: Median filter αj to obtain αjfilt.

Step 4: Dxl;jþ1 ¼ Hl Dxstatic Tα l;j rð Þ� n o

Do for l=0…L−1 (resample slice l)

The computation times for field corrected iterativereconstruction andMSVare 1.5 and 3 min/slice, respectively.

3.2. Motion, functional activation and geometric distortionsimulation in EPI time series

To evaluate the effectiveness of the CFMMSV method inrecovering accurate motion parameters, forming accuratefield-corrected intensity images and detecting functionalactivation regions, two time series datasets, labeled A and B,were simulated with different applied motion. The groundtruths available for head motion, activation regions andnondistorted image intensities enable quantitative evaluationof the correction method. We started with two perfectlyregistered T1- and T2-weighted image datasets (matrix size:256×256×124; voxel size: 0.8×0.8×1.5 mm) derived fromInternational Consortium of Brain Mapping (ICBM) data.The T1 volume served as the anatomical reference for MSVand the T2 volume was the “baseline” volume from which thetime series datasets were simulated. The spatial coordinates x,y and z for each volume denote in-plane image column(sagittal planes), in-plane image row (coronal planes) andthrough-plane slice (axial planes) directions, respectively. Arotation about a given axis denotes a rotation about the centerof the volume perpendicular to that axis. To simulatefunctional activation, an “activated” T2 volume was createdby increasing the T2 ICBM dataset intensity by 5% inpredefined ellipsoidal regions as shown in Fig. 7E and J. Sixbaseline-activation cycles, each of which was formed byconcatenating 10 baseline and 10 activated T2 volumes, were

Fig. 2. Simulated field-map slices from a single volume with significant field-inhvalues range from −64 to +320 Hz to simulate a maximum field inhomogeneity o

assembled to form a 120-volume time series. In addition, asimulated brain static field map was created by adding three3DGaussian blobs located at the inferior frontal and temporallobes to a 3D third-order polynomial (Fig. 2). This field mapwas scaled such that the off-resonance values range from −64to +320 Hz to simulate a maximum field-inhomogeneity of5 ppm at 1.5 T.

Dataset A is a geometrically distorted EPI time series withsimulated motion in tx, ty, tz and θz (translations and in-planerotation), while Dataset B is a geometrically distorted EPItime series with simulated motion in θx, θy and θz (rotations).To generate Dataset A, temporally smooth translational andin-plane rotational motion (tx, ty, tz and θz) were applied toboth the T2-weighted baseline-activation time series andsimulated field-map volumes. Sequential 5.6-mm-thickslices were then resampled to form 120-volume intensityand field-map time series datasets (volume matrix size:128×128×14). Each resampled slice has its own set ofmotion parameters. The applied motion has maximum valuesof 7.20 mm, 8.00 mm, 3.51 mm and 4.70° for tx, ty, tz and θz,respectively. The T2-weighted volumes obtained are hence-forth referred to as the time series image intensity groundtruth without geometric distortion. These were used tocompute the image normalized root mean square error(RMSE) values at various correction cycles to measure theaccuracy of the field-corrected reconstructed images. Theapplied motion did not change the orientation of the air–tissue interface with respect to B0 and thus is unlikely tochange the field-map significantly except for the respectivetranslation or in-plane rotation. Thus, forward distorting theT2 volume with the rotated-translated field map is reasonableas long as out-of-plane rotations θx and θy are not applied.However, to test the effectiveness of the CFMMSV frame-work in the presence of out-of-plane motion while assuming

omogeneity near frontal lobe and inferior temporal lobe regions. Field-mapf 5 ppm at 1.5 T.


the field map moves with a rigid body transformationfunction together with the head, a second pair of 120-volumetime series intensity and field-map datasets (dataset B) weresimulated with temporally smooth motion in three rotationparameters θx, θy and θz. The assumption that the static fieldmap moves with the head, while simplistic in the presence oflarge out-of-plane motion, provides a way to generate anotherwise realistic time series dataset with out-of-planemotion and subsequent field inhomogeneity-induced geo-metric distortion. Datasets A and B are not meant to becompared with each other. They illustrate the performance ofthe algorithm for different types of applied motion, not fordifferent types of motion-induced field map changes. Thesimulated rotational motion has maximum values of 5.0°,8.6° and 8.1° for θx, θy and θz, respectively. Slice acquisitioninterleaving was incorporated when generating both timeseries datasets.

