2712 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 53, NO. 5, OCTOBER 2006

Estimation of the Rigid-Body Motion FromThree-Dimensional Images Using a Generalized

Center-of-Mass Points ApproachB. Feng, Member, IEEE, P. P. Bruyant, Member, IEEE, P. H. Pretorius, Associate Member, IEEE,

R. D. Beach, Senior Member, IEEE, H. C. Gifford, Member, IEEE, J. Dey, Member, IEEE, M. A. Gennert, andM. A. King, Senior Member, IEEE

Abstract—We present an analytical method for the estimationof rigid-body motion in sets of three-dimensional (3-D) SPECTand PET slices. This method utilizes mathematically defined gen-eralized center-of-mass points in images, requiring no segmenta-tion. It can be applied to compensation of the rigid-body motionin both SPECT and PET, once a series of 3-D tomographic imagesare available. We generalized the formula for the center-of-massto obtain a family of points comoving with the object’s rigid-bodymotion. From the family of possible points we chose the best threepoints which resulted in the minimum root-mean-square differencebetween images as the generalized center-of-mass points for usein estimating motion. The estimated motion was used to sum thesets of tomographic images, or incorporated in the iterative recon-struction to correct for motion during reconstruction of the com-bined projection data. For comparison, the principle-axes methodwas also applied to estimate the rigid-body motion from the sametomographic images. To evaluate our method for different noiselevels, we performed simulations with the MCAT phantom. Weobserved that though noise degraded the motion-detection accu-racy, our method helped in reducing the motion artifact both vi-sually and quantitatively. We also acquired four sets of the emis-sion and transmission data of the Data Spectrum Anthropomor-phic Phantom positioned at four different locations and/or orien-tations. From these we generated a composite acquisition simu-lating periodic phantom movements during acquisition. The sim-ulated motion was calculated from the generalized center-of-masspoints calculated from the tomographic images reconstructed fromindividual acquisitions. We determined that motion-compensationgreatly reduced the motion artifact. Finally, in a simulation withthe gated MCAT phantom, an exaggerated rigid-body motion wasapplied to the end-systolic frame. The motion was estimated fromthe end-diastolic and end-systolic images, and used to sum theminto a summed image without obvious artifact. Compared to theprinciple-axes method, in two of the three comparisons with an-thropomorphic phantom data our method estimated the motionin closer agreement to the Polaris system than the principal-axesmethod, while the principle-axes method gave a more accurate es-timation of motion in most cases for the MCAT simulations. Asan image-driven approach, our method assumes angularly com-

Manuscript received November 11, 2005; revised July 17, 2006. This workwas supported in part by the National Institute for Biomedical Imaging and Bio-engineering Grant R01 EB001457 and the National Heart, Lung, and BloodGrant R01 HL 50349.

B. Feng, P. P. Bruyant, P. H. Pretorius, R. D. Beach, H. C. Gifford, J. Dey, andM. A. King are with the Department of Radiology, University of MassachusettsMedical School, Worcester, MA 01655 USA (e-mail: Bing.Feng@umassmed.edu).

M. A. Gennert is with the Department of Computer Science, Worcester Poly-technic Institute, Worcester, MA 01655 USA (e-mail: Michaelg@sequoia.wpi.edu).

Color versions of Figs. 1–6 available online at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TNS.2006.882747

plete data sets for each state of motion. We expect this method tobe applied in correction of respiratory motion in respiratory gatedSPECT, and respiratory or other rigid-body motion in PET.

Index Terms—Center-of-mass, emission tomography, motioncompensation.

