Dynamic Three-Dimensional Undersampled DataReconstruction Employing Temporal Registration

Pablo Irarrazaval,1,2 Redha Boubertakh,1 Reza Razavi,1 and Derek Hill1

Dynamic 3D imaging is needed for many applications such asimaging of the heart, joints, and abdomen. For these, the con-trast and resolution that magnetic resonance imaging (MRI)offers are desirable. Unfortunately, the long acquisition time ofMRI limits its application. Several techniques have been pro-posed to shorten the scan time by undersampling the k-space.To recover the missing data they make assumptions about theobject’s motion, restricting it in space, spatial frequency, tem-poral frequency, or a combination of space and temporal fre-quency. These assumptions limit the applicability of each tech-nique. In this work we propose a reconstruction techniquebased on a weaker complementary assumption that restrictsthe motion in time. The technique exploits the redundancy ofinformation in the object domain by predicting time frames fromframes where there is little motion. The proposed method iswell suited for several applications, in particular for cardiacimaging, considering that the heart remains relatively still dur-ing an important fraction of the cardiac cycle, or joint imagingwhere the motion can easily be controlled. This paper presentsthe new technique and the results of applying it to knee andcardiac imaging. The results show that the new technique caneffectively reconstruct dynamic images acquired with an under-sampling factor of 5. The resulting images suffer from littletemporal and spatial blurring, significantly better than a slidingwindow reconstruction. An important attraction of the tech-nique is that it combines reconstruction and registration, thusproviding not only the 3D images but also its motion quantifi-cation. The method can be adapted to non-Cartesian k-spacetrajectories and nonuniform undersampling patterns. MagnReson Med 54:1207–1215, 2005. © 2005 Wiley-Liss, Inc.

Key words: 3D; reconstruction; undersampling; temporal regis-tration

Magnetic resonance imaging (MRI) is increasingly beingused for dynamic imaging, not only in two but also in threedimensions. There is no doubt about the clinical utility ofprocuring the temporal evolution of the MRI signal. Exam-ples of areas in which dynamic imaging is important arecardiac imaging (1–3); breast imaging (4,5); abdomen im-aging (6), particularly liver (7); functional MRI (8); jointimaging (9,10); and vascular imaging (11,12).

The necessity of dynamic three-dimensional imagingposes a difficult challenge for the current MRI technology,

due to the need to acquire large quantities of data in shorttime intervals. Several MRI sequences have been devel-oped for speeding the acquisition as much as permitted bythe hardware. The idea is to fill the required k-space as fastas possible. This has been done by shortening the repeti-tion times of sequences that use the traditional 2DFT tra-jectories; by employing more efficient k-space trajectoriesand therefore reducing the number of required readouts;and by speeding up the acquisition by only sampling partof the required k-space. The method we propose in thispaper falls into this last category: undersampled acquisi-tion. There are several possible methods for reconstructingimages from undersampled k-space data. Most of them tryto recover the missing data from prior information or fromthe information redundancy present in a dynamic data set.We may classify the primary reconstruction methodsbased on the assumptions they make about the imagedobject. These assumptions are made either in the temporalor in the spatial dimension where they are described eitherin the space and time domain or in their Fourier counter-parts. The reduced field of view (rFOV) (13) and keyhole(14,15) techniques are based on restrictions imposed onspace. The rFOV technique assumes that the time-varyingsignal is confined in image space to only a fraction of theFOV. This assumption enables undersampling of k-spaceduring dynamic acquisition and later undoing the aliasingby subtracting the aliased static image to obtained thedifference image for each frame. Its dual version, the key-hole technique, assumes that the time-varying signal isconfined in Fourier space. Changes are mostly occurring inthe center part of k-space and therefore only that part isupdated during the dynamic acquisition. Obviously, thisis a simplistic classification since there are several tech-niques that are based on assumptions that are made inmore than one domain. For instance, the hybrid techniqueproposed by Parrish and Hu (16) explicitly takes advantageof restriction in movement in both image and Fourierspace. Other techniques are based on restrictions imposedin the time and space dimension, assuming that the timevariations are band limited for some spatial positions,allowing for a more efficient packing in the x-f space. TheUNFOLD technique (17) undoes the aliasing introduced bythe undersampling by filtering the signal in time. A moregeneralized approach is k-t BLAST, proposed by Tsao andcolleagues (18,19), where the reconstruction is done bysolving a linear system.

