PROMO: Real-Time Prospective Motion Correction inMRI Using Image-Based Tracking

Nathan White,1 Cooper Roddey,2 Ajit Shankaranarayanan,3 Eric Han,3 Dan Rettmann,3

Juan Santos,4 Josh Kuperman,5 and Anders Dale2,5*

Artifacts caused by patient motion during scanning remain aserious problem in most MRI applications. The prospectivemotion correction technique attempts to address this problemat its source by keeping the measurement coordinate systemfixed with respect to the patient throughout the entire scanprocess. In this study, a new image-based approach for pro-spective motion correction is described, which utilizes threeorthogonal two-dimensional spiral navigator acquisitions,along with a flexible image-based tracking method based onthe extended Kalman filter algorithm for online motion mea-surement. The spiral navigator/extended Kalman filter frame-work offers the advantages of image-domain tracking withinpatient-specific regions-of-interest and reduced sensitivity tooff-resonance-induced corruption of rigid-body motion esti-mates. The performance of the method was tested using off-line computer simulations and online in vivo head motionexperiments. In vivo validation results covering a broad rangeof staged head motions indicate a steady-state error of lessthan 10% of the motion magnitude, even for large compoundmotions that included rotations over 15 deg. A preliminary invivo application in three-dimensional inversion recoveryspoiled gradient echo (IR-SPGR) and three-dimensional fastspin echo (FSE) sequences demonstrates the effectiveness ofthe spiral navigator/extended Kalman filter framework for cor-recting three-dimensional rigid-body head motion artifactsprospectively in high-resolution three-dimensional MRI scans.Magn Reson Med 63:91–105, 2010. VC 2009 Wiley-Liss, Inc.

Key words: motion artifact; motion correction; Kalman filter;real-time; rigid-body; navigator

Artifacts caused by patient motion during scanningremain a serious problem in most clinical and researchMRI applications. In fast single-shot sequences, such asdynamic two-dimensional (2D) echo-planar imaging (EPI),between-scan motion can introduce significant motion-related variance to the voxel-time courses and disrupt thespin excitation history of the acquisition (1,2). In multi-

shot 2D and three-dimensional (3D) sequences, within-

scan patient motion results in k-space data inconsisten-

cies, causing artifacts such as ghosting, blurring, and

ringing in the images themselves. Offline image registra-

tion can mitigate most between-scan motion artifacts in

time-series data (3–5) but cannot correct for changes in

the spin excitation history caused by through-plane

motion. In addition, while some within-scan motion

artifacts can be corrected retrospectively using knowl-

edge of the motion history derived from either navigator

scans (6,7) or overlapping k-space data (8,9), most of

these methods are limited by the inability to (1) fully

correct for through-plane motion in 2D sequences and

(2) avoid k-space data inconsistencies caused by inter-

polation errors.

An alternative approach to motion correction, which

shares none of these drawbacks, is modify the pulse-

sequence online, in real-time, during the acquisition

itself. Some of the first real-time prospective motion cor-

rection methods used straight-line navigators to correct

for linear translations of organs in the chest (10–12).

Since then, navigators with more sophisticated k-space

trajectories have been developed for higher-order motion

correction. Ward et al. (13) used orbital (circular) naviga-

tors acquired in three orthogonal 2D planes for prospec-

tive correction of full 3D rigid-body motion artifacts in

2D EPI. To account for motion outside the plane of each

navigator (through-plane motion effects), multiple orbital

navigators are acquired in series, where each orbital nav-

igator provides an adjustment to the subsequent orbital

navigator planes. Spherical navigators, which generalize

the orbital navigator trajectory to 3D shells in k-space, do

not suffer from through-plane motion effects and can be

used to estimate 3D rigid-body motion in a single naviga-

tor (14–16). However, the spherical technique works best

for rotations along the navigator trajectory as apposed to

rotations across them (14). In addition, a real-time appli-

cation of the spherical navigator remains missing due in

part to the slow iterative procedure required to register

spherical navigator magnitude data (14). One of the most

recent real-time navigator techniques, introduced by van

der Kouwe et al. (17), used an optimized ‘‘cloverleaf’’ k-

space trajectory for prospective correction of 3D rigid-

body motion in 3D spoiled gradient echo sequences.

Compared to the orbital and spherical trajectories, the

cloverleaf method offers the advantages of a more rapid

k-space readout, together with improved translation esti-

mation by including linear segments through the center

of k-space (17).In some applications, prospective motion correction can

be achieved through self-navigation. Thesen et al. (18)

1Department of Cognitive Science, University of California, San Diego, LaJolla, California, USA2Department of Neurosciences, University of California, San Diego, La Jolla,California, USA3Global Applied Science Lab, GE Healthcare, Waukesha, Wisconsin, USA4Department of Electrical Engineering, Stanford University, Palo Alto,California, USA5Department of Radiology, University of California, San Diego, La Jolla,California, USA

Grant sponsor: National Institutes of Health; Grant numbers:1R01AG031224, 1U01AG024904-02, 5 U24 RR021382-04, U54 NS056883-01

*Correspondence to: Anders Dale, Ph.D., #8950 Villa La Jolla Drive, SuiteC101, La Jolla, CA 92037. E-mail: amdale@ucsd.edu

Received 19 March 2009; revised 1 July 2009; accepted 23 July 2009.

DOI 10.1002/mrm.22176Published online in Wiley InterScience (www.interscience.wiley.com).

Magnetic Resonance in Medicine 63:91–105 (2010)

VC 2009 Wiley-Liss, Inc. 91

described a self-navigated approach for 2D EPI called Pro-spective Acquisition CorrEction, or PACE, where each indi-vidual time-series volume is reconstructed online and regis-tered to the first volume in the sequence. While the image-base tracking (registration) procedure in PACE providesaccurate 3D rigid-bodymotion estimates, the correction itselfis performed on a slow volume-by-volume basis, or approxi-mately every 4 sec (18). Zaitsev et al. (19) and Speck et al.(20) developed an optical tracking device for improved real-time slice-by-slice correction in 2D EPI. In general, theadvantage of optical tracking is that it does not require modi-fication of the pulse sequence. However, this comes at aprice of requiring significant additional hardware to imple-ment, including an MRI-compatible camera system mountedinside the scanner room and repeated calibration for optimaltracking performance (21).