To forward distort the T2 time series images from bothdatasets, simulated blipped EPI Cartesian k-space data of thedistorted images were generated in conjunction with therespective field-map time series with motion using Eq. (1).The distorted images (Fig. 3B) were then reconstructed fromthis k-space data using a system model with a field map set tozero [16]. The simulated readout time was 43.8 ms andthe pixel bandwidth in the phase encoding direction was22.8 Hz. In subsequent sections, Datasets A and B will referto the final geometrically distorted time series with therespective applied motion.

3.3. Activation detection with random permutation test

After repositioning all the EPI time series slices intovolumes, MSV yields time series volumes that may haveempty voxels. This results in variable sample sizes fordifferent voxels for statistical analysis. The nonparametricstatistical method of voxel-wise random permutation, usingthe averaged difference between activation and rest imagesas the test statistic, was used for significance testing ofdifferences in voxel intensities in the simulated datasets[12,21]. This statistical technique is simple, robust andindependent of sample size variability [22]. Random draws

Fig. 3. (A) T2 ICBM slice before simulated geometric distortion. (B) T2

ICBM slice after simulated geometric distortion with a peak fieldinhomogeneity of 5 ppm at 1.5 T.

of 2000 permutations of activated and rest periods were usedto form a permutation distribution for each voxel from whichactivated regions are identified by testing the null hypothesisof no activation at a fixed threshold of P=.001. To obtainROC curves, we vary the threshold α values from 10−4 to 1.0to obtain a set of activation maps and, together with theground truth activation map, compute the true positive andfalse alarm rates. The area under each ROC curve (AUC) isused to measure how accurately the activation regions havebeen detected.

4. Results

In our experiments, the concurrent correction schemewas evaluated on both simulated EPI time series datasets.Both datasets have known ground truths for the appliedslice-wise motion parameters, enabling quantitative eva-luation of MSV performance. In addition, the nondistortedtime series intensity images with applied motion serve asimage intensity ground truths to evaluate the performanceof the field-corrected image reconstruction process.

We investigated three performances indices. The firsttwo indices, MSV RMSE (Table 1) and image normalizedRMSE (Fig. 4), evaluate MSV motion correction and field-corrected reconstruction performance, respectively. A finalperformance index, area under the activation detection ROCcurve (Table 3 and Fig. 5), evaluates activation detectionaccuracy. In Table 1, the MSV RMSE for correctioncycle κ for each motion parameter, e.g., translation in y, iscomputed with

RMSE ty; j� � ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1L

XL�1

l¼0

t l;jy � tly;true� 2

;

vuut ð10Þ

where t l;jy and ty,truel are the estimated and true translation

parameters, respectively, for slice l. In Table 2, usingtranslation in y as an example again, the standard deviationof the error el;jty ¼ t l;jy � tly;true

� across all slices in the time

series is computed with

rerror ty; j� � ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

L� 1ð ÞXL�1

l¼0

el;jty � Pe jty

� 2;

vuut ð11Þ

where Pe jty

is the mean registration error of ty acrossall slices.

Tables 1 and 2 list the MSV RMSE and error standarddeviation values of the estimated MSV motion parameters.With median filtering of the estimated MSV parameters, theerrors decrease as κ increases and empirically converge torelatively small values. This implies that registrationaccuracy has improved with the concurrent correctionmethod compared to a single-cycle correction methodwhere a static field map is used for geometric distortioncorrection prior to MSV (cycle 0). The observed empirical

Table 1RMS error (Eq. ( 10)) of raw MSV estimates for simulated EPI Datasets A and B

Correctioncycle

RMSE (mm and °)

Dataset A (motion applied: tx, ty, tz and θz) Dataset B (motion applied: θx, θy, θz)

tx ty tz θx θy θz tx ty tz θx θy θz

Distorted 1.05 8.10 0.96 1.63 1.35 0.92 1.17 7.59 2.52 2.92 1.58 1.04Cycle 0 (κ=0) 0.57 3.14 0.77 1.69 0.88 0.44 0.42 3.15 0.42 2.91 2.60 0.44Cycle 1 (κ=1) 0.42 0.54 0.29 0.67 0.30 0.31 0.41 0.43 0.24 0.23 0.24 0.13Cycle 2 (κ=2) 0.41 0.43 0.29 0.27 0.16 0.09 0.40 0.41 0.23 0.21 0.22 0.10Cycle 3 (κ=3) 0.44 0.47 0.34 0.27 0.19 0.13 0.40 0.42 0.23 0.22 0.22 0.11


convergence strongly suggests that the algorithm is relativelystable as κ increases.