In this paper we will present a new technique to recon-struct undersampled data, which exploits the informationredundancy of the object from time frame to time frame. Inthe general case the object’s changes in time are “small”and many times “predictable” from previous frames. Theproposed technique is based on predicting how the imageshould look like for a particular time and based on the

1Division of Imaging Sciences, Guy’s Hospital, Kings College London, Lon-don, United Kingdom.2Departamento de Ingenierıa Electrica, Pontificia Universidad Catolica deChile, Santiago, Chile.Grant Sponsor: FONDECYT; Grant Number: 1030570; Grant Sponsor:EPSRC/MRC funded Medical Imaging and Signals IRC (UK).*Correspondence to: Pablo Irarrazaval, Departamento de Ingenierıa Electrica,Pontificia Universidad Catolica de Chile, Vicuna Mackenna 4860, Macul,Santiago, Chile. E-mail: pim@ing.puc.clReceived 27 May 2004; revised 16 June 2005; accepted 16 June 2005.DOI 10.1002/mrm.20671Published online 26 September 2005 in Wiley InterScience (www.interscience.wiley.com).

Magnetic Resonance in Medicine 54:1207–1215 (2005)

© 2005 Wiley-Liss, Inc. 1207

incompletely sampled data correcting the prediction tomatch the actual object. The method works by reconstruct-ing first a rough estimate of the images (with a slidingwindow reconstruction (20)), which is assumed to be agood estimate for some of the frames, when there is littlemovement. These well-estimated frames are the base usedto predict the badly estimated frames, when there is largemotion. As a “by-product” of the reconstruction we alsohave the motion vectors, which could facilitate tasks assegmentation and motion quantification.

METHODS

To explain the proposed reconstruction algorithm, in thissection we will describe the acquisition strategy, the reg-istration and prediction stage, and finally the procedurefor assuring data consistency.

We employed a regular undersampling pattern for ac-quiring the k-space data. Since we are basing the recon-struction algorithm on the idea of redundancy of informa-tion, we need to ensure that we acquire a “little bit fromeverywhere.”

From the undersampled data, in k-space and time, wereconstruct an initial estimate of the image using a simplesliding window reconstruction. The assumption of themethod is that this initial estimate will be as good as fullysampled data for some of the time frames where there islittle motion. The motion of the subject is then estimatedby registering the time frames of this initial estimate toeach other. The time frames where there is little motion aretransformed, according to this registration, to form a pre-dicted frame for times where there is greater motion.

Finally, we ensure that the reconstructed image is per-fectly consistent with the acquired samples by substitutingthose samples in the predicted data and slightly modifyingthe new samples to avoid aliasing artifacts. At the end, weobtain estimates for the nonacquired samples in k-spacethat are as consistent as possible to the estimated motion.

Acquisition

Let m(x,t) be the desired 3D movie we are interested in,and let M(k,t) be its Fourier transform along the spatialdimensions. The undersampled acquisition can be repre-sented by

MU(k,t) � S(k,t)�M(k,t),

where S�k,t� � �i��k � ki,t � ti� represents the samplingfunction that samples the data at positions (ki,ti). Theproportion of sampled points from all points, as defined bythe Nyquist criteria and image parameters (field of viewand resolution), is the inverse of the undersampling factor, q.

The distribution of the sampling points in the k-t spacecan be arbitrarily defined as long as it allows the recon-struction of one fully sampled time frame from data ac-quired during a limited time. A simple way to do this is tohave one sample for every q frames for each k-space posi-tion. It is better if that sample is not in the same frame forall k-space positions. We used a pattern as shown by theblack dots in Fig. 1 for ky or kz versus time. Note thatnormally the readout direction kx does not need to be

undersampled. This pattern restricts the extension of thevoids of information in k-space and time (18).

Registration and Prediction

The registration and prediction stage of the algorithm isdone in the following four steps: computation of an initialestimate; frame registration; frame classification; andframe prediction. These steps are summarized in Fig. 2.

Initial Estimate

From the undersampled data MU (k,t) an initial estimate ink-space MT(k,t) and in object space mT(x,t) are computed.This is done by convolving in time the data with a trian-gular kernel of size 2q � 1 (a simple linear interpolation, asrepresented by the horizontal arrows in Fig. 1), effectivelysmoothing it in time:

MT(k,t) � ��k) � (t/�2q � 1��*MU(k,t).