I. INTRODUCTION

I N SPECT and PET, patient motion during imaging causesreconstruction artifacts and degrades diagnostic accuracy if

not correctly compensated [1]–[4]. There are numerous paperspublished on this subject, and more can be expected. Basedon the motion-detection technique employed, these papers canbe divided into the categories of data-driven methods [5]–[12]and external tracking system methods [13]–[21]. Each categoryof methods has advantages and disadvantages. The externaltracking system methods can employ an IR tracking devicesuch as the Polaris, or a video-tracking system (VTS) to detectthe motion of isolated markers. This offers high accuracy whenestimating the motion of markers, but in some cases (for ex-ample respiratory motion) does not directly reflect the motionof the organ imaged. The data-driven methods estimate themotion from the data and may use iterative approaches basedon various objective functions. Converging or robust algorithmsare not easy to obtain.

We propose a simple analytical method of detecting rigid-body motion from a series of sets of three-dimensional (3-D)tomographic images. Other analytical algorithms for rigid-bodyregistration have been published and successfully applied in themedical imaging field. Examples of such analytical methods in-clude the use of tensor-based moment functions [22], and theprinciple-axes method [23], [24]. Our method is a generaliza-tion of the conventional center-of-mass approach which givesonly information for rigid-body translation [25]. For estimatingrotation in addition to translation we defined mathematically afamily of “generalized center-of-mass points.” The motion ofthese points matches the motion in the images. These general-ized points can be used to correct rigid-body motion in imagereconstruction, as in the case of using the rigid-body motion de-tected by an external tracking system [15]–[17]. One distinctionbetween our method and the methods utilizing the tensor-basedmoment functions or the principle-axes method is that the gen-eralized center-of-mass points are linear in terms of the coor-dinates, while the other methods mentioned use tensors whichare nonlinear in terms of the coordinates. An additional fea-

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FENG et al.: RIGID-BODY MOTION FROM 3-D IMAGES 2713

ture of our method is the optimization process, implemented tochoose the power functions and presmoothing Gaussian func-tions which lead to the optimal alignment between two sets ofimages. For comparison, the principle-axes method [23] wasalso implemented and applied to our test data.

One limitation of our method is that a series of 3-D tomo-graphic images, one for each state of motion of the object, mustbe available. Usually, a SPECT acquisition cannot provide sucha series of tomographic images. However, in a respiratory gatedacquisition, one set of images can be reconstructed for eachstage of respiration. In PET, it is possible to divide acquisitiondata into independent time bins and thereby reconstruct a se-ries of images [26]. Our method might be applied to the abovecases. In reality respiratory motion has been shown to be com-plex and nonlinear [27], we expect our method could be em-ployed to compensate for the rigid-body part of the respiratorymotion affecting the heart.

II. METHODS

A. Detection of the Rigid-Body Motion From the GeneralizedCenter-of-Mass Points and by the Principle-Axes Method

We assume a series of tomogaphic reconstructions of amoving object are available, from which the rigid-body motionof the object can be determined. Each of the reconstructionsdescribes the 3-D intensity distribution of the object at a certaintime. The th voxel ( , where is the numberof voxels) has intensity . From time to , we assumethat the center of the th voxel moves from coordinates

to , due to a rigid-body motion. Thecoordinate transformation for this is

where is the rotation matrix, is the center of ro-

tation, and is the translation vector. We define generalized

center-of-mass points at time as

(1)

where function gives a cost to , and a set of functionsgives a set of points. The conventional center-of-mass pointuses . Higher order moments of can be usedto provide sets of functions . Note that the definition islinear with coordinates . At time , the general-ized center-of-mass points are

(2)

Keeping in mind that the rotation matrix and translation vectorconsist of unknown constants, it is straightforward to prove

, which means that

the generalized center-of-mass points have the same rigid-bodymotion as the object. Function should be chosen underthe constraint of , otherwise it leads to change of theimage support and truncation of the transformed image .One simple choice of that satisfies is a powerfunction as employed herein.