In fMRI, the accuracy of the reconstructed images'intensity values is of key importance since brain activationmaps are computed from the change in image intensityvalues in the time series. To measure image quality, thenormalized RMSE values between corrected EPI slices andtheir corresponding T2-weighted ground truth images werecomputed and averaged over the 120 volumes. Fig. 4 showsthat the average normalized RMSE of each slice in the 14-slice T2-weighted volume decreases as κ increases. Empiri-cal truth images were computed by correcting the simulatedEPI time series images with the exact same dynamic fieldmap that was used to distort them. The empirical truthimages contain errors inherent to the image reconstructionand MSV processes, and represent the best images that canbe obtained using these correction methods if the simulated

Fig. 4. Normalized RMSE values for each EPI slice in the same position in the head averaged over 120 volumes for various correction cycles for (A) Dataset Awith applied tx, ty, tz and θz motion, and (B) Dataset B with applied θx, θy, θz motion.

dynamic field map were known exactly. These empiricalground truth images are not the known ground truth intensityimages that subsequently corrected datasets are compared toin computing the image intensity normalized RMSE values.It is observed that when κ≥2, the corresponding normalizedRMSE values converge to the normalized RMSE values ofthe empirical truth images which implies that the CFMMSVmethod yielded updated field maps that are very close to theground truth dynamic field maps. Fig. 6 shows reconstructedEPI slices from the same position in the head and theircorresponding absolute error images when compared tocorresponding T2-weighted ground truth images as κincreases. The field-corrected image errors are greatlyreduced when κ≥3.

As κ increases, the field-corrected EPI images becomemore similar to the ground truth T2 images. However, it ispossible to reduce image normalized RMSE without

Table 2Standard deviation of the error (Eq. ( 11)) of raw MSV estimates for simulated EPI Datasets A and B

Correctioncycle

Standard deviation of error of median filtered MSV (mm and °)

Dataset A (motion applied: tx, ty, tz and θz) Dataset B (motion applied: θx, θy, θz)

tx ty tz θx θy θz tx ty tz θx θy θz

Distorted 1.61 15.66 3.45 11.80 11.80 3.26 2.53 20.46 20.53 27.03 11.52 2.21Cycle 0 (κ=0) 1.77 4.13 3.87 15.31 10.28 0.99 0.25 2.70 2.06 1.71 0.42 0.08Cycle 1 (κ=1) 0.22 2.37 0.24 6.75 1.94 3.10 0.13 0.17 0.10 0.09 0.12 0.03Cycle 2 (κ=2) 0.08 0.50 0.41 1.92 0.09 0.02 0.11 0.15 0.09 0.08 0.08 0.02Cycle 3 (κ=3) 0.34 0.50 0.69 1.47 0.40 0.15 0.11 0.15 0.10 0.09 0.08 0.02


improving activation detection because the simulatedintensity increase is only 5% and applied to a relativelysmall subset of activated voxels. Thus, we applied thestatistical random permutation test on Datasets A and B at allstages of the CFMMSV correction process and computedROC curves and respective AUC values to verify thatactivation detection performance improves as κ increases.Fig. 5A–B shows the ROC curves for both datasets at severalstages of the CFMMSV correction process and Table 3shows the corresponding AUC values. For both datasets, theAUC values increase significantly between cycle 0, whichare the results one would obtain if the same static field mapwas used to correct all the time series data, and cycle 3,which are the final results after applying the CFMMSVmethod. The improvement in activation detection is evenmore significant upon considering that fMRI statistics areoften computed directly from EPI datasets withoutgeometric distortion correction, i.e., compare the AUC

Fig. 5. ROC curves showing activation detection performance for (A)Dataset A and (B) Dataset B at several stages in the CFMMSVcorrection process.

Table 3Area under ROC curve (AUC) values for activation detection of Datasets Aand B at various stages in CFMMSV correction

Correctioncycle

AUC

Dataset A Dataset B

Distorted 0.8880 0.9043Cycle 0 (κ=0) 0.8664 0.9295Cycle 1 (κ=1) 0.8990 0.9382Cycle 2 (κ=2) 0.9053 0.9519Cycle 3 (κ=3) 0.9208 0.9521Ground truth T2 0.9209 0.9659

values between the distorted dataset and cycle 3. Fig. 7shows the activation maps obtained (P=.001) for two slicesfrom Datasets A and B at several stages in the CFMMSVcorrection process. Comparing the activation maps forcycle 0 (second row) and cycle 3 datasets (third row), wesee that the latter has more true positives and fewer falsepositives. The ground truth activation regions for the twoslices are shown in Fig. 7E and J. Activation maps obtainedby applying the random permutation test on the groundtruth time series images for both datasets are also shown(fourth row) to illustrate the best performance obtainablewith the statistical analysis method used.