If data from multiple coils are available, one image for eachcoil is estimated and the registration step that follows usesthe root mean square of them.

Frame Registration

The idea is to register, nonrigidly, every frame to everyother, such as to obtain all the transformation parametersthat allow the conversion of one frame into any other.Since registration algorithms are computationally intense(slow), we only register a fixed frame, mT(x,t0), usually thefirst one, to all the rest. We did not mask the images tospeed up the registration since we want to keep themethod as general as possible. We used the VTK CISG 3D

FIG. 1. Sampling pattern example for an undersampling factor of 5.The same pattern is used for ky and kz versus time. Black dotsrepresent acquired samples and white dots the estimated samplesfor one frame. For the initial estimate the samples are computed asa linear interpolation from the acquired samples (represented by thehorizontal arrows).

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Registration Toolkit (21), which performs a voxel-basednonrigid (free form deformation) registration on the basisof B-spline interpolation with a normalized mutual infor-mation similarity measure (22). The obtained transforma-tions are then used to register a grid of reference points inmT(x,t0). These registered reference points allow the trans-formation of any frame into any other. Note that mT(x,t0) isonly used for defining the grid of reference points and theimportant information obtained from the registration is therelative displacement of these points between frames,which is independent of the starting frame. In this paperwe used trilinear interpolation to compute the intensityvalues for every voxel inside each of the tetrahedronsdefined by the registered reference points. Thus, B-splineinterpolation is used for finding the reference points andtrilinear for the voxels in between. The assumption behindthe use of the registration stage is that the only changebetween frames is a geometrical deformation, and there isno intensity change for the same anatomic location.

The registration step is done with the root mean squarecombination of all coils, if more than one has been used.

Frame Classification

Using the registered reference points, we compute a mo-tion index for each frame. The motion index is computedas follows: (1) the standard deviation for the three coordi-nates (x, y, and z) of each reference point is calculated; (2)

we consider only those points with a standard deviationgreater than one pixel in at least one coordinate; (3) foreach considered point we find the time frames for which itcrosses the mean value (this is a rough estimation of thetimes of maximum speed); and finally the index is thecount of points crossing the mean for each time frame.This index is used to classify the frames as either movingor not-moving with a simple threshold. The threshold wasmanually set, trading off two typically conflicting objec-tives: to correctly classify the not-moving frames and tomaximize the number of not-moving frames in order tohelp the prediction by placing not-moving frames as closeas possible to every moving frame. As a rule of thumb weset it slightly above the plateau formed by the not-movingframes.

Frame Prediction

The purpose of this step is to substitute the moving frameswith a predicted frame. Let ms

P(x,t) be the predicted framefor time t computed by transforming the initial estimateframe s, mT(x,s). The final predicted frame, mP(x,t), iscomputed as the median of several predictions (we usedfive in the examples shown in this paper) from differenttimes s, always using the same registered reference points.These times are chosen such that frame s is a not-movingframe and is as close as possible to time t. This step is doneon a coil-by-coil basis for multicoil data.

FIG. 2. Graphical summary of the predic-tion and registration stage of the proposedmethod: the undersampled data are con-volved in time with a triangular kernel;frames are registered to each other and amotion index is computed; frames are clas-sified as moving or not-moving; movingframes are substituted by the median of thepredicted frames from neighboring not-moving frames.

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Data Consistency

To ensure that the result of the reconstruction is consistentwith the data actually acquired, the following mechanismwas used: The output images of the registration and pre-diction stage were Fourier transformed to obtain predictedraw data. These predicted raw data were used to completethe missing samples from the actually acquired data. Inother words, all acquired data were directly used and thepredicted data are only used for the nonsampled k-spacepositions

MA(k,t) � MP(k,t) � S�k,t� � �MU(k,t� � MP(k,t)}.

This direct substitution presents a problem when the pre-diction differs significantly from the actual object: sharpdiscontinuities appear in k-space, i.e., an acquired sam-pled may be surrounded by very different predicted sam-ples. The discontinuities give rise to aliasing artifacts. Toreduce this problem we estimate an “error” of predictionin k-space, which consists of the difference between theacquired samples and the predicted samples at the samelocations (not used otherwise),

ME(k,t) � S�k,t� � �MU(k,t� � MP(k,t)}.

