k-t PCA: Temporally Constrained k-t BLASTReconstruction Using Principal Component Analysis

Henrik Pedersen,1,2* Sebastian Kozerke,3 Steffen Ringgaard,1 Kay Nehrke,4 andWon Yong Kim1

The k-t broad-use linear acquisition speed-up technique(BLAST) has become widespread for reducing image acquisi-tion time in dynamic MRI. In its basic form k-t BLAST speeds upthe data acquisition by undersampling k-space over time (re-ferred to as k-t space). The resulting aliasing is resolved in theFourier reciprocal x-f space (x � spatial position, f � temporalfrequency) using an adaptive filter derived from a low-resolu-tion estimate of the signal covariance. However, this filteringprocess tends to increase the reconstruction error or lower theachievable acceleration factor. This is problematic in applica-tions exhibiting a broad range of temporal frequencies such asfree-breathing myocardial perfusion imaging. We show thattemporal basis functions calculated by subjecting the trainingdata to principal component analysis (PCA) can be used toconstrain the reconstruction such that the temporal resolutionis improved. The presented method is called k-t PCA. MagnReson Med 62:706–716, 2009. © 2009 Wiley-Liss, Inc.

Key words: k-t BLAST/SENSE; parallel imaging; principal com-ponent analysis

The need for higher spatial and temporal resolution indynamic MRI has prompted the development of severaldedicated imaging speed-up techniques. While traditionalmethods achieve faster imaging by exploiting either spatialcorrelations (1–3) or temporal correlations (4–7) in thedata, new techniques have emerged that exploit both spa-tial and temporal correlations to achieve even faster imag-ing. These methods include UNFOLD (8,9) and k-t BLAST(10), both of which have been combined with parallelimaging to aid the reconstruction (10–13). Essentially,these methods are based on the observation that dynamicobjects exhibiting a high degree of spatiotemporal correla-tions have a compact representation in x-f space (x �spatial position, f � temporal frequency). Consequently,when k-space is sampled sparsely over time t the resultingaliasing in x-f space shows only little overlap of the signalreplicas, thus facilitating the process of recovering the trueobject.

The aliasing is removed by temporal filtering in x-f spacebased on either general assumptions (UNFOLD) or explicitprior knowledge (k-t BLAST) about the imaged object.However, the inherent temporal filtering of these methodstends to reduce temporal fidelity, leading to either in-creased reconstruction error or lower achievable reductionin data acquisition. This article addresses the temporalfidelity problem in the context of k-t BLAST and its par-allel imaging counterpart (k-t SENSE) (10). These methodsresolve the signal aliasing by an adaptive filtering processin which the aliased signal is distributed according to alow-resolution estimate of the signal covariance, as de-rived from a set of training images. The appealing propertyof this approach is that it allows some overlapping of thealiased object, thereby permitting higher acceleration fac-tors than, e.g., UNFOLD.

One major limitation of k-t BLAST, however, is that thereconstruction problem is inherently underdetermined(i.e., there are fewer equations than unknowns), implyingthat the method depends entirely on the estimated signalcovariance. If multiple receive coils are available, sensitiv-ity encoding (3) can be incorporated, potentially leading toan overdetermined reconstruction problem. This is the k-tSENSE method. In general, these methods perform bestwhen there is as little aliasing as possible, since there willbe fewer signal overlaps to separate. The number of signaloverlaps can be somewhat reduced by optimizing the sam-pling pattern in k-t space (14), but in applications wherethe imaged object has a large extent in x-f space (e.g., ifmany pixels exhibit broad temporal bandwidth) this strat-egy results in little improvement.

The method presented in this article is a generalizationof k-t BLAST that is more suitable for such applications. Itallows the use of higher acceleration factors by constrain-ing the reconstruction using a standard data compressiontechnique, called principal component analysis (PCA).Higher acceleration is achieved because the constrainedreconstruction problem in its basic form is overdeter-mined, as opposed to k-t BLAST. Therefore, the signalcovariance and sensitivity encoding provide additionalinformation to resolve the aliasing. This is the main dif-ference between k-t BLAST and the proposed method,called k-t PCA. For clarity, we write k-t PCA/SENSE whensensitivity encoding is included.

THEORY

The basic principle of k-t PCA is illustrated in Fig. 1. Rawdata are undersampled with a given acceleration factorsuch that the acquired k-t space locations conform to asheared grid (Fig. 1a). The consequence of this k-t sam-pling pattern is a sheared point spread function (PSF) (Fig.

