Data Convolution and Combination Operation (COCOA)for Motion Ghost Artifacts Reduction

Feng Huang,1* Wei Lin,1 Peter Bornert,2 Yu Li,1 and Arne Reykowski1

A novel method, data convolution and combination operation,is introduced for the reduction of ghost artifacts due tomotion or flow during data acquisition. Since neighboring k-space data points from different coil elements have strongcorrelations, a new ‘‘synthetic’’ k-space with dispersed motionartifacts can be generated through convolution for each coil.The corresponding convolution kernel can be self-calibratedusing the acquired k-space data. The synthetic and theacquired data sets can be checked for consistency to identifyk-space areas that are motion corrupted. Subsequently, thesetwo data sets can be combined appropriately to produce a k-space data set showing a reduced level of motion inducederror. If the acquired k-space contains isolated error, the errorcan be completely eliminated through data convolution andcombination operation. If the acquired k-space data containwidespread errors, the application of the convolution also sig-nificantly reduces the overall error. Results with simulatedand in vivo data demonstrate that this self-calibrated methodrobustly reduces ghost artifacts due to swallowing, breathing,or blood flow, with a minimum impact on the image signal-to-noise ratio. Magn Reson Med 64:157–166, 2010. VC 2010Wiley-Liss, Inc.

Key words: data consistency; parallel imaging; imagereconstruction; nonrigid motion; ghost artifacts; GRAPPA

In MRI, it usually takes seconds to minutes to acquirethe data needed for image reconstruction. Any patient/system motion during the acquisition will introduce arti-facts in the reconstructed images. Severe motion artifactscould lead to misdiagnosis or reduced patient through-put due to the necessity to repeat scans. Therefore, overrecent years, numerous techniques have been developedto reduce motion artifacts.

Many techniques have been designed to compensatefor pseudoperiodic (1,2) and rigid-body motion (3–7) in aprospective or retrospective manner. Some of these tech-niques have been implemented in commercial MR sys-tems and have been clinically validated. However, theyare inefficient in the case of random or nonrigid motion,which is still a serious problem practically. In someapplications, such as cervical spine MRI, swallowingand coughing may introduce random nonrigid motionartifacts. This has created a demand for techniques thatcan efficiently reduce random nonrigid motion artifacts,with a minimum impact on imaging protocols and imagequality.

One approach to avoid motion artifacts is to speed

up the MR data acquisition in the hope that the neces-

sary data can be acquired before problematic subject

motion occurs. Single-shot imaging methods such as

echo-planar imaging (8) can be used to reduce motion

artifacts but suffers from a limited spatial resolution.

Partially parallel imaging (PPI) techniques (9,10) repre-

sent a very useful alternative, allowing for fast imaging.

Since parallel acquisition provides redundant informa-

tion, PPI techniques have been used for motion detec-

tion and correction. Besides techniques for rigid motion

detection and correction (3,4), PPI techniques take

advantage of the redundancy in the multicoil dataset to

enforce data consistency (11,12). This can also be

applied for nonrigid motion compensation. In Winkel-

mann et al. (11), this idea is pursued based on the ob-

servation that motion artifacts are also sensitivity

encoded. The origin of the motion artifact can be esti-

mated in the spatial domain, using consistency of sensi-

tivity maps and an extended SENSitivity Encoding

(SENSE) reconstruction method can be used to correct

the motion artifact. In Fautz et al. (12), multiple copies

of the full k-space data are reconstructed from multiple

subsets of the acquired full k-space data set. Assuming

that the motion artifacts are incoherent among these

copies, averaging of these copies will reduce the overall

motion artifact level. However, an intrinsic problem

with data consistency–based methods is that they might

be concomitant with a certain signal-to-noise ratio

(SNR) reduction. In Winkelmann et al. (11), regularized

SENSE was proposed to reduce the influence on SNR

at the cost of some residual motion artifacts. Using the

multiple-average method proposed in Fautz et al. (12),

the reconstructed image is compromised by the noni-

deal g-factor (9). When the undersampling rate is high

to produce a sufficiently large number of copies, the g-

factor could introduce significant reduction of SNR.