This error is smoothed with a linear interpolation filterand added to the predicted raw data for the nonsampledk-space positions. If the prediction is perfect the error willbe zero and the prediction will not be modified, but if theprediction is really “bad,” for instance, all zeros, the errorwill be equal to the acquired samples and the smoothederror will be the sliding window solution. Therefore, forany reasonable prediction the result will be something inbetween the sliding window and the perfect reconstruc-tion. The data consistency algorithm is also done on acoil-by-coil basis in case of multicoil data.

RESULTS

The proposed algorithm was employed to reconstructthree different kind of movie data: 3D knee images, 2Dcardiac images, and 3D cardiac images (presented in orderof complexity). All experiments were done on a PhilipsIntera 1.5 T scanner. The subjects were volunteers andpatients who gave consent to participate in local researchethics approved studies.

3D Knee Images

A 3D movie of the knee is an ideal application for theproposed reconstruction algorithm. It is relatively easy tocontrol the movement and it is an interesting applicationfrom the clinical point of view. First we tested the algo-rithm from fully sampled data, simulating the undersam-pling at the reconstruction stage. And second, we recon-structed data collected with an undersampled acquisition.

The first reconstruction was done acquiring continu-ously fully sampled 3D images of a volunteer’s movingknee. The sequence parameters are as follows: 3D FFEdynamic study, 50° flip angle, TE � 2.6 ms, TR � 5.3 ms,256 � 100 � 10 acquisition matrix, 40 frames (5.3 s each),

430 � 168 � 30 mm3 FOV (1.68 � 1.68 � 3 mm3 resolu-tion), with a quad knee coil. The acquired data were un-dersampled by a factor of q � 5, according to the patterndescribed in Fig. 1. The volunteer was asked to hold theknee still in a straight position for 1 min (10 frames), in aslightly bent position for 1 min, and finally another minutein a straight position again. The transitions from one po-sition to the other took approximately 30 s (5 frames), fora total imaging time of 3 min 32 s.

The results of the reconstruction are shown in Fig. 3(slices through z for a moving frame) and in Fig. 4 (evolu-tion of a particular line through time). We only recon-structed the middle 215 mm (128 pixels) in the longitudi-nal direction, where there is significant signal. As ex-

FIG. 3. Reconstructed images, moving frame 26, from a 3D kneeacquired with simulated undersampling. The images show a slicethrough the center of the z dimension. (a) Initial estimate. (b) Pre-dicted. (c) predicted and data consistent. (d) Fully sampled. (e) Initialestimate difference with the fully sampled. (f) predicted and dataconsistent difference with the fully sampled. Note that the aliasingresidual is reduced and the edge definition recovered, althoughsome edge displacement persists.

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pected the initial estimate for not-moving frames was notblurred but show considerable blurring and aliasing formoving frames—see frame 26 in Fig. 3a. The predictedimage deblurs very well the most important features of theimage as can be appreciated in Fig. 3b or c. This last one isconsistent with the acquired data: its Fourier transformpasses exactly through the acquired samples. For compar-ison purposes we have included the image reconstructedusing the fully sampled data in Fig. 3d and the differ-ences with the undersampled data in Fig. 3e and f. Theunblurring in time can be better appreciated in Fig. 4,which shows the time evolution for a line in space forthe four reconstructions. Notice that, as explained in theDiscussion, the difference images could be misleadingsince small displacements of the edges result in highvalued differences and the unblurring effect is hard toappreciate.

The second reconstruction was done acquiring continu-ously undersampled 3D images of a volunteer’s movingknee. The sequence parameters are as follows: 3D FFEdynamic undersampled study, 50° flip angle, TE �1.79 ms, TR � 5.3 ms, five-times undersampled 256 �

100 � 10 acquisition matrix, 40 frames (1.06 s each), 430 �168 � 30 mm3 FOV (1.68 � 1.68 � 3 mm3 resolution), witha quad knee coil. The pulse sequence was implemented bySebastian Kozerke while at King’s College London as partof his k-t BLAST work (19). The acquisition was under-sampled by a factor of q � 5, according to the patterndescribed in Fig. 1, i.e., only 200 repetition times wereacquired. The volunteer was asked to hold the knee still ina straight position for 10 s (10 frames), in a slightly bentposition for 10 s, and finally another 10 s in a straightposition. The transitions from one position to the othertook approximately 5 s (5 frames), for a total imaging timeof 42 s.