In this study, we present an alternative image-basedtechnique for prospective motion correction called‘‘PROMO’’ (standing simply for PROspective MOtion cor-rection). The PROMO approach utilizes three orthogonal2D spiral navigator acquisitions, along with a flexibleimage-based tracking method based on the extended Kal-man filter (EKF) algorithm (22,23) for real-time motionmeasurement. The Kalman filter framework for image-based tracking in MRI was introduced by White et al. (24)and applied in 3D spiral-navigated image sequences byShankaranarayanan et al. (25,26) and Roddey et al. (27),and in MR spectroscopy by Keating et al. (28). In addi-tion, Sprincemaille et al. (29) recently introduced a novelapplication of the Kalman filter for real-time separation ofcardiac and respiratory components in navigator data.Here, we describe the PROMO motion correction methodin detail and provide some preliminary results in 3Dinversion recovery spoiled gradient echo (3D IR-SPGR)and 3D fast spin echo (3D FSE) sequences.

MATERIALS AND METHODS

Spiral Navigator Acquisition

The navigator acquisition included three orthogonal low-flip-angle, thick-slice, single-shot spiral acquisitions (SP-

Navs), with the following pulse-sequence parameters:echo time/pulse repetition time ¼ 3.4/14 ms, flip (d) ¼8� (to minimize the impact of signal saturation on theacquired 3D volume), BW ¼ �125 kHz, field of view ¼32 cm, effective in-plane resolution ¼ 10 � 10 mm,reconstruction matrix ¼ 128 � 128, slice thickness ¼10 mm. Spiral readouts were selected because they pro-vide an efficient k-space coverage and allow for image-based tracking with reduced sensitivity to distortion.The off-resonance effect is somewhat mitigated by theoptimization of spiral length since the requirement forthe in-plane resolution of the navigators is not very high.Initial experiments were done without any trajectorymeasurement. However, it should be noted that in caseof severe off-resonance (>�100 Hz), the measurementerrors will increase significantly. To avoid this, we per-form center frequency correction (prescan software onthe scanner) before every scan. The orientation conven-tion adopted in this work defined the x-axis as the left-right direction, the y-axis as the anterior-posterior direc-tion, and the z-axis as the inferior-superior directionwith respect to the patient. The axial, sagittal, and coro-nal planes of each SP-Nav spanned the x-y, y-z, and thex-z axes, respectively. The SP-Nav pulse-sequence andexample reconstructed image is shown in Fig. 1.

Integration in 3D IR-SPGR and 3D FSE

SP-Navs were integrated into a 3D IR-SPGR and 3D FSEsequence (Fig. 2). To enable prospective correction with-out impacting scan time, multiple SP-Navs were playedout during the intrinsic longitudinal (T1) recovery timeof the two sequences. Immediately after acquisition, eachSP-Nav was reconstructed and used as input to the EKFmotion tracking and correction algorithm (describednext). The time required to collect a single SP-Nav wasapproximately 42 ms, followed by 6 ms for image recon-struction of all three planes. To allow ample time forestimation and feedback, the repetition time for each SP-Nav was programmed for 100 ms. A total of J ¼ 5 SP-Navs (�500 ms) were acquired during the T1 recoverytimes of both sequences, which were programmed for

FIG. 1. SP-Nav pulse-sequence and reconstructed image used for motion tracking.

92 White et al.

approximately 700 ms and 1200 ms, respectively. For the3D IR-SPGR sequence, a recessed elliptical centric ky-kzordering was used for the imaging segment (30), with anentire ky-kz plane acquired after each inversion pulse.The thickness of the IR pulse was twice that of the imag-ing slab. The 3D IR-SPGR pulse-sequence parameters wereecho time/pulse repetition time ¼ 3.9/8.7 ms, inversiontime ¼ 270 ms, flip (a) ¼ 8�, BW ¼ �15.63 kHz, field ofview ¼ 24 cm, voxel size ¼ 1.25 � 1.25 � 1.2 mm, num-ber of k-space segments ¼ 192. For the 3D FSE sequence,an extended echo train acquisition (32) was used, with thefollowing parameters: echo time/pulse repetition time ¼110/2500 ms, flip ¼ 90�, BW ¼ �31.25 kHz, field of view¼ 24 cm, voxel size ¼ 1.25 � 1.25 � 1.2 mm, echo trainlength ¼ 90, number of k-space segments ¼ 90. The over-all scan times of the 3D IR-SPGR and 3D FSE sequenceswere 8 min and 4.5 min, respectively. All scans were col-lected using a 1.5T GE Signa HDx system (Waukesha, WI),using an eight-channel head coil.

EKF Motion Tracking and Correction

Dynamic State Space Model

Real-time 3D rigid-body motion tracking and correctionon the SP-Nav data were performed using the well-known EKF algorithm (22), which provides recursive

state estimates in nonlinear dynamic systems perturbedby Gaussian noise. The basic dynamic state-space modelimplemented here is shown in Fig. 3 and can be writtenas a set of system equations of the form

xk ¼ Axk�1 þ w; P wð Þ;Nð0;QÞ [1]

yk ¼ hðxkÞ þ v; P vð Þ;Nð0;RÞ: [2]

xk is the (unobserved) state of the dynamic system at time-step k and yk are the (observed) measurements. Alsoshown in Fig. 3 are the a priori x̂kjk�1 and a posteriori x̂kjkstate estimates, whose calculation are described later.

For image-based tracking of rigid-body motion, xk ¼[tx,ty,tz,yx,yy,yz]

T is a six-dimensional vector of motion pa-rameters, where the first three and last three elements aretranslations and rotations in x-y-z space, respectively, and ykis an Nv-dimensional vector of voxel intensities of the k-thSP-Nav, where Nv is the total number of voxels in all threenavigator planes. The time evolution of the state vector isgiven by the 6� 6 state transition matrixA and the Gaussianprocess noise by w, with mean zero and covariance Q.Because of the unpredictable nature of patient head motionsover the sampling interval of the SP-Navs, the dynamicswere modeled using a simple random walk, with A as theidentity matrix. hð�Þ;R6�!RN is a nonlinear vector-valuedfunction of the current state called the measurement

FIG. 2. Spiral-navigated (a) 3D IR-SPGR and (b) 3D FSE pulse-sequence. A train of J¼ 5 SP-Navs is played out prior to each of theM inversionsor excitation pulses of the two sequences, respectively. EKFmotion correction is performed after each individual SP-Nav using the EKF algorithm.