1MR Research Centre, Aarhus University Hospital Skejby, Aarhus, Denmark.2Functional Imaging Unit, Glostrup Hospital, Copenhagen, Denmark.3Institute for Biomedical Engineering, University and ETH Zurich, Zurich,Switzerland.4Philips Research Laboratories, Hamburg, Germany.Grant sponsor: Philips Medical Systems; Grant sponsor: University of AarhusGraduate School of Health Sciences; Grant sponsor: Danish Agency forScience, Technology and Innovation.*Correspondence to: Henrik Pedersen, MSc, PhD, Functional Imaging Unit,Glostrup Hospital, University of Copenhagen, Ndr. Ringvej 57, DK-2600Glostrup, Copenhagen, Denmark. E-mail: klaverhenrik@gmail.comReceived 3 July 2008; revised 11 February 2009; accepted 26 March 2009.DOI 10.1002/mrm.22052Published online 7 July 2009 in Wiley InterScience (www.interscience.wiley.com).

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© 2009 Wiley-Liss, Inc. 706

1b) in the Fourier reciprocal x-f space. For 4-fold acceler-ation, the true x-f object (Fig. 1c) is overlapped four times,resulting in the aliasing pattern shown in Fig. 1d. Basi-cally, k-t PCA constrains the appearance of the true objectsuch that each temporal frequency must be a linear com-bination of a set of predetermined basis functions (Fig. 1e).The basis functions are derived from the training imagesusing PCA (15,16). When combined with knowledge aboutthe sampling PSF and the signal covariance, the con-strained reconstruction separates the overlapped signalsvery accurately, thereby maintaining good temporal fidel-ity. For simplicity, k-t PCA will be described for 2D imag-ing in the kx-ky-t space, where kx denotes the frequency-encoding direction and ky is the phase-encoding direction.Also, we will assume a 4-fold undersampling throughoutthe Theory section.

Data Acquisition

Although some applications permit acquisition of thetraining data in a separate scan, this study adopts a vari-

able density k-t sampling pattern in which training dataand undersampled data are collected simultaneously. Thetraining data consist of a few central phase-encoding linesor training profiles sampled for each point in time (Fig. 2a).The corresponding images have low spatial resolution, butthe temporal resolution is maintained. This is an impor-tant observation, because it justifies deriving the temporalbasis functions from the training data. The undersampleddata are confined to a sheared grid, as illustrated in Fig. 2b.The combined variable density k-t sampling pattern isdepicted in Fig. 2c.

It is important to note that the number of acquired train-ing profiles represents a trade-off between lowering the netacceleration factor (i.e., the total number of points in k-tspace divided by the number of sampled points) and im-proving the quality of the training data. This trade-off wasstudied for k-t BLAST by Hansen et al. (17), who foundthat 10–20 training profiles were sufficient for cardiacCINE imaging. Later studies have established a “conven-tion” of 11 training profiles (18,19). This convention wasadopted in the present study. However, while previousstudies of k-t BLAST and k-t SENSE have focused primar-ily on 5-fold acceleration, our study goes a step further andcompares with k-t PCA using 8-fold data acceleration.

Principal Component Analysis

PCA is a commonly used method that reduces highlydimensional datasets to lower dimensionality by extract-ing and exploiting relevant correlations within the data.Assuming that the image series contains nx pixels alongthe frequency-encoding direction, ny pixels along thephase-encoding direction, and nf time frames, the trainingdata in x-f space can be rearranged in an nxny�nf matrix,denoted Ptrain. PCA decomposes this matrix, such that:

FIG. 1. Illustration of the signal aliasing and data constraint in k-tPCA. The k-t space is undersampled on a sheared grid (a), resultingin a sheared point spread function (PSF) in the Fourier reciprocal x-fspace (b). The reconstructed x-f object (d) exhibits an aliasingpattern that corresponds to the convolution of the true x-f object (c)with the PSF. For 4-fold acceleration, four pixels in the true objectfold onto a single pixel in the aliased object. k-t BLAST reconstructsthe true object by distributing the signal of each aliased pixels to itsfour aliasing pixels based on an estimate of the signal covariance. Ink-t PCA, this process is further facilitated by the constraint that eachfrequency profile in the true object must be a linear combination ofa number of basis functions, called principal components (e).

FIG. 2. The k-t sampling pattern used for k-t PCA is composed of atraining set (a) and an undersampled set (b), which are acquiredsimultaneously (c).