The results of the reconstruction are shown in Fig. 5(slices through z for a moving frame) and in Fig. 6 (evolu-tion of a particular line through time). We only recon-structed the middle 215 mm (128 pixels) in the longitudi-nal direction, where there is significant signal. As ex-pected the initial estimate for not-moving frames wasnot blurred but showed considerable blurring and alias-ing for moving frames—see frame 18 in Fig. 5a. Thepredicted image deblurs very well the most important

FIG. 4. Time evolution of the reconstructed im-ages from a 3D knee acquired with simulated un-dersampling. The images show the intensity of aline, i.e., the y-t plane. (a) Initial estimate. (b) Pre-dicted. (c) predicted and data consistent. (d) Fullysampled. (e) Initial estimate difference with the fullysampled. (f) Predicted and data consistent differ-ence with the fully sampled.

FIG. 5. Reconstructed images, moving frame 18, from a 3D knee acquired with undersampled acquisition. The images show a slice throughthe center of the z dimension. (a) Initial estimate. (b) Predicted. (c) Predicted and data consistent.

3D Undersampled Reconstruction 1211

features of the image as can be appreciated in Fig. 5b orc. The unblurring in time can be better appreciated inFig. 6, which shows the time evolution for a particularline of the image.

2D Cardiac Images

To test the reconstruction algorithm we acquired a 2Dcardiac sequence of images, with the following parame-ters: 2D B-FFE, 50° flip angle, TE � 1.46 ms, TR � 3 ms,256 � 154 acquisition matrix, 50 frames (cardiac gated),400 � 320 mm2 FOV (1.56 � 2.08 mm2 resolution), slicewidth of 8 mm, with a 5-channel synergy cardiac coil. Theacquired data were undersampled by a factor of q � 5during the reconstruction, according to the pattern de-scribed in Fig. 1. The volunteer was asked to hold thebreath for about 25 s.

The results of the reconstruction are shown in Fig. 7(for a moving frame) and in Fig. 8 (evolution of a par-ticular line through time). As expected the initial esti-mate show some blurring (noticeable in the myocar-dium) and aliasing for moving frames—see frame 27 inFig. 7a. The predicted images deblur it as can be appre-ciated in Fig. 7b or c. The unblurring in time can bebetter appreciated in Fig. 8, which shows the time evo-lution for a particular line of the image. For comparisonpurposes we have included in Fig. 7d the image recon-structed using the fully sampled data and the differ-ences with the undersampled data in Fig. 7e and f.Notice that, as explained in the Discussion, the differ-ence images could be misleading since small displace-ments of the edges result in big difference values and theunblurring is hard to appreciate.

3D Cardiac Images

A more challenging application of the proposed algorithmis 3D cardiac images. We applied the reconstruction algo-rithm to data of patients who held their breath for approx-imately 20 s. The sequence parameters employed are asfollows: 3D B-FFE, 50° flip angle, TE � 1.56 ms, TR �3.01 ms, 256 � 100 � 16 acquisition matrix, 20 frames (0.6s each), cardiac gated, 440 � 260 � 96 mm3 FOV (1.72 �2.60 � 6 mm3 resolution) with a 5-channel synergy cardiaccoil. The acquisition was undersampled by a factor of q �5, according to the pattern described in Fig. 1.

The results of the reconstruction are shown in Fig. 9(slices through z for a moving frame) and in Fig. 10 (evo-lution of a particular line through time). The reconstruc-tion algorithm improves the time resolution, compared to

the initial estimate, as it can be better appreciated in Fig.10. There is only a slight improvement in spatial resolu-tion or reduction of aliasing.

FIG. 6. Time evolution of the reconstructed im-ages from a 3D knee acquired with simulated un-dersampling with simulated undersampling. Theimages show the intensity of a line, i.e., the y-tplane. (a) Initial estimate. (b) Predicted. (c) Pre-dicted and data consistent.

FIG. 7. Reconstructed images, moving frame 27, from a 2D cardiacimage with simulated undersampling. (a) Initial estimate. (b) Pre-dicted. (c) Predicted and data consistent. (d) Fully sampled. (e) Initialestimate difference with the fully sampled in the region of interest. (f)Predicted and data consistent difference with the fully sampled inthe region of interest. Note that the myocardium/blood edge defi-nition is recovered, although some edge displacement persists.