The purpose of this work is to reduce ghost artifacts

with minimum impact on SNR.It has been shown in k-space-based PPI techniques

(10,13,14) that signal from each k-space point can beapproximated by a linear combination of the signalsfrom its adjacent points from multiple coil elements. Thecombination weights are constant throughout the entirek-space, which is equivalent to convolution in k-space.Instead of a reconstruction with partially acquired data,in this work the convolution is used to produce an extrasynthetic k-space data set from the fully acquired k-spacedata. If motion happens during the data acquisition, thesynthetic and the acquired data sets will show differen-ces, which can be used to locate the motion-corrupteddata. An appropriately combined new k-space data set,which uses the signal either from the synthetic or the

1Invivo Corporation, Gainesville, Florida, USA.2Philips Research Europe, Hamburg, Germany.

*Correspondence to: Feng Huang, PhD, Invivo Corporation, 3545 SW 47thAvenue, Gainesville, FL 32608. E-mail: fhuang@invivocorp.com

Received 17 July 2009; revised 9 November 2009; accepted 21 December2009.

DOI 10.1002/mrm.22358Published online in Wiley InterScience (www.interscience.wiley.com).

Magnetic Resonance in Medicine 64:157–166 (2010)

VC 2010 Wiley-Liss, Inc. 157

measured data sets at each k-space location, has loweroverall artifact level than the individual ones. By opti-mizing the convolution kernel and the combinationscheme, an image with reduced motion artifacts andwell-preserved SNR can be reconstructed. This proce-dure is called data convolution and combination opera-tion (COCOA) in this paper.

THEORY

Error Redistribution Due to Convolution

From parallel imaging, it is known that any k-space sig-nal from multiple coil elements can be approximated bya linear combination (14). Equation 1 demonstrates thisapproximation.

XJ

j¼1

XL

l0¼1

WDkj ;l0Sl0 ðkr � DkjÞ ¼ SlðkrÞ ½1�

Here, J is the number of neighboring k-space signalsused in the linear combination, L is the number of coilelements. Sl0(kr � Dkj) denotes k-space signals from ele-ment l0, with shift Dkj used to approximate signal Sl(kr)from the coil element l. The weights WDkj,l

0 depend onthe coil element sensitivities (14) and the relative shiftDkj. This operation defined by Eq. 1 is a convolution ink-space, with the convolution kernel defined by WDkj,l

0,which can be either calculated from sensitivity maps(14) or through data fitting (10,15).

Now assume that there is an error DSl(kr) at location krand Sl(kr) becomes Sl(kr) þ DSl(kr). If the neighboring k-space signals in Eq. 1 are accurate and if a convolutionkernel is available, then Sl(kr) can be accurately recov-ered by the application of Eq. 1. However, the applica-tion of Eq. 1 will spread the error DSl(kr) to its neighbor-ing signals in the synthetic k-space. Without loss ofgenerosity, a specific convolution kernel using twoupper and two lower adjacent neighbors is used as anexample to explain the error distribution. Using this con-volution kernel, Eq. 1 becomes Eq. 2.

X4

j¼0;j 6¼2

XL

l0¼1

Wj;l0Sl0 ðkr � ðj � 2ÞDkyÞ ¼ SlðkrÞ; ½2�

where Dky is the k-space signal distance along the phase-encoding (PE) direction. And the application of the con-volution at kr � Dky will result in

X4

j¼0;j 6¼1;2

XL

l0¼1

Wj;l0Sl0 ðkr � ðj þ 1� 2ÞDkyÞ

þXL

l0¼1

W1;l0 ðSl0 ðkrÞ þ DSl0 ðkrÞÞ ¼ S0lðkr � DkyÞ: ½3�

Hence, the signal at location kr � Dky in the synthetic k-space would be Slðkr � DkyÞ þ

PLl0¼1 W2;l0DSl0 ðkrÞ. Signals at

locations kr � (j � 2)Dky, j ¼ 0, 3, 4 will be influenced simi-larly and become Slðkr � ðj � 2ÞDkyÞ þ

PLl0¼1 W4�j;l0DSl0 ðkrÞ.

This means the error at location kr is dispersed to its

neighbors in a way that is dependent on the definitionof the convolution kernel. Since the magnitude of theseconvolution kernel weights WDkj,l

0 is far less than 1 ifmore than one neighbor is used from each coil element(14), the error level at each neighbor after convolutionwill be lower than DSl(kr). Therefore, the differencebetween the two k-space data sets before and after con-volution, for a given coil element, will have higher valueat kr and lower value at its neighbors. The magnitude ofthis difference is named convolution difference in thiswork since it is introduced by convolution.