We need at least three generalized center-of-mass pointsin each image to track 6-degree-of-freedom rigid-body mo-tion. This is an analog to the case of fixing an object in3-D where you need at least three pins. That is, one pointcan determine translation, and a second can fix two ro-tational degrees of freedom about the first. However, athird point is needed to establish the third rotational de-gree-of-freedom. By choosing the function in (1) aspower functions , gener-alized center-of-mass points are obtained. Generally differentpoints will be obtained for different power functions, since thehigher the power function index, the closer the generalizedcenter-of-mass point is to the hottest point in the image. Sincemore than three points are available, we optimize the methodby using the “best” three points. The criterion employed is theroot-mean-square difference (RMSD) between the image attime and the image at time t with its location moved by therigid-body motion that is determined from three of points.Once any three points (characterized by the indexes of powerfunctions) are chosen, we employ a least-squares algorithm[28] that uses singular value decomposition (SVD) to estimatethe rigid-body motion. The RMSD is calculated by

(3)

where is the number of voxels in each image, is theimage intensity at th voxel measured at time , and isthe image intensity at the same voxel on the image that is mea-sured at time t and shifted with the estimated rigid-body motion.Note that both and represent the original unsmoothedimages. RMSD is dependent on the estimated rigid-body mo-tion, therefore it is dependent on the power functions chosen togenerate the generalized center-of-mass points. The time for op-

timization was approximately proportional to

. was chosen as 5 in our study since a higherwould slow down the optimization significantly.

Generally, images are degraded by noise which is caused bythe random nature in the imaging process. Therefore, we investi-gated whether it was preferable to apply presmoothing to the im-ages before calculation of the generalized center-of-mass points.The presmoothing function we chose is a 3-D Gaussian func-tion, whose sigma is chosen from a few discrete values (from 0up to 5.5 pixels with a step size of 0.5 pixel). The optimal pres-moothing Gaussian function is chosen as the one that gives theminimum RMSD.

2714 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 53, NO. 5, OCTOBER 2006

By minimizing RMSD, the rigid-body motion is estimatedfrom images at times and . In the case of more than twotime points, a baseline configuration is chosen (for example, thefirst time point), and the motion from the baseline can be esti-mated for other time points. The estimated motion can be usedpostreconstruction to generate a single summed set of slicesfrom the series of slices for different states of motion. Oncethe motion is estimated from the set of 3-D tomographic slices,it can also be used to model the motion in reconstruction ofall of the projection data as a whole dataset [20]. In this case,the reconstruction goes through all the motion states at eachprojection angle modeling the motion relative to some refer-ence position. These two motion compensation strategies (in-telligent summing and modeling motion in reconstruction) willboth be evaluated through the MCAT simulations and experi-mental measurements.

The principle-axes [23] method was also implemented to pro-vide a comparison of the generalized center-of-mass approachto an existing method. Based on the classical theory of the rigid-body motion, the principle-axes were defined as the eigenvec-tors of the inertia tensor. Defined in (9) in Faber et al. [23], theinertia tensor is symmetric and its eigenvectors are orthogonalto each other if the three eigenvalues are nondegenerate. Theseorthogonal principle-axes rotate with the object in the case ofthe rigid-body rotation. The rotation matrix is equal to thesum of the tensor products of the unit eigenvectors (denoted by

) at time (before motion) and their counterparts (de-noted by ) at time . To write explicitly

(4)

where denotes the tensor product of two vectors. For eacheigenvector, there is freedom of going positive or negative di-rection. Eigenvectors and are chosen to formright-handed sets. In addition, the angles between and , be-tween and are forced to be no greater than 90 degrees toavoid ambiguity. Since for any vectors , and , we have

(5)

where denotes the dot product. It is simple to show that

(6)

which means that rotates to , respec-tively.

The translation was calculated from the center-of-masses attime and time .