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DISCUSSION

The main advantages of the proposed reconstructionmethod are that (1) it does not require the motion to bespatially localized (no restrictions on the FOV); (2) it doesnot require any particular undersampling pattern; (3) it isperfectly applicable to any trajectory; and (4) it also pro-vides motion information.

The only restriction that the proposed method places onthe data are that there exists enough time to reconstructwith a sliding window at least one frame free of motionblur or aliasing. There is no restriction on the fraction ofthe FOV that is moving. This makes the method useful fora great variety of applications, in particular for those inwhich it is easy to control the movement, like joint imag-ing. On the other hand, it places a limit on the undersam-pling factor for situations in which the object is still onlyfor a limited time, for instance, the heart. One disadvan-tage of the proposed reconstruction algorithm is the com-putational load imposed by the registration stage. It tookbetween 30 min and 3 h to reconstruct the images of thispaper on a regular PC, although this time could be consid-erably reduced if one used an easier function to calculatesimilarity measurement such as correlation instead of thenormalized mutual information function or by reducingthe number of control points for the spline interpolation ordegrees of freedom (we used one every five pixels in eachdimension).

The results presented in this paper show how the recon-struction algorithm works. In general, the images are spa-tially well defined and have good temporal resolution. Forthe patient 3D heart, the improvement, compared to thesliding window reconstruction, is relatively minor. This isexplained because the method assumption is not com-pletely fulfilled. The not-moving frames show some alias-ing and blurring that are not corrected by the algorithm,probably coming from irregular motion of the heart, mo-tion from not well maintained breath hold, or nonperiodicbeating. We believe that an undersampling strategy lesssensitive to aliasing could fix some of these problems.

We used the root mean square error (RMSE) to compareour reconstruction and the sliding window with the fullysampled images for the two data sets where they wereavailable: knee and 2D heart. For the knee, our reconstruc-tion has an RMSE of 3.82%, whereas the sliding window is

FIG. 8. Time evolution of the reconstructed images from a 2Dcardiac image with simulated undersampling. The images show theintensity of a line, i.e., the y-t plane. (a) Initial estimate. (b) Predicted.(c) Predicted and data consistent. (d) Fully sampled. (e) Initial esti-mate difference with the fully sampled in the region of interest. (f)Predicted and data consistent difference with the fully sampled inthe region of interest.

FIG. 9. Reconstructed images, moving frame 11, from a cardiac 3D undersampled data. The images show a slice through the center ofthe z dimension. (a) Initial estimate. (b) Predicted. (c) Predicted and data consistent.

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4.01%. For the 2D heart our reconstruction has a poorerRMSE value than the sliding window (6.71% versus5.35% in frame 28, for instance). These numbers do notreflect the visual appearance of the images and it is ex-plained by the fact that a pixel-by-pixel comparison, likethe RMSE, penalizes heavily small displacements in theedges. Since our reconstruction algorithm is based on theestimation of the pixel motion, the errors are manifestedprecisely as displacements. The algorithm in general re-covers well the shape of the intensity profiles but with anerror in its location for some time frames. Figure 11 showsan example for frame 28 of the heart data, where theregistration is not working well. It also explains the bigdifferences in images 3, 4, 7, and 8. We speculate that thismisalignment is due to a tendency of the nonrigid regis-tration to underestimate large motions and the extra diffi-culty to register the initial estimate that already has tem-poral blurring. Evidently, the more control points used inthe registration, the more able it will be to pick up large orextremely localized motions.

Although we do not think that it is the case for our datasets, another source of error for the registration could bethe changes of intensity and/or phase from frame to framefor the same anatomic location. These would be evident ina dynamic contrast experiment or in 2D acquisitions withthrough-plane motion. We do not visualize an easy exten-sion of our method, without changing the framework, toaccount for these changes. In any case, it has been shownthat the normalized mutual information is a good choicefor small intensity changes (23,24).

We found that it is fairly easy to select the threshold fordefining the moving and not-moving frames from the mo-

tion index and that the rule of thumb works well (settingthe threshold slightly above the relatively flat valleys ofthe index). If the threshold is set too high, i.e., movingframes are incorrectly classify as not-moving, the pre-dicted images will have some of the aliasing and blurringpresent on those frames. On the other hand, if the thresh-old is set too low, i.e., times with little motion are incor-rectly classified as moving, the predicted images will de-teriorate because they will be based on frames furtheraway in time and therefore probably more different.