If the acquired k-space data set is corrupted by contin-uous motion, and all k-space signals contain similarerror level, then the convolution will disperse the errorat each location to its neighbors in a lower level. Sincethese errors at different k-space locations are related todifferent motion states and since the impact of each spe-cific motion state will be reduced by the convolution,the overall artifact level will be reduced. In this sce-nario, the convolution difference will not vary signifi-cantly among k-space locations because the errors arewidespread.

Combination of the Acquired and the Synthetic k-Space

It has been shown in the previous section that theacquired and the synthetic k-space data sets have differ-ent error distribution. Therefore, it is possible to producea k-space with error level lower than each individualdata set through a proper combination of two k-spaces.From the knowledge of PPI and the deduction in the pre-vious sections, it can be seen that the convolution canpotentially introduce two kinds of side effects: redistrib-uted error and noise amplification. Therefore, theacquired k-space data should be used in the combined k-space at locations without error to reduce the effect ofnoise amplification. This requires distinguishing thelocations of the motion-corrupted signals. In this work,the convolution difference is proposed to pinpoint whenthe motion has occurred. A combination scheme canthen be decided accordingly. For simplicity, the convo-lution difference is added along coil elements and fre-quency encoding (FE) directions. Hereafter, the convolu-tion difference means the magnitude of the differencebetween the synthetic and the acquired k-space data setsafter summation.

Let var be the ratio between the standard deviationand the mean of the convolution difference. A small varmeans the error levels at all locations are similar. Thisimplies that there is continuous motion or no motion. Abig var means the error levels are greater at some loca-tions. This could be related to sudden motion or periodicmotion/flow. Let err be the ratio between the convolu-tion difference and the magnitude of the correspondingsignal. A small err means that the overall convolutiondifference is low relative to the signal. This could be thecase if the error due to motion/flow is insignificant. Con-trarily, a large err means that the error due to motion/flow is considerable. If both var and err are small, theacquired data should be used as the combined k-space topreserve SNR since the error due to motion is

158 Huang et al.

insignificant. When var is small and err is large, thisindicates that continuous motion. Hence, the syntheticdata should be used as the combined k-space to reduceartifacts. When var is big but err is small, it indicatesthat insignificant sudden or periodic motion/flow hasoccurred. Since the convolution will not introduce sig-nificant redistributed error, the synthetic data should beused as the combined k-space to sufficiently reduce arti-facts. When both var and err are large, it means that sig-nificant sudden or periodic motion/flow has occurred.To reduce the redistributed error due to convolution andto preserve SNR, the combined k-space should use thesynthetic data at locations with large differences formotion reduction, and use the acquired data at otherlocations to preserve SNR and avoid the redistributederror. Based on this analysis, Table 1 is designed as arule of thumb for a dedicated combination scheme. Theempiric values given in Table 1 are used as thresholdsfor the images in our experiments.

Schemes for Minimization of Artifacts and Noise

The previous two sections introduced the procedures ofCOCOA: convolution and combination in k-space. Thereare two kinds of errors in the combined k-space: residualmotion artifacts and amplified noise. Both kinds of errorsare related to the design of convolution kernels. There-fore, the optimization of convolution kernel is firstconsidered.

Different from k-space-based PPI techniques, the goalof convolution here is not to estimate unacquired data.Instead, a separate copy of k-space data is reconstructedfrom the acquired k-space data. Due to this difference,the design of the convolution kernel is different fromthose used in Generalized Autocalibrating Partially Par-allel Acquisitions (GRAPPA) (10,15). To reduce the errordue to motion/flow, the convolution kernel should belarge enough to contain sufficient motion-free data ordata with incoherent motion. Moreover, the kernelshould not contain data acquired in the same or adjacentpulse repetition times (TR) since they are prone to simi-lar coherent motion artifacts, which depend on theactual TR and the physiologic motion in focus. Toreduce the noise amplification due to k-space interpola-tion, the convolution kernel support should contain datawith strong correlations with the data being generated.Usually, closer neighbors have stronger correlation.Therefore, the convolution kernel support should con-tain the closest neighboring k-space points wheneverpossible. In this work, data fitting with the calibrationsignal is used to calculate the convolution kernels. Prac-tically any fraction of the potentially motion-corrupted

k-space data can be used for calibration; therefore, extradata acquisitions are unnecessary. Since the signal nearthe k-space center has a higher SNR than outer regions, anumber of central PE lines from the potentially motion-corrupted data are used for calibration in this work.