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Ptrain � WtrainB, [1]

where Wtrain and B are matrices of size nxny�npc andnpc�nf, respectively, and npc represents the number ofprincipal components (PCs) used in the subsequent recon-struction. The rows of B contain the PCs, which are spa-tially invariant basis functions with which to reconstructevery temporal frequency profile present in the trainingdata. Each row of Wtrain contains the (temporally invariant)weighting coefficients of the PCs for a specific spatiallocation in the training data. We refer to this representa-tion as the x-pc space representation. The transformationbetween x-f space and x-pc space is illustrated in Fig. 3 fora fully sampled dataset.

PCA is optimal in the sense that it minimizes the meansquared error between the true and compressed data forany given data reduction. Thus, most of the variationwithin the data is typically modeled within the first fewPCs. By restricting the x-pc representation to the first fewPCs, the corresponding x-f data can be represented in amuch sparser fashion (i.e., with fewer degrees of freedom).In this way, k-t PCA can put very strict constraints on thedata reconstruction if necessary. The example in Fig. 3euses only the first 10 PCs.

Signal Encoding

The most fundamental assumption underlying k-t PCA isthat the true x-f data P are given by:

P � WB [2]

where B is derived from the training data as explainedabove, and W is the x-pc representation of P. In otherwords, we exploit that the training data are acquired withfull temporal resolution and use the PCs derived fromthese data. By explicit calculation of Eq. [2] it follows thatthe signal at location (x, fm) is given by:

P�x, fm� � �i�1

npc

W�x, i�B�i, fm� � w�x� � b�fm� [3]

where w(x) denotes the row of W corresponding to thespatial location x, and b(fm) is the fm’th column of B.

As described in Ref. (10), the aliased intensity at loca-tion (x, fm) is given by the sum of four pixel intensities inthe true object in accordance to the sampling PSF:

Palias�x, fm� � 1�P�x1, fm,1�P�x2, fm,2�P�x3, fm,3�P�x4, fm,4�

�, [4]

where 1 � [1 1 1 1], and (xi, fm,i) denotes the position of thei’th aliasing pixel in the true object. The exact locations ofthe aliases depend on the sampling PSF, which in turndepends on the rotation and shearing of the k-t sampling

FIG. 3. Overview of the linear spaces used in k-t PCA for a typical series of free-breathing myocardial perfusion images. Inverse Fouriertransform of the k-space data (a) yields the image series (b). A time profile for one frequency-encoding position is shown in (c). Applyinginverse Fourier transform along time t yields the x-f representation (d). Applying principal component analysis (PCA) yields the x-pcrepresentation (e), which consists of a set of spatial weighting coefficients and a set of principal components. Each temporal frequencyprofile in x-f space is given by a linear combination of the principal components (i.e., the sum of the principal components, weighted by theirrespective weighting coefficient).

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pattern (14). Equation [4] is the signal encoding equationused in k-t BLAST. Clearly, this equation is underdeter-mined.

Combining Eqs. [3] and [4], we get:

Palias �x, fm� � 1Bmwx [5]

where

Bm � �bT�fm,1� 0 0 0

0 bT�fm,2� 0 00 0 bT�fm,3� 00 0 0 bT�fm,4�

� [6]

and

wx � �wT�x1�wT�x2�wT�x3�wT�x4�

�. [7]

Here, w(xi) is the row of W corresponding to the spatialposition of the i’th aliasing pixel, and analogously b(fm,i) isthe column of B containing the location of the temporalfrequency of the i’th aliasing pixel. Superscript T denotesthe matrix transpose. Importantly, since the coefficientmatrix W is temporally invariant, there is an expressionsimilar to Eq. [5] for each temporal frequency. The result-ing system of equations can be written in matrix form as:

Palias,x � �Palias�x, f1�Palias�x, f2�···Palias�x, fnf�

� � �1B1

1B2···1Bnf

�wx � Ewx. [8]

This is the signal encoding equation of k-t PCA. Compar-ison with Eq. [4] shows that k-t BLAST encodes one posi-tion in the aliased x-f space at a time, whereas k-t PCAencodes the aliasing simultaneously for all temporal fre-quencies at a given spatial location. Thus, the signal en-coding equation of k-t PCA is overdetermined if the num-ber of PCs (npc) is chosen sufficiently low. This is the maindifference between the two methods.

Image Reconstruction

Equation [8] can be solved using any least squares method.Denoting the signal covariance matrix Mx

2 (see definitionbelow), we used the regularized least squares solution:

wx � Mx2EH�EMx

2EH � �I��Palias,x [9]

where superscript � indicates the Moore–Penrose pseudo-inverse, and superscript H is the conjugate transpose. Idenotes the identity matrix, and � is a regularization pa-rameter used to stabilize the matrix inversion. Large valuesof � will smooth the solution, whereas too low valuesamplify noise. Therefore, the regularization parametermust be chosen with care. The signal covariance was cal-culated from the training data:

Mx2 � diag��wtrain,x�2� [10]

where wtrain,x is constructed from the rows of Wtrain, as inEq. [7].