There are situations in which it may be convenient toundersample the k-space in a nonuniform way, for exam-ple, following a pattern in the style of keyhole acquisitionsor when using non-Cartesian trajectories. The proposedreconstruction method can easily be modified to take thatinto consideration. It would be necessary to compute theinitial estimate differently, for example, changing thewidth of the triangular kernel as a function of k-spaceposition (it does not make sense to blur in time more thannecessarily regions that are fully sampled).

The algorithm can easily be modified to take into con-sideration non-Cartesian k-space trajectories. Besides thecomputation of the initial estimate, it would also be nec-essary to modify the way the predictions are made consis-tent with the acquired data. This will probably involveinterpolation steps in order to place the predicted data inthe non-Cartesian positions of k-space. Another possibleapplication of this technique is for motion-compensatedreconstructions (25).

The obvious question regarding undersampled acquisi-tion strategies is how much undersampling it is possible totolerate without losing useful information. This limit isgiven by the amount of information redundancy present inthe data. It is not easy to measure this redundancy, notonly because the complexity of the models but also be-cause of the dependency on the application and subject.Still, it is useful to have a rough estimate of a lower boundfor it. Using concepts of information theory, we know thatthe information content is less than the entropy (computedassuming independence between pixels and frames). Forthe 2D heart the entropy is 4.87 bits/pixel in the imagedomain and 2.18 in the Fourier domain. The question nowis how much of this information is redundant. A loss-lesscompression of each frame with JPEG gives an average of0.57 bits/pixel, which suggests that at least 4.3 bits/pixel

FIG. 10. Time evolution of the reconstructed images from a cardiac3D undersampled data. The images show the intensity of a line, i.e.,the y-t plane. (a) Initial estimate. (b) Predicted. (c) Predicted anddata consistent.

FIG. 11. Intensity profile for a blurred frame (28)where the algorithm produces an error in the po-sition but a good estimation of the shape.

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are redundant with neighboring pixels. The same datacompressed with loss-less MPEG (as a movie format)yields an average of 0.448 bits/pixel. That is an extra 0.11bits from redundancy with pixels in other frames. In otherwords, there is at least a factor of 10 of redundancy. If wealso consider subjective redundancy and compress eachframe with lossy JPEG (quality factor of 85%), the averageinformation is 0.14 bits/pixel (redundancy factor of 34).That number is further reduced to 0.036 bits/pixel if weuse lossy MPEG compression, a redundancy factor of 135.This analysis does not take into account how the informa-tion is distributed throughout the image, but it gives anindication that theoretically it is reasonable to think inundersampling factors above 10 without losing informa-tion.

CONCLUSION

We have presented a method for reconstructing under-sampled data of 3D movies. The method does not requirethe motion to be restricted to a fraction of the FOV or to afraction of the temporal frequency. It does require that it ispossible to reconstruct a “good” frame of the movie whenthere is little motion of the object. We believe that this canbe assumed for a wide variety of applications.

Other advantages of the proposed reconstructionmethod are that it also provides motion information; itdoes not require any particular undersampling pattern;and it is perfectly applicable to any k-space trajectory,including non-Cartesian trajectories.

We tested the method with data from a moving knee (3D)and the heart (2D and 3D) with an undersampling factor of5. In all cases the method was able to reconstruct very wellthe object, eliminating aliasing and blurring in space andtime.

ACKNOWLEDGMENTS

The authors thank Jo Hajnal, Bill Crum, Simon Arridge,Vivek Muthurangu, Jeffrey Tsao, and Sebastian Kozerkefor helpful discussions.

REFERENCES

1. Sakuma H, Takedaa K, Higgins CB. Fast magnetic resonance imaging ofthe heart. Eur J Radiol 1999;29:101–113.

2. Rzedizan RR, Pykett IL. Instant images of the human heart using anew, whole-body MR imaging system. Am J Roentgenol 1987;149:245–250.

3. Chapman B, Turner R, Ordidge RJ, Doyle M, Cawley M, Coxon R,Glover P, Mansfield P. Real-time movie imaging from a single cardiaccycle by NMR. Magn Reson Med 1987;5:246–254.