PROMO Real-Time Motion Correction 93

model, which describes how the states are observedthrough the noisy measurements. For image-based tracking,h(�) describes an interpolation into a fixed reference naviga-tor yref (collected at the beginning of the scan) at the loca-tions specified by xk. The mathematical form of h(�) andthe details of the interpolation are provided in the Appen-dix. To improve the robustness of the tracking to potentialvariations in the mean signal level of each SP-Nav, themeasurements yk were forced to have unit norm at eachtime-step. The error in the measurement process is givenby the Gaussian measurement noise v, with mean zero andcovariance R. In the dynamic state-space model above, thenoise processes are time invariant and statistically inde-pendent (i.e., E(wvT) ¼ 0).

From a Bayesian point of view, the a posteriori filter-ing density P(xk|y0:k) of the state given the complete his-tory of observations y0:k ¼ {y0,y1,…,yk} constitutes thecomplete solution to the motion tracking problem.The EKF algorithm provides recursive estimates ofthe conditional mean of the filtering densityx̂kjk ¼ E xk jy0:k

� � ¼ RxkP xk jy0:k

� �dxk using a two-stage

recursion of prediction and correction at each time-step.Thus, x̂kjk is the corrected a posteriori state estimate attime-step k, given all the measurements up to andincluding yk. For prospective motion correction, an addi-tional scanner update step is included in the recursion.

Prediction

In the prediction step, the state transition matrix A andassumed process noise covariance Q are used to predictthe state and error covariance at each time-step. With Aas the identity matrix, the predicted state and error co-variance become

x̂kjk�1 ¼ x̂k�1jk�1 [3]

P̂kjk�1 ¼ P̂k�1jk�1 þQ; [4]

where x̂kjk�1 is the predicted or a priori state estimate attime-step k, given the a posteriori state estimate x̂k�1jk�1

at the previous time-step k-1 (see Fig. 3).P̂kjk�1 ¼ E xk � x̂kjk�1

� �xk � x̂kjk�1

� �Th iis the a priori state

estimation error covariance at time-step k, given the a pos-teriori state estimation error covariance P̂k�1jk�1 at k � 1.

Update

In PROMO, the predicted state x̂kjk�1 of the EKF is usedto update the scanner pulse-sequence prior to collectingthe measurements yk. This includes updating the slice,

phase, and frequency encoding of the imaging volume,as well as the navigators, given the 3D translations androtations in x̂kjk�1. A real-time communication system, asdescribed in Ref. (32) and implemented in Ref. (33), wasused for efficient and near-real-time communication ofSP-Nav data, EKF state estimates, and associated meta-data between the EKF module and the scanner pulse-sequence/receive chain.

Correction

Immediately following the scanner update, the measure-ments yk are acquired and used to correct the predicted statex̂kjk�1 and error covariance P̂kjk�1. The basic idea of EKFcorrection is to approximate the nonlinear function h(�)using a Taylor series expansion around x̂kjk�1 and applythe standard linear Kalman filter equations to the new line-arized model. In this study, the first-order iterated EKF cor-rection was implemented (22), which truncates the Taylorseries to first order and iterates the correction iterates thecorrection around the updated state. The first-order iteratedEKF correction is analogous to performing an onlineGauss-Newton optimization of the following nonlinear costfunction (34)

Ci xkð Þ ¼ yk � h xkð Þ� �TR�1 yk � h xkð Þ� �

þ xk � x̂ikjk�1

� �TP̂�1

kjk�1 xk � x̂ikjk�1

� �½5�

for i ¼ 0, 1, … , N iterations. When N is set to zero, thefirst-order iterated correction reduces to the standard EKFcorrection. Details of how the cost function Ci(xk) is mini-mized given the online updates to the scanner pulse-sequence are provided in the Appendix.

The first term on the left-hand side of Eq. 5 is the

‘‘data prediction error’’, while the second term on the

right-hand side of Eq. 5 is the ‘‘model estimation error’’.

Note that most offline intensity-based image registration

algorithms are based on iteratively minimizing the ‘‘data

prediction error’’, or some variant of this function. The

EKF can therefore be viewed as a generalization of tradi-

tional intensity-based image registration that incorpo-

rates an additional model for the dynamics and is

uniquely suited for online applications because (1) it is

causal in nature and (2) it does not require iterating the

cost function (N can be set to zero), which is important

in time-sensitive applications. Thus, in the limit of infin-

itely large process noise covariance Q, the ‘‘model esti-

mation error’’ is effectively ignored when minimizing the

cost, and the EKF is analogous to traditional image-based

least-square registration with N iterations. The weighting

of the ‘‘data prediction error’’ and ‘‘model estimation

error’’ to the overall cost depends on both error terms Qand R. Selection of these priors is an important step in

optimal EKF performance and is described next.

Filter Tuning

Because the EKF is unchanged by scaling Q and R bythe same factor, it is often convenient to define a singlescalar that controls the relative weighting of the ‘‘data

FIG. 3. Dynamic state space model of the EKF. The a priori stateis used to update the pulse sequence at each time-step.

94 White et al.

prediction’’ and ‘‘model estimation errors’’. To do this,we factor Q and R into

Q ¼ r2w �Q [6]

R ¼ r2v � R; [7]

where r2w and r2v are scalars, and Q and R are matrices ofunit norm containing the structure of the respective noiseprocesses. The ratio S ¼ r2v /r

2w then controls the effective

‘‘bandwidth’’ of the filter, with increased tracking abilityon the one hand (high weighting of data information,with low values of S) and increased noise suppression onthe other (high weighting of model information, with highvalues of S). Throughout the remainder of this work, werefer to S as the ‘‘smoothness factor.’’

The noise covariance structure, or entries of Q and R,can be selected to utilize prior knowledge about theobject’s dynamics and measurement errors, respectively.Given the physical constraints of the patient inside theMRI environment, some of the off-diagonal entries of Qwill likely be nonzero. However, for the sake of simplic-ity, we assume each motion parameter is independent;thus, Q is diagonal and can be written

Q ¼ diagð½12 12 12 ðp=180Þ2 ðp=180Þ2 ðp=180Þ2 �TÞ;[8]

where the first three and last three elements have unitsmm2/TN2 and rad2/TN2, respectively, with TN being thetime between the onset of two consecutive SP-Navs. Inaddition, we assume the measurement errors are white;thus, R is diagonal, and the inverse can be written

R�1 ¼

d1 0 0 00 d2 0 0

0 0 . ..

00 0 0 dNv

26664

37775: [9]

The scalars along the diagonal of R�1

0 � fdigNv

i¼1 � 1 pro-vide a means of weighting the contributions of eachvoxel to the motion estimate, and thus setting the appro-priate weights di to zero provides a principled approachfor motion tracking within a priori regions-of-interest(ROIs). This procedure is described next.