B. MCAT Phantom Simulations With Different Noise Levels

To experimentally investigate our method under controlledconditions we employed an MCAT phantom which containedonly the heart and liver as sources. We created two configura-tions: (1) before the motion (at time ) and (2) after the motion(at time ). The motion consisted of a 15-degree rotationaround the vertical Y axis and a translation (

and ). The projection data were generated

for the two configurations by using a ray-driven projector whichmodeled attenuation and distance-dependent blurring. Each setof projection data contained 60 frames, simulating a 180-degreeacquisition on a SPECT system of a Tc-99 m perfusion imaging-agent. Four noise-levels were simulated: noise-free, 1 million(M), 0.5 M, and 0.1 M counts in the heart, with Poisson noiseadded. This resulted in 4 motion-free datasets for each config-uration. After reconstructing these datasets with compensationsfor attenuation and distance-dependent blurring, we calculatedthe generalized center-of-mass points on the reconstructed im-ages, with optimization of the presmoothing Gaussian functionand the three power functions in terms of the minimum RMSD.The estimated motion was used to move the image at time

back and sum it into the image at time . Three-dimen-sional Gaussian interpolation [20] was employed to move im-ages. To test usefulness of derived motion in an iterative re-construction with motion compensation, four other datasets (forfour noise levels, respectively) were created by combining dataof the two configurations, simulating an acquisition of a pe-riodic motion which consisted of only two motion states (ateach angle the phantom moved between configurations 1 and2 back and forth). Using the OSEM algorithm which modelsthe known rigid-body motion with the 3-D Gaussian interpola-tion [20], we reconstruct the motion-present datasets with themotion derived from the generalized center-of-mass points. Toevaluate the quantitative accuracy of the reconstruction, we cal-culated the root-mean-square error (rmse) which has an expres-sion similar to (3). In other words, the rmse was defined as theRMSD between a certain reconstruction and the truth. For eachnoise-level, the truth was assumed to be the image reconstructedwith compensation for the true motion.

The principle-axes method was also applied to the images atand which had been reconstructed from the motion-free

datasets to allow comparison to this established method. Therigid-body motion obtained from the principle-axes method wassubstituted into (3) to calculate the RMSD.

C. Experimental Study With an Anthropomorphic Phantom

We also acquired four datasets of the Data Spectrum anthro-pomorphic phantom with cardiac insert on an IRIX system. OnemCi, 10.2 mCi, and 6.3 mCi of Tc-99 m was added into heart,liver, and body of the phantom, respectively. This resulted inapproximate concentrations of 1.0, 0.1, and 1.0 in these threestructures. A Polaris (Northern Digital Inc.) tool consistingof four spherical markers was attached tightly to the top ofthe anthropomorphic phantom. Between each acquisition thephantom was translated and or rotated. The orientation wasrecorded by Polaris. In each acquisition, two heads of the IRIXacquired a total of 204 degree data with 102 degree gantryrotation at 34 angular steps. A scatter window (5% widthcentered at 120 keV) was also acquired. After reconstructingeach individual acquisition, we calculated the generalizedcenter-of-mass points and the rigid-body motion, by optimizingthe presmoothing Gaussian function and determining whichthree power functions to be employed based on minimizing theRMSD. Summed images were created with and without takinginto account the motions. A mixed dataset was generated tosimulate an acquisition in which the phantom moved period-ically at each angle in an exaggerated “respiratory motion”

FENG et al.: RIGID-BODY MOTION FROM 3-D IMAGES 2715

manner. We reconstructed the mixed dataset with and withoutmodeling the motion in the iterative reconstruction.

The principle-axes method was again utilized to estimate therigid-body motions in the images reconstructed from the fouracquisitions to provide a comparison to our method.