5. Discussion

The proposed CMFMSV framework is a retrospectivecorrection framework that incorporates slice-to-volumeregistration, field-map updating and geometric distortioncorrection for the purpose of improving activation detectionperformance in fMRI. In this work, we performed iterativefield-corrected reconstruction to remove geometric distortionand employed a field-map updating scheme that applies rigidbody MSV motion parameters to a high-resolution staticfield map of the object. Our results on two simulated EPItime series datasets show that as the number of correctioncycles increases, the field-corrected image quality and theaccuracy of recovered MSV motion parameters improve andempirically converge to the ground truths. This is evident inthe decreasing MSV RMSE, MSV error standard deviationand image normalized RMSE values as κ increases (Tables 1

Fig. 6. (A–E, K–O) Intensity and (F–J, P–T) absolute difference images with respect to ground truth images for two sample slices from Dataset A at variousstages in the CFMMSV correction process. (Top row) Geometrically distorted dataset, (second row) cycle 0, (third row) cycle 1, (fourth row) cycle 2, (fifth row)cycle 3. All images are displayed on the same normalized intensity scale ranging from 0 to 1.


and 2 and Fig. 4). Improved activation detection per-formance, as inferred from the increasing AUC values inTable 3, is also observed as κ increases. These results de-monstrate that, under the simulated conditions, the CFMMSVmethod can provide effective correction of motion artifactsthat are complicated by the field effects induced by rigid headmotion. It is interesting to note that for Dataset A, the AUCvalue for cycle 0 actually decreased when compared to theAUC value for the distorted dataset. The AUC valuesincreased subsequently in cycles 1 through 3. This illustratesthat, in the presence of significant motion, using a static fieldmap to correct an entire time series can lead to degradedactivation detection performance, even after applying motion

correction to the time series data. This is because all motion-induced field-map changes were ignored in cycle 0.

Some outlier MSV motion estimates in the time seriesmay result from local minimum solutions obtained by theNelder–Mead implementation of the MI-based registrationalgorithm. This occurs infrequently and is likely due to theinherent reduced information content of slice-to-volumeregistration. Using a median filter on the motion estimatesfor the purpose of updating the field map is akin to throwingout wrong estimates and “interpolating” a value to replacethem, given the prior assumption that head motion iscorrelated in time. Since the goal in fMRI is activationdetection accuracy and not necessarily perfect motion


correction accuracy, it is conceivable that an alternative, andperhaps conventional, approach to handle outlier estimates inslice-wise registration, e.g., topmost slices of the brain whereinformation content is greatly reduced, is to just excludethose slices from statistical analysis for activation detection.That is a common statistical data analysis approach whenoutliers are present (data cleaning). However, this wouldmean fewer temporal samples available for statisticalanalysis for each voxel. Temporal filtering (median filter inour case) attempts to retain these outlier slices for statisticalanalysis for activation detection. In all likelihood, a betterapproach to remove such outliers exists and is a subject forfuture work.

In the first concurrent correction cycle, i.e., κ=0, theestimated translations in y were not used to update the fieldmap since ty includes geometric distortion effects in additionto spatial motion in y by the object. It might be argued thatrotations in z should also be ignored since they give rise toin-plane translation components. However, θz does not needto be omitted in the field map update process when κ=0because after the registration is done in MSV, rotations arecomputed (relatively accurately) from the final control pointsbefore the translations are estimated. If estimates of θz areomitted in the zeroth iteration, the algorithm is unnecessarilydeprived of a piece of relatively accurate information forimproving the next field map estimate.

Dataset A was simulated with relatively realistic assump-tions of how the susceptibility-induced field-map changeswith translations and in-plane rotation. Thus, the resultsobtained from that dataset are a reasonable indication of theperformance of the concurrent correction method on well-shimmed real MR data with such motion. For real MR data,the field map may include other contributions like post-shimsystem-induced field inhomogeneity that remain stationarywith respect to the head. Since the focus of the simulationstudy is on susceptibility-induced artifacts, we have assumedthat these additional field-map contributions are lessdominant and can be characterized separately, if necessary.This is a reasonable assumption for data collected from awell-shimmed magnet. Dataset B was simulated with similarassumptions as Dataset A except that local field-mapchanges that may arise due to out-of-plane rotations are notmodeled exactly. Thus, the results from Dataset B are lessindicative of the CFMMSV method's performance on realMR data in the presence of larger out-of-plane rotation.However, for small out-of-plane motion, the local field-mapchanges may be small enough [6] to justify the assumptionthat the dynamic field map can be approximated with a rigid-body transformation of the static field map.