ROI Tracking

The advantage of image-based ROI tracking is that theestimation can focus on a particular region in objectspace while ignoring other regions that may corrupt therigid-body motion estimates. In this study, a brain ROI isadapted to each patient to remove the effects of non-rigidmotion of the neck and jaw. The general procedure isshown schematically in Fig. 4 and can be separated intotwo principal phases. During the first phase (phase I), atrain of 20 SP-Navs is played out in rapid succession im-mediately prior to the start of the ‘‘dummy’’ scans. Dur-ing collection of this SP-Nav train, the EKF is used toregister a fixed 3D proton density weighted head atlas tothe patient. Once the registration is complete, the entriesof R

�1(data weights) are set to the voxel values in a 3D

binary brain mask, defined in atlas space. The construc-tion of both the 3D head atlas and associated brain maskwas performed offline and is described in detail in Rod-dey et al. (27). Immediately after the brain ROI is speci-fied, a series of four ‘‘dummy’’ scans are acquired toallow equilibration of the longitudinal magnetization.Once the ‘‘dummy’’ scans are complete, the final EKF pa-rameters, such as the Jacobian H (see the Appendix for adescription of the Jacobian and how it is calculated), ref-erence navigator yref (which is set to the last SP-Navprior to scan start), and covariance matrices Q and R, arecalculated and stored in memory. The entire duration ofphase I is approximately 10 sec. An example referencenavigator, registered head atlas, and brain ROI are shownin Fig. 5.

In the second phase (phase II), the ‘‘tuned’’ EKF is runonline during the acquisition of the image data. Becausethe SP-Navs are only acquired during the T1 recovery pe-riod of the pulse sequence, no correction is performedduring the imaging segment itself, which typically lastsabout 2 sec. To account for this, motion-corrupted k-space segments were reacquired at the end of the scan.To determine which segments needed to be reacquired, asimple ‘‘rescan metric’’ was defined

FIG. 4. Online EKF schematic for patient-specific ROI tracking.

xappk is the ‘‘applied’’ state used to collect the measure-ments yk and is described in the Appendix.

FIG. 5. Results from phase I of the EKF tracking procedure show-ing the SP-Nav reference image (bottom row), registered head

atlas (top row), and brain ROI (white lines). All voxels outside ofthe brain ROI in the SP-Nav are ignored during tracking.

PROMO Real-Time Motion Correction 95

qðnÞ ¼ x̂�ðnÞ � x̂þðnÞ�� ���� ��2; [10]

where x̂�ðnÞ and x̂þðnÞ denote the a posteriori motionestimates (in mm and deg) immediately prior to and im-mediately after the collection of the nth k-space segment,respectively. All k-space segments with q(n) greater thanor equal to a user-specified threshold T are then markedfor rescanning. In this study, the rescan threshold wasselected based on in vivo experiments performed duringno motion and is described in the section titled ‘‘In vivovalidation experiment.’’

Offline Simulations

Offline Monte Carlo simulations were used to test theperformance of the EKF for motion tracking in the SP-Nav sequence. For each Monte Carlo run, a time series of10 SP-Navs was synthesized by interpolating into a sin-gle isotropic 3D SP-Nav volume (collected with similaracquisition parameters to the online 2D SP-Navs). Tosimulate online motion correction, the orientation ofeach synthesized SP-Nav was determined by combiningboth the simulated motion with the predicted state ofthe EKF at each time point. Two different simulationexperiments were performed. The first tested simpleone-dimensional (1D) motions in each parameter sepa-rately and the second tested compound random ampli-tude motions in all parameters simultaneously. In allexperiments, step functions were used to simulatemotion (starting at the forth time point) so that the con-vergence behavior of the EKF could easily be visualizedusing different filter parameters. Both ‘‘large’’ and‘‘small’’ motions were tested using either a 10 mm/deg or1 mm/deg step (for the 1D simulation experiment) orconstraining the vector norm across all parameters (inmm and deg) to be either 10 or 1 (for the compoundmotion experiment), respectively. Prior to estimation,random Rician noise was added to each image, with vari-ous signal-to-noise ratios (SNRs). The measurement noiselevel r2v of the EKF was determined empirically for eachSNR level, using the mean (within brain) signal magni-tude estimated from the native 3D SP-Nav volume. Thus,when testing the effect of different ‘‘smoothness factors’’S, only the process noise level r2w of the EKF was varied.All simulations were performed using Matlab (Math-works, Inc., Natick, MA).

Online In Vivo Experiments

Three different types of in vivo experiments were per-formed to test the online performance of the SP-Nav/EKFtracking framework in the 3D IR-SPGR and 3D FSEsequences. Prior to scanning, all subjects providedinformed consent in accordance with the University ofCalifornia, San Diego institutional review board.

In Vivo Validation Experiment

The first in vivo experiment was designed to assess thesteady-state accuracy of the SP-Nav/EKF motion esti-mates. The basic procedure consisted of modifying the3D IR-SPGR sequence (Fig. 2a) to collect the same three

volumes back to back (V1, V2, and V3). During the scan,a highly motivated volunteer was instructed to eitherremain motionless throughout (‘‘no-motion’’ trial) ormake a single brief motion during the middle of the sec-ond volume (V2) and remain in that position until theend of V3 (‘‘motion’’ trial). No instruction was given asto what type of motion to perform, only to vary the mag-nitude and direction of motion from trial to trial. Duringeach trial, motion correction was performed with respectto the reference navigator of V1. After the scans werecomplete, three quantities were estimated for each trial:(1) the steady-state EKF estimate, calculated by averagingthe online motion estimates during V3; (2) the steady-state error, calculated offline by registering V1 to V3using an iterative 3D rigid-body registration algorithm;and (3) the ‘‘true’’ motion of the subject, calculated bycombining the steady-state EKF estimate with the steady-state error. Prior to computing these quantities, allmotion plots were inspected to verify the subject did notmove during the collection of V1 or V3. Each trial of thevalidation experiment required approximately 24 min ofscanning, and a total of 15 trials were performed (two‘‘no motion’’ trials and 13 ‘‘motion trials’’). No rescan-ning was performed during the validation experiment,but the two ‘‘no motion’’ trials were used to determinethe rescan threshold T for the remaining two in vivoexperiments. The mean q(n) during both ‘‘no motion’’ tri-als was 0.31, with a minimum of 0.07 and a maximum of0.68. Thus, to be conservative, the rescan threshold Twas set to 1 for all subsequent in vivo experiments.