Besides the design of convolution kernel, the level ofresidual motion artifact is also influenced by the data ac-quisition scheme. Based on the reasoning in the previoussection, it can be seen that the ideal scenario for thismethod is that the motion-corrupted signals are uni-formly distributed in k-space with sufficient distance. Inthis scenario, the errors due to motion in the synthetic k-space will be evenly distributed, and this will result inan incoherent error in the image space. To pursue theideal scenario, it is proposed to acquire data in an inter-leaved manner. This means that PE lines are dividedinto several equally spaced subsets covering the entire k-space and are acquired one by one. Figure 1 shows oneexample of the acquisition scheme. Each color is for onesubset. Each subset could be acquired in one or morecontinuous TRs. The distance between adjacent lines ineach subset is called the interleaving factor. In this way,the k-space signals acquired in each subset, which arecorrupted by similar motion, are evenly distributed in k-space. It should be mentioned that the acquisitionscheme in Fig. 1 is preferred but not necessary for theproposed artifact reduction technique.

Following these rules, a convolution kernel can bedesigned, as shown in Fig. 1. Here it is assumed that thedata set is acquired with an interleaving factor of 4. InFig. 1, the FE direction is perpendicular to the paper. Toenhance the performance of the operation, the convolu-tion kernel could be extended to multiple data pointsalong the FE direction. In this work, three adjacentsource data points along the FE direction are used in thekernel. From Fig. 1a, it can be seen that the reconstruc-tion is analogous to PPI, with an acceleration factor of1.3. Therefore, it is expected that the second kind oferror due to parallel-imaging-based data interpolationwill be very low.

Combination of COCOA and PPI for FurtherReduced Artifacts

Partially parallel acquisition has been routinely used forfast imaging and motion reduction. The proposed opera-tion can be directly combined with PPI and applied topartially acquired data set. We propose to apply COCOAbefore other reconstruction steps. The same rules for ker-nel design provided in the previous section can beadopted. Figure 1b shows the example when the inter-leaving factor is 4 and the acceleration factor is 2.

Table 1Combination Scheme

var (standard deviation/mean) err (difference/signal) Combination scheme

Small (�0.06) Small (�0.01) Acquired

Small (�0.06) Big (>0.01) SyntheticBig (>0.06) Small (�0.01) Synthetic

Big (>0.06) Big (>0.01) Use synthetic data at locations with big difference (>mean);use acquired data at other locations

COCOA for Motion Compensation 159

Different from GRAPPA, COCOA does not need extraautocalibration signal lines for convolution kernel calcu-lation. Hence, it can be easily adopted with the data forSENSE. After the application of the proposed operator, amotion-corrected partial k-space data set can be gener-ated. The PPI reconstruction technique can then beapplied to the corrected data set subsequently. Since theoriginal k-space data set is partially acquired, highernoise amplification due to convolution is expected.

MATERIALS AND METHODS

Data Simulation and Acquisition

The experiments were divided into two groups. In thefirst group of experiments, the simulated data sets wereused to quantitatively show the performance of the CO-COA approach in comparison to the multiple-averagemethod (12). The second group of experiments used invivo data sets to show the performance of the proposedmethod for several applications.

Two data sets were simulated using the Shepp-Loganphantom and the sensitivity maps of an eight-channelhead coil (Invivo Corporation, Gainesville, FL). Thesimulated matrix size was 256 � 256. It was assumedthat the data were acquired with four interleaved subsetsin both data sets, as shown by the different color in Fig.1a. In the first data set (simulated data set 1), the firstout of the four subsets corresponded to the phantom

with a moved object, shown by the arrow in Fig. 2b. Tosimulate nonrigid motion, the vertical and horizontalaxes of that object were extended 2.5 pixels and 5 pixels,respectively, during acquisition of the first subset. Nonoise was added to these data. In the second data set(simulated data set 2), it was assumed that there was nomotion. Only random noise with gaussian distributionwas added to simulate the degradation due to noise.