By solving Eq. [8] for all positions x, we can determinethe x-pc representation W from which the true x-f objectP � WB can be recovered. An important observation aboutEq. [8] is that if we substitute B with the unit basis (i.e., theidentity matrix), then W � P and Eq. [8] becomes mathe-matically equivalent to the solution of the k-t BLAST prob-lem (10). This shows that k-t PCA and k-t BLAST arerelated simply by the choice of temporal basis function.

Assuming that the coil sensitivities are time-invariant,sensitivity encoding is incorporated in k-t PCA simply byreplacing the vector 1 in Eq. [5] with the sensitivity matrixS, as defined in (3):

Palias�x, fm� � S�P�x1, fm,1�P�x2, fm,2�P�x3, fm,3�P�x4, fm,4�

� � SBmwx. [11]

This results in a system of equations similar to Eq. [8],which can be solved using Eq. [9]. This is the reconstruc-tion equation for k-t PCA/SENSE. If the coil sensitivitiesare time-varying, they can be estimated as described byKellman et al. (11), and Eq. [11] can be modified accord-ingly.

Implementation Details

In order to reduce ringing artifacts, the training data werefiltered in k-space along the phase-encoding direction us-ing a Hamming filter. No temporal filtering was applied tothe training data. The interleaved training acquisition usedin this study does in principle allow substituting the train-ing data into the reconstructed images for improved dataconsistency. However, to keep the results more general,training plug-in was not utilized in the present study.

The regularization parameter � was calculated accordingto:

� � �1nf

�i�1

npc

�BHB�i,i, [12]

where � is the noise variance in x-f space, and (BHB)i,i

denotes the i’th diagonal element of BHB. This equationreflects the noise amplification occurring when transform-ing data from x-f space to x-pc space using the first npc

principal components.

MATERIALS AND METHODS

Numerical Simulations

In order to identify applications where k-t PCA is prefer-able over k-t BLAST, and vice versa, a numerical phantomwas implemented that allowed simulation of three impor-tant cases from cardiac MRI: 1) ECG-gated, breath-hold,CINE imaging; 2) ungated, free-breathing, real-time imag-ing; and 3) ECG-gated, free-breathing, myocardial perfu-

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sion imaging. The phantom was designed to emulate thetwo main components of a typical 2D slice, namely, thestationary chest wall and the dynamic heart, as shown inFig. 4a. Fig. 4b shows a selected time profile for one fre-quency-encoded position for each of the three cases. TheCINE images show the beating motion of the heart over onecardiac cycle. The real-time images exhibit cardiac andrespiratory motion over eight cardiac cycles and two re-spiratory cycles. Finally, the perfusion images depict thecontrast bolus passage while the heart is moving due torespiration. Figure 4c shows the x-f representation of eachtime profile with full sampling of k-t space. Evidently thethree datasets show a similar extent in x-f along the spatialdimension, but their temporal frequency contents are dif-ferent. These differences result in different degrees of sig-nal overlap in the aliased x-f space, as demonstrated in Fig.4d for 8-fold undersampling. In particular, the CINE datahave a narrower temporal bandwidth than the two otherdata sets and consequently demonstrates fewer signaloverlaps.

Each of the three image series was generated as if ac-quired with a matrix size of 192 � 192 and 48 time frames.

A single receive coil with uniform sensitivity was as-sumed. The fully sampled k-space data were formed byapplying the Fourier transform along both spatial dimen-sions of the image series. Normally distributed noise witha standard deviation of 10% of the mean intensity ink-space was then added to the data. Based on these k-spacedata, the reduced datasets used in the subsequent simula-tions were assembled by collecting phase-encoding linesfrom appropriate timepoints. To compare different accel-eration factors, we performed reconstructions with 11training profiles and acceleration factors ranging from 2 to16. Similarly, to compare the influence of training dataresolution, images were reconstructed with 8� accelera-tion using 3, 5, 7, 11, 21, 48, 96, and 192 training profiles.All k-t PCA reconstructions were conducted with 10 PCs.In each case the reconstruction error was reported as theroot-mean-square (RMS) error relative to the fully sampleddata.

In Vivo Simulations

To investigate whether the findings of the numerical sim-ulations apply to in vivo experiments, we compared the

FIG. 4. Numerical phantom usedfor the numerical simulations. Thecompartments of the phantomare shown in (a). A time profile forone frequency-encoding positionis shown in (b) for each of thethree cardiac cases: CINE imag-ing, real-time imaging, and perfu-sion imaging. The x-f representa-tions are shown below with fullsampling (c) and 8-fold under-sampling (d).