Repeated Staged Motion Experiment

The second in vivo experiment was designed to assessthe performance of the SP-Nav/EKF framework for reduc-ing 3D rigid-body head motion artifacts in the MRI scansthemselves under relatively controlled conditions. Threetypes of scans were performed on a single subject: (1) ascan without motion correction, where the subject wasasked to perform repeated staged motions continuouslyduring the scan (PROMO OFF condition); (2) a scan withmotion correction, where the subject was asked to repeatthe same motions as in (1) (PROMO ON condition); and(3) a baseline scan without motion correction, where thesubject was asked to stay as still as possible (NOMOTION condition). Two types of staged head motionswere performed: (1) ‘‘side-to-side’’ motion and (2) ‘‘nod-ding’’ motion.

Real-Life Experiment

The final in vivo experiment was designed to assess theoverall performance of the SP-Nav/EKF framework forreducing 3D rigid-body head motion artifacts in the MRIscans under realistic conditions. To do this, two healthyyoung males were scanned (ages 10 and 11) with the 3DIR-SPGR sequence both with and without PROMO cor-rection twice (for a total of four scans per subject).Young males were recruited because of their knowntendency to move during the scan sequence despitebeing instructed not to. No specific instruction was givento either subject, only to remain ‘‘as still as possible.’’

96 White et al.

RESULTS

Offline Simulations

EKF tracking results for the 1D steps along each motionparameter are shown in Fig. 6 for both ‘‘large’’ (10 mm/deg) and ‘‘small’’ (1 mm/deg) motion magnitudes with anSP-Nav SNR of 10 (which approximates the actual SNRof the online SP-Navs). The results are shown in ‘‘ma-trix’’ format where the columns indicate the simulatedparameter and the rows indicate the estimated parame-ter. Not surprisingly, the error variance in the estimatesdue to noise is more apparent for the ‘‘small’’ subvoxelmotions than the ‘‘large’’ motions.

EKF tracking results for compound random amplitudemotion steps are shown in Fig. 7. For all plots, theaverage norm error �standard deviation over 500 MonteCarlo runs is shown. Fig. 7a shows the norm error fortranslations and rotations. Fig. 7b shows the effect ofSNR on the overall (norm across all six parameters)EKF tracking error. Fig. 7c shows the effect of addingadditional iterations (N) on the overall tracking error fora fixed SNR of 10. Only N ¼ 2 iterations are shownbecause increasing the number of iterations beyond thispoint had little effect on the tracking results. In Fig.7a–c, the ‘‘smoothness factor’’ is set arbitrarily low (S ¼6.1 � 103) to remove the weight of the ‘‘model estima-tion error’’ on the Kalman cost function, and reducingS any further had minimal effect on the tracking per-formance. Because only the ‘‘data prediction error’’ isused during estimation, one can interpret the iteratedEKF in Fig. 7c as a standard image-based least-squaresregistration with N iterations. Figure 7d shows theeffect of increasing the ‘‘smoothness factor’’ (i.e.,

increasing the weight on the ‘‘model estimation error’’)on the overall tracking error for ‘‘small’’ motions with afixed SNR of 10.

In Vivo Validation Experiment

Results from the in vivo validation experiment areshown in Fig. 8. Online motion estimates for a singlerepresentative ‘‘motion-trial’’ are shown in Fig. 8a.Note how the subject made a single brief motion duringthe middle of V2 and remained fixed in this positionthroughout the remainder of the scan. The steady-stateEKF motion estimates are plotted against the ‘‘true’’motion of the subject for all 15 trials in Fig. 8b. Themean absolute steady-state error (MAE) � standard devi-ation and mean magnitude of the ‘‘true’’ motion acrossall trials are shown in Fig. 8c. The MAE � standarddeviation was calculated to be tx ¼ 0.14 � 0.15 mm, ty ¼0.35 � 0.39 mm, tz ¼ 0.17 � 0.15 mm, yx ¼ 0.39 � 0.31deg, yy ¼ 0.25 � 0.20 deg, and yz ¼ 0.27 � 0.30 deg. Themean magnitude of the ‘‘true’’ motion was calculated tobe tx ¼ 1.34 mm, ty ¼ 1.22 mm, tz ¼ 2.11 mm, yx ¼ 4.33deg, yy ¼ 2.55 deg, and yz ¼ 7.03 deg. The overall MAE(mean across all parameters and trials) was 0.26 and theoverall magnitude of motion (mean across all parametersand trials) was 3.09, which yields an expected overallsteady-state error of less then 10% of the true motion.

Repeated Staged Motion Experiment

Online EKF motion estimates obtained during the ‘‘side-to-side’’ and ‘‘nodding’’ motion experiments for the two

FIG. 6. EKF tracking results for simulated 1D motion steps. The true motion is shown in gray and the estimated motion in black. A total

of six trials (columns) were used to create each 6 � 6 matrix. Off-diagonal plots highlight the ‘‘crosstalk’’ error of the EKF.

PROMO Real-Time Motion Correction 97

PROMO ON 3D IR-SPGR scans are shown in Fig. 9. Asshown in the figure, ‘‘side-to-side’’ motion is character-ized by rotations around the z-axis and translationsaround the x-axis, while ‘‘nodding’’ motion was charac-

terized by rotations around the y-axis and translationsaround the z-axis. Although not shown, similar motionestimates were evident during the other ‘‘side-to-side’’and ‘‘nodding’’ motion experiments.

FIG. 7. Average EKF tracking errors (over 500 Monte Carlo runs) for compound random amplitude motion steps plotted on a log scale.a: Translation and rotation error for ‘‘large’’ and ‘‘small’’ motion steps. b: Overall error (norm across all six parameters) as a function of

SNR for ‘‘large’’ and ‘‘small’’ motion steps. c: Overall error as a function of EKF iteration (N) for ‘‘large’’ and ‘‘small’’ motion steps with anSNR of 10. d: Overall error as a function of ‘‘smoothness factor’’ S for ‘‘small’’ motion steps with an SNR of 10. In all plots, the arrowindicates the onset of the motion. Time point 1 is omitted as the predicted state of the EKF is zero.

FIG. 8. Results from the in vivo validation experiment. a: Online motion estimates for a single representative ‘‘motion’’ trial. Inset showsa close-up of the tracking estimates during the motion onset interval. b: The ‘‘true’’ vs steady-state EKF estimates over all 15 experi-

mental trials. c: The mean absolute steady-state error (MAE, dark gray) compared with the mean magnitude of the ‘‘true’’ motion (lightgray) for each parameter across all trials.