D. Estimation of the Rigid-Body Motion in Presence ofNonrigid-Body Motion

The end-diastolic (ED) and end-systolic (ES) frames of agated MCAT phantom [29] were chosen for this study. Onlythe heart (the left and right ventricles) of the phantom was usedin our test. The cardiac motion between the ED and ES framesconsisted of shortening of the left ventricle (LV) and thickeningof the LV wall. The shape of the heart changed from the end-di-astole to the end-systole, thus the cardiac motion in the gatedMCAT phantom was a nonrigid-body motion. We applied an ex-aggerated translation ( cm, andcm) and rotation (15 degrees around Y axis) to the ES image.The motion between the ED and moved ES was estimated withthe generalized center-of-mass points approach. The estimatedmotion was compensated in summing the ES image into EDimage. It should be noted that this will not result in a full correc-tion as only a rigid-body correction is being made of a motionwhich is in part nonrigid. However, this serves as a test of cor-rection of the rigid-body component of motion when there isa nonrigid component also present. For comparison, a directlysummed image was also generated without compensating forthe rigid-body motion. Also to serve as a comparison, the prin-ciple-axes method was applied to estimate the rigid-body mo-tion between the ED image and the moved ES image.

III. RESULTS

A. MCAT Phantom Simulations With Different Noise Levels

In the MCAT simulations, the rigid-body motion was esti-mated from the images before and after the motion. In calcu-lation of the generalized center-of-mass points and estimationof the rigid-body motion, the optimal presmoothing Gaussianfunction and the optimal three power functions were chosenbased on the minimum RMSD defined in (3). The results for theoptimal parameters and estimated motion are listed in Table I.As the counts decreased, the accuracy in motion estimation alsodecreased. This was also reflected as an increase in the min-imum relative RMSD for the image with lower counts. The rel-ative RMSD is the RMSD scaled by the counts.

Applied to the same images, the principle-axes method gavemore accurate estimates of the rigid-body motion for all thefour noise-levels (Table II). The motion-corrected summed im-ages using the principle-axes method show no visual differ-ence from those images using our method. However, the RMSDfrom the principle-axes method was significantly larger for eachnoise-level than the RMSD from the generalized center-of-masspoints method. In simulations of the projection data, attenuationand distance-dependent blurring were modeled in the numer-ical projector. These degradation factors might have affectedthe accurate reconstruction of the images and thus the deter-mination of the motion. The smaller RMSD from our method

TABLE IIN THE MCAT PHANTOM SIMULATIONS, THE MOTION WAS ESTIMATED FROM

IMAGES FOR 4 NOISE-LEVELS, USING THE SMOOTHING FUNCTION AND THREE

GENERALIZED CENTER-OF-MASS POINTS DETERMINED TO MINIMIZE THE

RMSD. NOTE: ROT_Y IS Y AXIS ROTATION

TABLE IIIN THE MCAT PHANTOM SIMULATIONS, THE MOTION WAS ESTIMATED FROM

IMAGES FOR 4 NOISE-LEVELS, USING THE PRINCIPLE-AXES. NOTE: ROT_YIS Y AXIS ROTATION

could also be the result of the optimization process. The gener-alized center-of-mass points method being selected on the basisof minimizing the RMSD as opposed to minimizing the error inthe estimate of the motion. The smaller the RMSD, the better theoverall alignment of the two sets of images, and the less severemotion artifact in the summed image. Thus, this was selected asour criterion for optimization of the parameters of the general-ized center-of-mass points method.

Due to the optimization process involved, our methodwas slower than the principle-axes method. On a 2.8-GHZXeon computer running Linux, the time to align two sets of128 128 128 images was 14 seconds for the principle-axesmethod, and 50 s for the generalized center-of-mass pointsapproach.

The motion estimated from the generalized center-of-masspoints was used to move the image at time back and sumit with the image at time t (Fig. 1). As shown in Fig. 1, themotion compensated summing generated artifact-free images.To better differentiate different noise-levels, no postsmoothingwas applied to the summed images.