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performance of k-t PCA, k-t BLAST, k-t PCA/SENSE, k-tSENSE by simulations using a set of myocardial perfusionimages acquired in a pig. The data were acquired on a 1.5T whole-body MR system (Gyroscan Achieva, PhilipsHealthcare, Best, the Netherlands) equipped with a five-element cardiac receive coil. We imaged the first-pass ofthe contrast agent in the short-axis plane using a saturationrecovery, spoiled gradient echo sequence (field of view[FOV] � 256 � 256 mm2, slice thickness � 10 mm, matrixsize � 128 � 96, TR � 2.6 ms, TE � 1.3 ms, flip angle �15°). Imaging continued for 48 consecutive cardiac cycles.The pig was fully anesthetized during the whole experi-ment and the contrast agent was administered slowlythrough a vein in the outer ear. The animal respirator wasswitched on during the entire experiment, thereby induc-ing respiratory motion of the heart. This type of data hasproven demanding to reconstruct with k-t SENSE, partic-ularly when the patient fails to hold his/her breaththroughout the entire experiment (19). The pig experi-ments were conducted as part of an already ongoing studyapproved by the local ethical committee.

As for the numerical phantom, the in vivo simulationswere performed using 11 training profiles and accelerationfactors ranging from 2 to 16. To compare the influence ofdifferent training resolutions, we also performed recon-structions with 8-fold acceleration using 3, 7, 11, 21, 48,and 96 training profiles. In addition, k-t PCA and k-t PCA/SENSE reconstructions were performed with 1, 5, 10, 15,20, and 48 PCs to assess the influence of varying number ofPCs. In this case we used 8-fold acceleration and 11 train-ing profiles. For k-t PCA and k-t BLAST the reconstructionwas first performed individually for each of the five coils.In order to achieve the optimum signal-to-noise ratio, thefinal coil-combined dataset was obtained by summing theindividual coil datasets, normalized by their correspond-ing coil sensitivity map. In all cases, including k-t PCA/SENSE and k-t SENSE, the coil sensitivities were calcu-lated using the sum-of-squares coil combination of thetemporal average images as the reference. This procedureis similar to the one described in Ref. (12).

Experimental Verification

To verify the proposed method on undersampled in vivodata, an additional myocardial perfusion experiment wasconducted on another pig using the same MR hardware.The same imaging sequence was used, except that the FOVand matrix size were set to 288 � 288 mm2 and 256 � 256,respectively. This resulted in an increase of the spatialresolution from 2.0 � 2.67 mm2 to 1.25 � 1.25 mm2. Thetraining data and undersampled data were collected with11 training profiles and 8-fold acceleration, resulting in anet acceleration factor of 6.15. A total of 64 frames wereacquired. The coil-combined reconstructions of k-t PCAand k-t BLAST were performed as outlined above, and thecoil sensitivities were estimated using the same procedure.

RESULTS

Numerical Simulations

Figure 5 shows representative results of the numericalsimulations using 8-fold acceleration and 11 training pro-

files. The results were obtained from the perfusion case. Asshown in Fig. 5a k-t PCA causes less aliasing than k-tBLAST when viewing the data in the spatial domain. Thearrow indicates the location of an alias that is almostcompletely removed with k-t PCA, but not with k-tBLAST. The time profiles shown in Fig. 5b confirm thatthe aliasing reduction obtained with k-t PCA is consistentacross the entire time series. This is particularly clear inthe ventricular blood pool, as indicated by the arrows.Finally, the corresponding x-f data in Fig. 5c show that k-tPCA reconstructs the temporal frequency content moreaccurately than k-t BLAST. In particular, k-t PCA recon-structs the higher temporal frequencies, which are nulledby k-t BLAST (see arrows). This shows that k-t PCA causesless temporal blurring than k-t BLAST.

Figure 6 summarizes the results of the numerical simu-lations. With 11 training profiles and varying accelerationfactors (left column), the reconstruction error increasedwith the acceleration factor for both k-t BLAST and k-tPCA. However, the rate at which the error increased waslower for k-t PCA than k-t BLAST. For 2-fold acceleration,k-t BLAST had the overall lowest error, but for high accel-eration factors k-t PCA was superior. Considering acceler-ation factors 4–16, the overall improvement was mostpronounced for the perfusion case. No improvement wasseen in the CINE case below 10-fold acceleration.