98 White et al.

Reconstructed images from the repeated staged motionexperiment are shown in Fig. 10. In order to comparecorresponding slices, all reconstructed volumes were reg-istered and resampled (using high dimensional sincinterpolation) to the NO MOTION 3D FSE volume. Asevident in the figure, the PROMO corrected images havesignificantly reduced motion artifacts during both typesof staged motion to a level that is qualitatively similar tothe NO MOTION scans.

A close-up of the PROMO ON (‘‘nodding’’ motion) andNO MOTION IR-SPGR sagittal slice is shown inFig. 11. The arrows in the figure highlight an area of theneck outside the EKF brain mask that underwent non-rigid motion during the scan and was subsequentlyuncorrected in the final image.

Real-Life Experiment

Corresponding coronal sections through the PROMO ONand PROMO OFF 3D IR-SPGR scans for each of the twoyoung male subjects are shown in Fig. 12. Both sets ofPROMO ON and PROMO OFF scans were selected fromthe set of four to have qualitatively similar motion statis-tics as assessed with the ‘‘rescan metric’’ q(n), alsoincluded in the figure. Again, the images with PROMOcorrection have significantly reduced motion artifactscompared to the images without correction. The numberof rescanned k-space segments for Subject 1 and Subject2 using a threshold of T ¼ 1 were 23 and 38,respectively.

DISCUSSION

This work describes a new technique called PROMO thatutilizes three orthogonal 2D SP-Navs, along with a flexi-ble image-based tracking method based on the EKF algo-rithm for online motion measurement. A preliminaryapplication of the SP-Nav/EKF tracking framework in 3DIR-SPGR and 3D FSE pulse-sequences demonstrates theeffectiveness of the approach for significantly reducing

3D rigid-body motion artifacts prospectively in thesesequences.

Offline Simulations

Offline simulations revealed some important propertiesof the SP-Nav/EKF framework for rigid-body motiontracking. Fig. 6 demonstrated how both ‘‘large’’ and‘‘small’’ 1D translations and rotations in all three degreesof freedom could be estimated with minimal transient‘‘crosstalk’’ error. The term ‘‘crosstalk’’ error is used hereto describe the apparent motion in one parameter due tomotion in another (i.e., the off-diagonal plots in Fig. 6).While there was a tendency for rotations to have slightlylarger ‘‘crosstalk’’ errors than translations, this effect wasminimal, and in general, both rotations and translationscould be estimated with similar accuracy and precision.

Figure 7 shows the performance of the SP-Nav/EKFframework for tracking compound random amplitudemotion steps using various filter parameters and SNRlevels. Consistent with the 1D motion simulation results,Fig. 7a shows how translations and rotations could beestimated with similar accuracy and precision. Reducingthe SP-Nav SNR increased the overall error due to noise(i.e., the baseline error when in the steady-state), but hadlittle effect on the overall tracking error due to ‘‘large’’and ‘‘small’’ compound motions, i.e., the filter responsetimes (Fig. 7b). Thus, the number of SP-Navs (post–motion onset) required to estimate ‘‘large’’ and ‘‘small’’compound motions was relatively insensitive to SP-NavSNR. Furthermore, increasing the number of EKF itera-tions (N) at each time point (which increases the compu-tation time) was generally unnecessary as it only margin-ally improved the overall tracking error for ‘‘large’’compound motions, but had no effect on the overalltracking error for ‘‘small’’ compound motions (Fig. 7c).An inspection of Fig. 7a-c revealed that on average, twoto three SP-Navs (post–motion onset) were required toestimate ‘‘large’’ compound motions, while only aboutone to two SP-Navs were required for ‘‘small’’ compound

FIG. 9. Online motion estimates for the PROMO ON 3D IR-SPGR scan during ‘‘side-to-side’’ and ‘‘nodding’’ motion.

PROMO Real-Time Motion Correction 99

motions, regardless of the number of EKF iterations. Thisresult suggests that the majority of the transient SP-Nav/EKF tracking error comes from through-plane motion ofthe SP-Navs, which is greater for ‘‘large’’ compoundmotions. Transient tracking errors caused by through-plane motion are also present in orthogonal 2D orbitalnavigators (13) and are likely to have contributed sub-stantially to the transient ‘‘crosstalk’’ errors in the 1Dsimulations. However, as demonstrated in Fig. 6 and 7,this effect is mitigated when collecting multiple naviga-tors back to back such that the through-plane motioncomponent is reduced iteratively in time. The set of fiveSP-Navs played out during the 3D IR-SPGR or 3D FSE

sequences should therefore be sufficient even in extremesituations where the patient makes a relatively ‘‘large’’transient motion during this interval.

Finally, varying the ‘‘smoothness factor’’ S (i.e., chang-ing the degree to which the motion model influences thetracking estimates) resulted in a tradeoff betweenincreased noise suppression on the one hand (low filterbandwidths with high values of S) and reduced trackingerror due to transient motion on the other (high filterbandwidths with low values of S). This is evidenced inFig. 7d. As the level of noise suppression was relativelysmall compared with the increase in overall trackingerror, low ‘‘smoothness factors’’ (and no EKF iterations)

FIG. 10. Reconstructed images for the repeated staged motion experiment. Only a single axial and sagittal slice through the volumes is

shown. The number of rescanned k-space segments for the PROMO ON scans using a threshold of T ¼ 1 was (from top to bottom) 20,10, 35, and 5, respectively. The two images highlighted with arrows are shown again in Fig. 11.

100 White et al.

were used for all subsequent in vivo scans. It should benoted that this finding is not surprising, given (1) thelarge number of measurement voxels in each SP-Nav(effectively about 16,000 after masking) compared to the

small number of rigid-body motion parameters to be esti-mated at each time point and (2) the relatively highimage SNR (SNR ¼ 10). Both of these factors suggest thatthe ‘‘data prediction error’’ should contribute more

FIG. 11. Comparison of the PROMO ON 3D IR-SPGR scan during ‘‘nodding’’ motion and the NO MOTION 3D IR-SPGR scan. White

arrows highlight a region of the neck outside the brain ROI of the EKF, which moved non-rigidly during the scan and is subsequentlyuncorrected in the image.

FIG. 12. Real-life images taken from two healthy young males with and without PROMO correction along with the ‘‘rescan metric’’ qðnÞ.