The motions estimated from the generalized center-of-masspoints were also compensated in the iterative reconstruction ofthe motion-present datasets. Modeling the motion in reconstruc-tion greatly reduced the motion artifact, compared with recon-struction with no motion compensation (Fig. 2). Using the true-

2716 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 53, NO. 5, OCTOBER 2006

Fig. 1. Summed images were generated for the MCAT phantom simulations.(Upper row) No motion compensation in summing images at t and t+dt.(Lower row) Motion compensated summing. From left to right, results for 4noise-levels are plotted. No postsmoothing was applied to the summed images.

Fig. 2. Reconstructions of the motion-present simulations for severalnoise-levels (Bottom row, from left to right), using the motion estimatedfrom the optimal three generalized center-of-mass points (listed in Table I).As a comparison, the same data were reconstructed with the ideal motioncompensation (Upper row) and without motion compensation (Middle row).No postsmoothing was applied to these images.

TABLE IIITHE RMSE AS CALCULATED FOR THE IMAGES IN THE MIDDLE AND LOWER

ROWS IN FIG. 2

motion compensated images (top row in Fig. 2) as the truth, therelative (scaled by counts) rmse is listed in Table III.

B. Experimental Study With an Anthropomorphic Phantom

The four sets of data were reconstructed using six iterationsof the OSEM algorithm. The first measurement was chosenas the baseline. Motion from the baseline was estimated bythe generalized center-of-mass points method and by the prin-ciple-axes method in the reconstructed image for other threecases (Table IV). In Table IV it can be seen that the generalizedcenter-of-mass points method more closely approximated themotion as estimated by Polaris than the principle-axes methodin datasets 2 and 4. Note that there were sub-pixel translations

TABLE IVTHE RIGID-BODY MOTIONS OF THE ANTHROPOMORPHIC PHANTOM WERE

ESTIMATED FROM IMAGES RECONSTRUCTED FROM THE FOUR DATASETS,BY USING THE GENERALIZED CENTER-OF-MASS POINTS AND THE

PRINCIPLE-AXES, AND COMPARED TOTHE MOTIONS DETECTED BY POLARIS

Fig. 3. (Left) Reconstruction of the motion-free baseline acquisition. (Middle)The same transaxial slice for the image summed for the four reconstructionsof the data acquired in four sequential acquisitions, without motion compen-sation in summing. (Right) The same slice with compensation for motion. Nopostsmoothing was applied to the images. Note that the summed images hadfour times the counts of the no motion image.

Fig. 4. (Left) Reconstruction of the motion-present data with compensation forthe motion which was detected by Polaris. (Middle) No motion Compensation.(Right) Motion compensation with the motion estimated from the generalizedcenter-of-mass points. No postsmoothing was applied to the images.

unreported in Table IV for datasets 3 and 4 The motions es-timated from the generalized center-of-mass points methodwere used to generate a summed image (Fig. 3) or compensatethe motion in the reconstruction of the motion-present data(Fig. 4). In each approach, compensation for the rigid-bodymotion greatly reduced the motion artifact, as shown on thepolar maps in Fig. 5.

C. Estimation of the Rigid-Body Motion in Presence ofNonrigid-Body Motion

We applied our method and the principle-axes method to theED and ES (with an exaggerated rigid-body motion) images.The motion estimated from the generalized center-of-masspoints was 2.47, 2.93, and 1.99 cm translations along X, Y, andZ direction, and 11.6 degree rotation by Y axis. The motionestimated by the principle-axis method was 2.47, 3.44, and 3.14cm translations along X, Y, and Z direction, and 15.8 degree

FENG et al.: RIGID-BODY MOTION FROM 3-D IMAGES 2717

Fig. 5. Polar maps generated for the images of anthropomorphic phantom shown in Fig. 4. (Left) Motion compensation with Polaris. (Middle) No motion Com-pensation. (Right) Motion compensation with the generalized center-of-mass points.

rotation by Y axis. The difference between the estimated motionand the true motion ( ,and a 15 degree rotation by Y axis) might have been causedpartially by the original displacement between the ED and ESimages. If the original displacement between the ED and ESimages was ignored, our method estimated the translationsmore accurately than the principle-axes method, while theprinciple-axes method gave more accurate rotation than ourmethod. The summed images with and without taking intoaccount the rigid-body motion estimated from our method wereplotted in Fig. 6. Summing with motion compensation greatlyreduced the motion artifact.