FIG. 5. Representative results of the numerical simulations. Thedata correspond to a free-breathing perfusion study acquired with8-fold acceleration and 11 training profiles (and 10 principal com-ponents for k-t PCA). The reconstructed anatomy corresponding toframe number 17 is shown in (a) for the true data, k-t BLAST, and k-tPCA. Time profiles aligned through the heart are shown in (b), andthe corresponding x-f representations are shown in (c). The arrowsindicate areas where k-t PCA demonstrates less aliasing or highertemporal resolution than k-t BLAST.

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The right column of Fig. 6 shows the reconstructionerrors for 8-fold acceleration and different amounts oftraining. As for k-t BLAST, most of the benefit of thetraining data was achieved for k-t PCA within the first10–20 training profiles. However, the error for any givennumber of training profiles was lower for k-t PCA. In theperfusion case, for example, k-t PCA with just five trainingprofiles resulted in a lower error than k-t BLAST with fulltraining (i.e., 192 training profiles). This shows that k-tPCA requires less training data to obtain the same level ofaccuracy as k-t BLAST.

In Vivo Simulations

Figure 7 shows the quantitative results of the in vivosimulations. With varying acceleration factors (Fig. 7a) thesame behavior was observed as for the numerical simula-tions, namely, that k-t BLAST performed best for 2-foldacceleration, whereas k-t PCA was superior at higher ac-celeration factors. The accuracy of the two methods wassimilar for 3-fold acceleration. As expected, k-t SENSEperformed slightly better than k-t BLAST, whereas theperformance of k-t PCA and k-t PCA/SENSE was verysimilar. This shows that the temporal constraint obtained

with PCA provides more vital information to the recon-struction than the coil sensitivities.

With varying number of training profiles (Fig. 7b), thereconstruction errors of k-t PCA and k-t PCA/SENSE wereconsistently lowest. In particular, k-t PCA and k-t PCA/SENSE with 7 training profiles corresponded in accuracyto k-t SENSE with full training (i.e., no reduction on dataacquisition). Also, the results show that k-t PCA and k-tPCA/SENSE were more accurate with just 3 training pro-files compared to k-t SENSE with 11 training profiles. Forthe matrix size used in the simulations, this suggests thatthe net acceleration factor can be increased from 4.4 withk-t BLAST/SENSE to 6.6 with k-t PCA or k-t PCA/SENSE,while still obtaining an improvement in accuracy.

Finally, Fig. 7c shows that the optimum number of PCsfor perfusion imaging was about 10. In fact, the reconstruc-tion error increased when using more than 10 PCs. Thisbehavior was expected, because PCA models the majorvariations of the data within the first few PCs, whereas thehigher numbered PCs primarily model noise.

Figure 8 shows representative data reconstructed with8-fold acceleration and 11 training profiles (and 10 PCs fork-t PCA and k-t PCA/SENSE). The selected image frame in

FIG. 6. Summary of the results ofthe numerical simulations. Eachplot shows the relative RMS errorof k-t BLAST and k-t PCA as afunction of the acceleration factor(left column) or the number oftraining profiles (right column).Each row corresponds to a differ-ent cardiac case. The improve-ment of k-t PCA over k-t BLAST ismost pronounced for the perfu-sion case (bottom row), whereasthe two methods perform almostequally well for the CINE case (toprow).

712 Pedersen et al.

Fig. 8a illustrates the typical differences that we observedbetween k-t BLAST/SENSE and their respective k-t PCAcounterparts. Apart from a bright spot (see leftmost arrow)the images reconstructed with k-t PCA and k-t PCA/SENSEshowed good resemblance with the true image. In contrast,the k-t BLAST and k-t SENSE images were less comparableto the true image and showed more systematic errors. Inparticular, the edges of the heart appeared blurred (seerightmost arrow). In this case, the blurring results from aslight left-to-right respiratory movement of the heart,which is not faithfully reproduced by k-t BLAST/SENSE.

Figure 8b shows a signal intensity time curve derivedfrom a region-of-interest (ROI) placed in the left ventricu-lar blood pool. The corresponding curve obtained from thetrue image series is overlaid as a dotted line for compari-

son. The results confirm that k-t PCA and k-t PCA/SENSEshowed good resemblance with the true data, whereastemporal smoothing was observed for both k-t BLAST andk-t SENSE (see arrows). As expected, the temporal smooth-ing was significantly worse for k-t BLAST than k-t SENSE.Again, k-t PCA/SENSE yielded only little improvementover k-t PCA. All four methods failed to exactly recon-struct the peak of the bolus passage, showing that the k-tPCA methods also suffer from mild temporal blurring.