PROMO Real-Time Motion Correction 101

information to the Kalman cost than the ‘‘model estima-tion error.’’ However, it should also be noted thatalthough the model information played a relativelyminor role in the current application of the Kalman filterfor tracking in the SP-Nav sequence, in applicationswhere (1) there is reduced information in the measure-ments through either reduced SNR or number of imagingvoxels (e.g., self-navigated head motion tracking in 2DEPI slice data), (2) there are more rapid image-basedmeasurements available online, (3) there are more thansix parameters in the state vector (e.g., modeling non-rigid motion or including velocity terms), and (4) thereare strong regularities over time in the state parameters(e.g., when tracking cardiac or respiratory motion),increasing the weight on the dynamic model will likelyplay a more important role during tracking.

In Vivo Validation Experiment

The steady-state error in the online SP-Nav/EKF motionestimates was quantified in vivo using real biologicstaged head motions (Fig. 8). In general, the accuracy ofthe in vivo estimates scaled with the magnitude ofmotion, with larger head motions leading to larger esti-mation errors. However, these errors were consistentlyless than one tenth of the overall magnitude, even for tri-als that included large compound motions with rotationsover 15 deg (Fig. 8c). While the accuracy of the estimatesfrom a single SP-Nav could not be assessed in vivo, aninspection of the tracking results suggests that thesteady-state level of accuracy was reached in about twoto three SP-Navs, consistent with the offline simulationresults (c.f. inset in Fig. 8a). It should be noted that therange of motion magnitudes tested in this study wasmore than double the range used to benchmark the or-bital (13), spherical (14), and cloverleaf (17) methodsand yet in all trials the error never exceed the 10% levelfor any parameter, and was typically well below thismark. It should also be noted that our validation proce-dure focused on motions that are more ‘‘likely’’ to beexperienced in practice, given the nature of the scannerenvironment. Thus, not all parameters were tested to anequal extent during the validation experiment becausethe head coil itself limited some types of motions frombeing performed. This is evidenced by the fact that thesubject made larger rotations vs translations during thevalidation experiment (c.f. ‘‘true’’ motion in gray inFig. 8c).

Repeated Staged Motion Experiment

Results from the repeated staged motion experiment dem-onstrated the effectiveness of the SP-Nav/EKF frameworkfor reducing periodic ‘‘nodding’’ and ‘‘side-to-side’’ headmotion artifacts in both the 3D IR-SPGR and 3D FSEsequences (Fig. 10). Because the EKF was ‘‘tuned’’ to esti-mate rigid-body motion within the brain ROI, areas out-side this region were ignored and subsequently uncor-rected in the final images. This is illustrated in Fig. 11,which shows a close-up of the PROMO ON 3D IR-SPGRscan during ‘‘nodding’’ motion and the NO MOTION 3D

IR-SPGR scan. The area highlighted with a white arrow inthe figure corresponds to a region in the neck outside thebrain ROI that moved non-rigidly and was subsequentlyuncorrected in the final image.

Real-Life Experiment

The results from the real-life experiments on the twohealthy young males again demonstrated the overall effec-tiveness of the SP-Nav/EKF framework for significantlyreducing 3D rigid-body head motion artifacts in 3D IR-SPGR scans under real-life conditions (Fig. 12). Qualita-tively similar motion statistics, as quantified using the‘‘rescan metric’’ q(n), suggest that the improvement inimage quality was not due to an overall difference in thelevel of motion between the two scans (Fig. 12).

Advantages/Limitations

The image-based SP-Nav/EKF tracking framework imple-mented in this study offers several advantages over exist-ing k-space-based orbital (13), spherical (14), and clover-leaf navigator techniques (17). All k-space-basednavigator methods require precise and reproducible k-space sampling trajectories from shot to shot such thatrotations and translations in image space can be estimatedfrom k-space magnitude and phase information, respec-tively. Consequently, any effect that violates the assump-tions of this Fourier model, such as (1) off-resonance dueto magnetic field inhomogeneities and susceptibility dif-ferences, (2) gradient instabilities, and (3) nonrigidmotion of the object, will degrade the accuracy and preci-sion of k-space-based rigid-body motion estimates. TheSP-Navs, on the other hand, provide reduced sensitivityto motion estimation inaccuracies due to off-resonance byblurring these effects equally in all directions in imagespace. Moreover, tracking in image space allows for mask-ing areas of the object that may corrupt the rigid-bodymotion estimates. In this study, we show how the EKFcan be ‘‘tuned’’ during the ‘‘dummy’’ acquisition period toprevent non-rigid motion of the neck and jaw from cor-rupting the rigid-body estimates of the brain and skull.This type of masking using a k-space-based approach isdifficult and requires tuning the radiofrequency pulse toexcite only the rigid-body tissue component (s). However,although the motion estimates from a single SP-Nav havereduced sensitivity to off-resonance, any slow change inSP-Nav image quality over time due to scanner instabil-ities or time-varying off-resonance effects will corrupt theSP-Nav rigid-body motion estimates and ultimatelyreduce the quality of the final reconstructed images.

In terms of time requirements, a single three-plane SP-Nav can be collected and reconstructed in approximatelythe same time (48 msec) as a three-plane orbital naviga-tor (33 msec) (13) and spherical navigator (27 msec) (14)but requires substantially more time than a single clover-leaf (4.2 msec) (17). Therefore, the cloverleaf methodmay be advantageous in certain steady-state sequenceswhere there is no intrinsic sequence ‘‘dead time’’ fornavigator acquisition. However, in non-steady-statesequences such as 3D IR-SPGR and 3D FSE, a train of

102 White et al.

SP-Navs can be played out during the intrinsic T1 recov-ery period of the acquisition without adding to the over-all scan time. However, no motion correction is per-formed during the imaging segment itself, whichtypically lasts about 2 sec. To account for this, corruptedk-space segments were rescanned at the end of thesequence, depending on whether an online ‘‘rescan met-ric’’ exceeded a user-specified threshold. In this study,all in vivo scans were performed using a rescan thresh-old of T ¼ 1, derived empirically from repeated nomotion scans. This type of rescan procedure should begenerally acceptable in most clinical applications, giventhat the number of rescanned k-space segments for thetwo young (and fairly uncooperative) males who partici-pated in this study only resulted in an additional scantime of 50 sec (Subject 1) and 90 sec (Subject 2),respectively.