IV. DISCUSSION

Our method of using the generalized center-of-mass pointsreduces the registration of sets of images to the registration ofsets of points. However, this method could be sensitive to thenoise and other imperfectness present in images, such as arti-facts caused by lack of modeling the physics of imaging in re-construction. By minimizing the RMSD between sets of images,our method optimizes the presmoothing Gaussian function andchooses the optimal three power functions that give three pointsused in estimation of the rigid-body motion. We applied thismethod to the images reconstructed with no attenuation com-pensation, and found it worked quite well for all noise-levels(from 1 to 0.1M) studied in this manuscript. An alternative ap-proach is to use more than three points. We found that usingmore than three points resulted in very similar results for mo-tion estimation compared to the approach of using the optimalthree points which we concentrated on in this paper.

We compared our method with the principle-axes method. Wefound that the principle-axes method in most cases gave moreaccurate estimate of the rotation than our method. In two ofthree cases, our method estimated the translations which agreedbetter with those measured by the Polaris system than the prin-ciple-axes method. One possible explanation could be that onlythe center-of-mass was used to calculate the translation in theprinciple-axes method, while our method adjusted the estimateusing an optimization process.

As a short-coming of the generalized center-of-mass pointsmethod, it does not work for the binary images, since only onepoint can be obtained by (1). The principle-axes method wasshown to be able to successfully align binary images [23].

We tested two motion-compensation strategies once the rigid-body was known. One is summing the 3-D reconstructions aftercorrecting the motion in each set postreconstruction. The second

Fig. 6. In simulations with the gated MCAT phantom, the end-diastolic (ED)(Upper left) and the end-systolic (ES) (Upper right) images were used. The ESimage was translated and rotated (Upper right), and summed into the ED imagewithout compensation for the motion (Lower left), and with compensation forthe motion estimated from the generalized center-of-mass points (Lower right).Note the full 3-D heart was employed in this study. Translation and rotationwere also 3-D. Thus the summed ED+ES slice at lower right is not the sum ofthe ED and ES slices shown at the top of the figure, but the sum of the ED sliceat top and its appropriate counter part after 3-D translation and rotation.

is reconstructing the motion-present data by modeling the mo-tion in the iterative reconstruction process. Both approacheshelped reduce the motion artifacts. The summing approach isfaster, while the reconstruction approach is slower but mighthandle the statistics more appropriately. Since iterative recon-struction is a nonlinear process, the two approaches generallyresult in different images. Further investigations are needed toevaluate these two approaches.

In our simulations, with the gated MCAT phantom, the rigid-body motion was extracted approximately in the presence of thenonrigid-body wall motion. In realistic clinic studies, differentorgans might move in different fashions. Segmentation of theregion-of-interest will be required before applying our method.

Like the degeneracy encountered in the registration of points,our method could face the problem of degeneracy if symmetryexists in some directions. In the case of existence of the sym-metry in one direction, motion along that direction is undeter-mined, but will not cause motion artifacts in that direction.

It is of most interest whether our method could apply to theclinical patient study. In PET, by diving the acquisition time into

2718 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 53, NO. 5, OCTOBER 2006

short intervals, a series of 3-D images could be available. Ourmethod could be applied to sum these images up with motionestimated and compensated. As future work, we will investigateapplication of this method in PET imaging.

V. CONCLUSION

With simulations and experimental measurement we testedthe method of using the generalized center-of-mass points todetect and correct the rigid-body motion in imaging, and weobserved our method helped in reducing motion artifacts.

ACKNOWLEDGMENT

The authors would like to thank the reviewers for their manyuseful comments which greatly improved this manuscript.

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