Experimental

Figure 9a shows a representative image frame recon-structed from the undersampled perfusion data using k-tBLAST and k-t PCA. To facilitate the comparison, thedifference image (scaled by a factor of two) is also shown.The corresponding images of k-t SENSE and k-t PCA/SENSE are shown in Fig. 9b. Due to the slow injection rateof the contrast agent, the temporal frequency content of thedata was located relatively close to the DC component,thus promoting sparseness in x-f space. For this reason, allreconstructions were fairly successful and no major alias-ing artifacts were visible. Yet we observed differencesbetween k-t PCA and k-t BLAST, as well as between k-tPCA/SENSE and k-t SENSE. First, there was an apparentintensity drop in both the left and right ventricular bloodpools for k-t BLAST and k-t SENSE. Also, the edges of theheart (see rightmost arrow) and spot-like objects (see leftarrow) appeared sharper using the presented method.These effects indicate that the k-t BLAST and k-t SENSEimages were temporally smoothed compared to their re-spective k-t PCA counterparts.

Figure 9c depicts the signal intensity curve derived froman ROI located in the left ventricular blood pool. The mostimmediate observation is that the k-t BLAST and k-tSENSE curves exhibit an intensity overshoot near the be-ginning of the image series and an undershoot toward theend of the series, whereas the k-t PCA and k-t PCA/SENSEcurves do not. This is a well-known characteristic of k-tBLAST and k-t SENSE that occurs if there is a temporaldiscontinuity between the first and last frames of the imageseries. In general, the intensity curves confirmed that k-tBLAST appeared temporally smoothed compared to k-tPCA. This effect was also seen for k-t SENSE and k-tPCA/SENSE, although it was less prominent.

DISCUSSION

We have presented the theory of k-t PCA and compared itsperformance with k-t BLAST based on simulations and invivo experiments. Both methods can be viewed conceptu-ally in the framework of k-t PCA simply by a change oftemporal basis. Specifically, k-t BLAST uses the unit basis,whereas k-t PCA utilizes a temporal basis tailored to thetraining data. The advantage of the unit basis is that it hasmore degrees of freedom which, in principle, allow mod-eling of arbitrary dynamic object. However, since the sig-nal encoding equation is always underdetermined, k-tBLAST reconstruction relies entirely on the estimated sig-nal covariance, which in turn compromises temporal res-olution. On the other hand, k-t PCA constrains the shape ofthe temporal frequency content, inducing an interdepen-

FIG. 7. Summary of the results of the in vivo simulations. The plotsshow the relative RMS errors for increasing acceleration factors (a),number of training profiles (b), and number of principal components(c).

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dence between the signal at different frequency compo-nents. This results in a signal encoding equation that iseffectively overdetermined.

In this work we focused on 2D imaging, but extendingthe concept of k-t PCA to 3D imaging is straightforward.Similar to k-t BLAST, k-t PCA also extends to non-Carte-sian sampling, since this only requires the existence of anencoding matrix that calculates the sampled points in k-tspace given the x-pc space representation. In addition, thepresented framework potentially allows incorporating anyprior knowledge in the form of temporal basis functions,such as those derived from respiratory navigators or thedesign matrices used in functional MRI. Also, k-t PCA canincorporate the concept of adaptive regularization pre-sented by Xu et al. (20), which addresses a well-knownsignal nulling problem that occurs if aliasing falls onto thetemporal DC.

The numerical simulations revealed several importantdifferences between k-t PCA and k-t BLAST. Most impor-tant, the reconstruction error of k-t PCA grows at a slowerrate at increasing acceleration factors. This implies thatwhen the acceleration factor exceeds a certain limit, theproposed method is more accurate. The limit depends onthe number of signal overlaps in the aliased x-f data, whichin turn depends on the spatiotemporal characteristics of

the imaged object. The simulations suggest that the limit ishigh for CINE images, but low for real-time images andfree-breathing myocardial perfusion images. Nevertheless,k-t BLAST consistently performed best at 2-fold accelera-tion, where the signal overlaps are almost completely sep-arated in the aliased x-f data. Such signals are fairly easy toseparate, and consequently the loss in accuracy with k-tPCA at 2-fold acceleration is due to the inaccuracy of thePCA basis relative to the unit basis.