4. Kuhl CK, Schild HH. Dynamic image interpretation of MRI of thebreast. J Magn Reson Imaging 2000;12:965–974.

5. Ercolani P, Valeri G, Amici F. Dynamic MRI of the breast. Eur J Radiol1998;27:S265–S271.

6. Edelman RR, Hahn PF, Buxton R, Wittenberg J, Ferruci JT, Saini S,Brady TJ. Rapid mri with suspended respiration: clinical application inthe abdomen. Radiology 1986;161:125–131.

7. Ichikawa T, Araki T. Fast magnetic resonance imaging of liver. Eur JRadiol 1999;29:186–210.

8. Cohen MS, Bookheimer SY. Localization of brain function using mag-netic resonance imaging. Trends Neurosci 1994;17:268–277.

9. Rebmann AJ, Sheehan FT. Precise 3D skeletal kinematics using fastphase contrast magnetic resonance imaging. J Magn Reson Imaging2003;17:206–213.

10. Gresalmer RP. Patellar malalignment. J Bone Joint Surg 2000;11:1639–1650.

11. Padhani AR. Dynamic contrast-enhanced MRI in clinical oncology:current status and future directions. J Magn Reson Imaging 2002;16:407–422.

12. Herborn CU, Goyen M, Lauenstein TC, Debatin JF, Ruehm SG, Coger K.Comprehensive time-resolved MRI of peripheral vascular malforma-tions. Am J Roentgenol 2003;181:729–735.

13. Hu X, Parrish T. Reduction of field of view for dynamic imaging. MagnReson Med 1994;31:691–694.

14. Jones RA, Haraldseth O, Muller TB, Rinck PA, Oksendal AN. K-spacesubstitution: a novel dynamic imaging technique. Magn Reson Med1993;29:830–834.

15. van Vaals JJ, Brummer ME, Dixon WT, Tuithof HH, Engels H, NelsonRC, Gerety BM, Chezmar JL, den Boer JA. “Keyhole” method for accel-erating imaging of contrast agent uptake. J Magn Reson Imaging 1993;3:671–675.

16. Parrish TB, XP H. Hybrid technique for dynamic imaging. Magn ResonMed 2000;44:51–55.

17. Madore B, Glover GH, Pelc NJ. Unaliasing by Fourier-encoding theoverlaps usingthe temporal dimension (unfold), applied to cardiacimaging and fMRI. Magn Reson Med 1999;42:813–828.

18. Tsao J, Boesiger P, Pruessmann KP. k-t BLAST and k-t SENSE: dynamicMRI with high frame rate exploiting spatiotemporal correlations. MagnReson Med 2003;50:1031–1042.

19. Kozerke S, Tsao J, Razavi R, Boesiger P. Accelerating cardiac cine 3Dimaging using k-t BLAST. Magn Reson Med 2004;52:19–26.

20. d’Arcy JA, Collins DJ, Rowland IJ, Padhani AR, Leach MO. Applica-tions of sliding window reconstruction with Cartesian sampling fordynamic contrast enhanced MRI. NMR Biomed 2002;15:174–183.

21. Hartkens T, Rueckert D, Schnabel JA, Hawkes DJ, Hill DLG. Vtk cisg registra-tion toolkit: An open source software package for affine and non-rigid regis-tration of single- and multimodal 3d images. In Bildverarbeitung fur dieMedizin 2002. http://www.bvm-workshop.org, Springer-Verlag, 2002; 96.

22. Maes F, Collignon A, Vandermeulen D, Marchal G, Suetens P. Multi-modality image registration by maximization of mutual information.IEEE Trans Med Imaging 1997;16:187–198.

23. Rao A, Chandrashekara R, Sanchez-Ortiz GI, Mohiaddin R, Aljabar P,Hajnal JV, Puri BK, Rueckert D. Spatial transformation of motion anddeformation fields using nonrigid registration. IEEE Trans Med Imaging2004;23:1065–1076.

24. Rueckert D, Sonoda LI, Hayes C, Hill DLG, Leach MO, Hawkes DJ.Nonrigid registration using free-form deformations: application tobreast MR images. IEEE Trans Med Imaging 1999;18:712–721.

25. Schaeffter T, Rasche V, Carlsen IC. Motion compensated projectionreconstruction. Magn Reson Med 1999;41:954–963.

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