For time series datasets that do not have ground truths, anappropriate stopping criterion is necessary to terminate theCFMMSV correction process automatically. In Table 1 andFig. 4, respectively, the MSV RMSE and image normalizedRMSE values remain relatively constant for κ≥2 for bothDatasets A and B which suggests that the procedure could beterminated earlier, thus reducing computation time. How-

ever, the increasing AUC values in Table 3 indicate thatactivation detection performance continues to improve forκ≥2. Thus, there is a computation time vs. activationdetection performance tradeoff that may influence the choicefor the stopping criterion. To obtain minimal computationtime at the expense of activation detection performance, theCFMMSV process can be automatically terminated whenthe time series' average normalized RMSE difference for theprevious two cycles is below a threshold value. For maximalactivation detection performance, the number of nonoverlap-ping activated voxels for the previous two cycles can be usedas a dissimilarity measure to automatically stop theCFMMSV process.

The iterative field-corrected reconstruction techniquecorrects for geometric distortion in both the phase encodingand read-out directions if the field map is accurate. Initially,the field map is not accurate as it does not track head motion.In the field-map update process in cycle 0 (κ=0), we use theprior knowledge that the phase encoding distortion is largerthan the read-out distortion to avoid using the phaseencoding direction MSV motion parameter (ty) to updatethe field map. Subsequently, as the updated field mapsbecome more aligned to the true field map (i.e., moreaccurate field map is obtained), the iterative field-correctedreconstruction algorithm corrects for both phase encodingand read-out geometric distortions more accurately.

The CFMMSV framework was formulated to jointlycorrect for motion and geometric distortions arising fromsusceptibility-induced field inhomogeneity without theexplicit acquisition of dynamic field maps. The frameworkcan be further improved by using field-map updatetechniques that account for susceptibility-induced field-map changes with out-of-plane rotation. For example, aconceivable modification would be to apply MSV motionparameters to a high-resolution 3D volume of a brain that hasbeen segmented into air, bone and soft tissue regions. Thesusceptibility-induced component of the dynamic field mapcan then be reestimated from the transformed 3D structuralvolume using numerical techniques to approximate solutionsto the magnetostatic scalar potential based on Maxwell'sequations [23,24]. This will increase the computational costbut may yield improved performance. Future work includesa susceptibility phantom study of the robustness ofCFMMSV with out-of-plane rotations.

We have focused this study as a validationwork using truthsavailable in the simulation data. While it is the ultimate goal ofthe continued investigation, studies dealing with human fMRIdata would pose additional problems that may complicate thevalidation process. The lack of known truths would be theprimary cause of difficulty in the evaluation. Other intricaciesinvolving human data analyses would be brief real motion,such as jerking or swallowing and the extent ofmotion that canbe corrected. In real subject motion, transientmotion spikes areoften present due to brief motions followed by return tobaseline (e.g., swallowing). Modeling of such spikes was notincluded in this study. The extent of motion that can be

Fig. 7. Activation detection maps (P=.001) overlaid on anatomical data of two sample slices from (A–D, F–I) Dataset A and (K–R) Dataset B at several stages inthe CFMMSV correction process. Each row of activation maps corresponds to a specific correction stage consisting of (top row) geometrically distorted, (secondrow) cycle 0, (third row) cycle 3 and (fourth row) ground truth time series images. The simulated activation maps applied to the two slices are shown in (E) and (J).


corrected by CFMMSV can be explored with real MR datausing a susceptibility phantom.

6. Conclusions

We have described a different approach, CFMMSV, toperform geometric distortion correction by using concurrentfield maps computed based on the previously developed

MSV motion correction for fMRI time series data. Thetechnique has been evaluated on two 120-volume simulatedtime series with different applied motion. Under thesimulated conditions, the CFMMSV method improved theaccuracy in recovering both the MSV motion parameters andthe field-corrected reconstructed images compared to thesimpler method of performing motion and geometricdistortion correction independently. Activation detectionperformance, quantified using the AUC values at various


stages in the CFMMSV correction process, also improvedwith the CFMMSV method. Although the CFMMSVmethod may currently be limited to small out-of-planerotations, it presents a correction framework that addressesgeometric distortion and head motion in each slice in a timeseries as a joint problem and shows the potential to performmore accurate motion related artifact correction.

Acknowledgments

This work was supported in part by the National Instituteof Health grant 1P01 CA87634 & R01 EB00309.

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