Although k-t PCA performs better at higher accelerationfactors, the accuracy of the method inevitably falls below aclinically acceptable level if the acceleration factor is toohigh. In our experience, data acquisition with 8-fold ac-celeration represents a good trade-off, where the improve-ment over k-t BLAST is remarkable, and the quality of thereconstructed images seems sufficient for clinical pur-poses. Focusing on this acceleration factor, the numericalsimulations showed another interesting observation,namely, that the reconstruction error with k-t PCA wasconsistently lower for any given training size. Compared tok-t BLAST, this property may translate to either improvedreconstruction accuracy for a given net acceleration factor(i.e., by maintaining the number of training profiles), or itmay yield a lower net acceleration factor with the same

FIG. 8. Results of the in vivo simulations using 8-fold acceleration and 11 training profiles (and 10 principal components for k-t PCA andk-t PCA/SENSE). The reconstructions of a representative frame, as well as the corresponding error images, are shown in (a) for k-t SENSE,k-t BLAST, k-t PCA/SENSE, and k-t PCA. The rightmost arrow indicates an area where the k-t SENSE and k-t BLAST images are blurred,whereas the leftmost arrow indicates a bright spot that all four methods fail to accurately reconstruct. The reconstructed signal curvescorresponding to a ROI placed in the left ventricular blood pool are shown in (b). The arrows indicate time points where the reconstructedtime curves are temporally blurred.

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reconstruction accuracy (i.e., by lowering the number oftraining profiles).

The results of the in vivo simulations support the obser-vations made from the numerical simulations of the per-fusion case. As expected, the image data in Fig. 8 showthat the improvement with k-t PCA manifests mainly asbetter temporal resolution. For perfusion imaging thismeans that the contrast agent passage is more accuratelyreproduced, but also that the images contain fewer arti-facts due to respiratory movement. In practice, both k-tPCA and k-t BLAST will benefit from reusing the trainingdata, i.e., by utilizing training plug-in. The improvementdepends, however, on how many training profiles areused. This was not investigated in the present study.

Another interesting finding of the in vivo simulations isthat k-t PCA does not appear to benefit significantly fromincorporating sensitivity encoding. This may have impor-tant practical implications, because the reconstructiontime of k-t PCA/SENSE tends to be quite long. For exam-ple, the reconstruction time for each frequency encodedposition of the in vivo simulations was 0.8 sec with k-tSENSE and 12 sec with k-t PCA/SENSE. Performing thereconstruction for each of the five coils separately (i.e.,using k-t PCA) reduced the computation time to 4 sec,without compromising the accuracy of the reconstruction.In general, the reconstruction time of k-t PCA/SENSE isdetermined by the time it takes to invert the matrix in Eq.[9]. This matrix has size (ncnf)�(ncnf), where nc denotes thenumber of coils.

The in vivo experiments demonstrated the feasibility ofk-t PCA and k-t PCA/SENSE in actually undersampleddata. We did not observe any residual aliasing artifacts,which was expected due to the slow injection of the con-trast agent. In comparison, the temporal resolution of theproposed methods seemed superior to k-t BLAST and k-tSENSE, in agreement with the findings of the simulations.We expect, therefore, that k-t PCA or k-t PCA/SENSE willimprove measurements of myocardial perfusion obtainedfrom accelerated data, but further studies are required toinvestigate this in detail.

CONCLUSIONS

The framework of k-t BLAST can be considerably im-proved by exploiting more of the relevant signal correla-tions available in the training data. These correlations arerepresented as temporal basis functions, derived usingPCA. The k-t PCA technique utilizes these basis functionsto reduce the number of degrees of freedom in the recon-struction, thereby permitting a better recovery of the over-lapping signals in the aliased object. The k-t PCA and k-tPCA/SENSE methods can be used for a wide range ofapplications, including those exhibiting a broad temporalbandwidth (e.g., free-breathing myocardial perfusion andbrain perfusion). The concept is applicable to other tem-poral basis functions than those derived using PCA, andk-t BLAST can be viewed conceptually in the framework

FIG. 9. Results of the in vivo ex-periment using 8-fold accelerationand 11 training profiles (and 10principal components for k-t PCAand k-t PCA/SENSE). A typical re-constructed frame is shown in (a)for k-t BLAST and k-t PCA. Alsoshown is the difference image be-tween the two reconstructions. Thecorresponding results for k-tSENSE and k-t PCA/SENSE areshown in (b). The arrows indicateareas where the k-t BLAST/SENSEreconstruction appears less sharpthan its k-t PCA counterpart. Thereconstructed signals correspond-ing to a ROI located in the left ven-tricular blood pool are shown in (c).The arrows indicate timepointswhere k-t BLAST/SENSE appearstemporally blurred compared withits k-t PCA counterpart.

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of k-t PCA simply by a change of temporal basis. Thus, k-tPCA provides a convenient framework for incorporatingexplicit prior knowledge into the reconstruction in theform of temporal basis functions.

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