One limitation of the EKF tracking algorithm, as imple-mented in this study, is the inability to correct (ratherthan just ignore) non-rigid motion. In principle, however,one can augment the EKF state vector to include higher-order motion parameters for nonlinear motion tracking.Velocity and acceleration terms can also be incorporatedin the dynamic model when more rapid measurement in-formation is available. For example, Sprincemaille et al.(30) recently applied the linear Kalman filter with first-order constant velocity dynamics for rapid (�15 msec)real-time separation of cardiac and respiratory waveformsfrom navigator least-squares motion estimates. In general,the flexibility afforded by the dynamic state-space modelframework of the EKF should allow for online image-based tracking of many different body parts using varioustypes of measurements including 2D slice information forself-navigated tracking in EPI time-series data (24).

CONCLUSIONS

In summary, the results presented in this paper demon-strate the effectiveness of the PROMO motion correctiontechnique for significantly reducing 3D rigid-bodymotion artifacts prospectively in high-resolution 3Dscans. While the current implementation focused on spi-ral-navigated 3D IR-SPGR and 3D FSE scans, the flexibleimage-based EKF tracking procedure in PROMO may beuseful in other applications such as self-navigatedPROMO in 2D EPI time-series data.

ACKNOWLEDGMENTS

Funding for this work was provided by GE HealthcareInc.

APPENDIX

To describe the online first-order iterated EKF algorithmfor prospective motion correction in the scannersequence, let M(x) be a 4 � 4 affine transform matrixthat is parameterized by the state variable x ¼ [tx,ty,tz,yx,-yy,yz]

T

M xð Þ ¼ T xð ÞR xð Þ; [11]where

T xð Þ ¼1 0 0 tx0 1 0 ty0 0 1 tz0 0 0 1

2664

3775; [12]

and

R xð Þ ¼cycz cycz sy 0

�sxsycz � cxsz �sxsycz þ cxcz sxcz 0�cxsycz þ sxsz �cxsycz � sxcz cxcy 0

0 0 0 1

2664

3775 [13]

and cx, cy and cz are the cosines and sx, sy and sz are thesines of the parameters yx, yy and yz, respectively. Fur-thermore, let X ;R4�4�!R6�1 be a function that describesthe inverse mapping between the affine transform matrixM(x) and the state vector x

X MðxÞð Þ ¼

m14

m24m34

sin�1ðm13Þatan2ðm23= cosðyyÞ;m33= cosðyyÞÞatan2ðm12= cosðyyÞ;m11= cosðyyÞÞ

2666664

3777775; [14]

where mij is the ijth entry of M(x) and atan2 is the four-quadrant inverse tangent.

Now let x0 ¼ [0,0,0,0,0,0]T be the initial state of the

EKF at time k ¼ 0 and yref ¼ h(x0) be the initial fixed ref-

erence SP-Nav. In the ideal online correction scenario,

the affine transform matrix M x̂kjk�1

� �associated with the

predicted state x̂kjk�1 is used to update the scannerpulse-sequence at each time-step prior to collecting themeasurements yk. However, to generalize the method incase this condition is not met (e.g., if real-time communi-cation is interrupted or the previous corrected statex̂k�1jk�1 is not obtained in the time allowed), let x

appk be

the actual ‘‘applied’’ state used to collect yk. Now, letþx̂kjk�1 ¼ X M�1 xappk

� � �M x̂kjk�1

� �� �be the ‘‘adjusted’’ state

that takes into account the potential mismatch betweenthe predicted and ‘‘applied’’ state at time-step k. Thus,when the predicted state is equal to the ‘‘applied’’ stateat each time-step, the ‘‘adjusted’’ state is simply the ini-tial state x0.

The goal of EKF correction step is to use the measure-ments yk to refine the predicted state x̂kjk�1 and error co-variance P̂kjk�1. Recall that this is accomplished byminimizing the cost

Ci xkð Þ ¼ yk � h xkð Þ� �TR�1 yk � h xkð Þ� �

þ xk � x̂ikjk�1

� �TP̂�1

kjk�1 xk � x̂ikjk�1

� �; ½15�

for i ¼ 0, 1,…, N iterations. The first-order iterated EKF(Gauss-Newton) solution to such an optimization prob-lem is the following update function:

PROMO Real-Time Motion Correction 103

x̂iþ1kjk�1 ¼ x̂kjk�1

þKi yk � h þx̂ikjk�1

� ��Hi x̂kjk�1 � þx̂ikjk�1

� �� �; ½16�

where þx̂ikjk�1 ¼ X M�1 xappk

� � �M x̂ikjk�1

� �� �, Hi is the Jaco-

bian of the measurement function h(�) evaluated at þx̂ikjk�1,and Ki ¼ HiR

�1HTi þ

�P̂kjk�1Þ�1HT

i R�1 is the Kalman gain.

Details of the derivation of Eq. 16 are provided in manytexts, including Ref. (23) and are omitted here due to spaceconstraints. At the end of all iterations, the a posterioristate x̂kjk is set to x̂Nþ1

kjk�1 and the a posteriori covarianceP̂kjk is set to I�KNþ1HNþ1ð ÞP̂kjk�1. Note that when no iter-ations are performed (i.e., N ¼ 0), Eq. 16 reduces to the

standard EKF (22). It should also be noted that when pre-

dicted state is equal to the ‘‘applied’’ state at each time step

and N ¼ 0, the Jacobian Hi is always evaluated around the

initial state x0, and thus can be calculated once prior to

scan start and stored in memory.To describe the functional form of the measurement

function h(�), let yax, ysa and yco be the respective 2D axial,sagittal, and coronal images of the concatenated referencenavigator vector yref ¼ vecðyaxÞT ; vecðysaÞT ; vecðycoÞT

� �T,

where the operator vecð�Þ;Rm�m�!R2m�1 concatenates a 2Dmatrix into a 1D vector. Furthermore, let rax, rsa, and rco be4 � M matrices of initial (homogenous) voxel coordinatesfor yax, ysa and yco, respectively. Then, the measurementfunction h(x) can be written

h xð Þ ¼vec yax Pax �M xð Þ � raxð Þð Þvec ysa Psa �M xð Þ � rsað Þð Þvec yco Pco �M xð Þ � rcoð Þð Þ

24

35; [17]

where MðxÞ � rax; MðxÞ � rsa, MðxÞ � rco are the transformed

coordinates, and Pax ¼ diag ½ 1; 1; 0; 1 �T� �

, Psa ¼ diag

½ 0; 1; 1;1 �T� �

and Pco ¼ diag ½1; 0; 1; 1 �T� �

are mat-

rices that project the transformed coordinates onto therespective planes of the reference navigator prior tointerpolation.

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