Four in vivo data sets were acquired on a 3.0-TAchieva scanner (Philips, Best, The Netherlands). Twocervical spine data sets were acquired using a 16-elementneurovascular coil (Invivo Corporation). T2-weightedmultislice turbo spin-echo acquisitions were performedwith an interleaving factor of 4, i.e., four interleaved sub-sets. The acquisition parameters for this data set includefield of view 200 � 248 mm2/160 � 248 mm2 in the sag-gital orientation, matrix size 256 � 256/200 � 248, TR/echo time 3314/120 ms, flip angle 90�, slice thickness 3mm, echo train length ¼ 16, PE direction anterior-poste-rior/superior-inferior. Each subset was acquired by fourcontinuous echo trains. During the acquisition of thefirst data set (spine data set 1), the volunteer was askedto swallow once every 10–15 sec. The volunteer wasasked to swallow, cough, and move his neck randomlyduring the next acquisition (spine data set 2).

One set of shoulder data was acquired by T1-weightedturbo spin-echo sequence with an interleaving factor of4. A four-channel shoulder coil (Philips) was used. The

FIG. 1. Examples of convolution kernels. It is assumed that the data set has signals from four coil elements. Data are acquired with aninterleaving factor of 4. Colors indicate the different subsets. Except for the shaded area, all of the source data points in the black boxare used to reconstruct a separate copy of the signal in the gray box through convolution. Solid circles are source data points. This pro-

cess is then repeated by stepping through the entire k-space to generate a synthetic copy of the data set with redistributed errors. a:Convolution kernel for fully acquired data set. b: Convolution kernel for partially acquired data set.

160 Huang et al.

acquisition parameters included field of view 130 �130 mm2 in the coronal orientation, matrix size 336 �336, TR/echo time 521/20 ms, flip angle 90�, slice thick-ness 3 mm, echo train length ¼ 3; PE direction was supe-rior-inferior. The volunteer was asked to breathe heavilyduring acquisition, which introduced shoulder motion.

To examine the performance of the proposed methodfor the reduction of flow artifacts, a non-Electrocardiog-raphy (ECG) synchronized abdominal breath-hold dataset was acquired, using a 32-element cardiac coil (InvivoCorporation). A dual-echo gradient echo imagingsequence (field of view 375 � 375 mm2, matrix size 204� 256, TR 90 ms, echo time 1/echo time 2 2.3/5.8 ms,flip angle 80�, slice thickness 7 mm) was used for dataacquisition. PE direction was anterior-posterior.

Implementation of Reconstruction Algorithms

Convolution kernels defined in Fig. 1 were used for alldata sets (with interleaving) except the abdominal dataset (without interleaving). Convolution kernels were cal-culated through data fitting with the central 64 k-spacelines from the potentially motion-corrupted data. Theempiric thresholds defined in Table 1 were used for thecombination scheme. By applying the convolution andcombination operation, the new k-space data sets wereproduced for each channel. After Fourier transform, finalimage reconstruction was achieved by using the squareroot of sum of squares approach for image combination.

For comparison, the GRAPPA-based multiple-averagemethod provided in Fautz et al. (12) was implemented.

The central 64 k-space lines were used as Auto-CalibrationSignal (ACS) lines for GRAPPA. The size of the GRAPPAconvolution kernel was 4 � 5. Each interleaf was used toreconstruct one copy of the data set for average.

To validate the performance of COCOA with partiallyacquired data, the second spine data set was artificiallydownsampled to simulate a GRAPPA acquisition. Thesimulated undersampling rate was 2 and the number ofACS lines for GRAPPA was 32. The undersampled datasets were processed by convolution and combinationoperation for motion reduction before final GRAPPAreconstruction. Figure 1b shows the design of the convo-lution kernel used for this purpose. The convolution ker-nel for COCOA was calculated using the central 64 k-space lines from the equally downsampled data. Thesize of the GRAPPA convolution kernel was also 4 � 5.

Root mean square error (RMSE) and error map wereused for quantitative evaluation for simulated data sets.The error map is the magnitude difference between thereconstruction and the motion/noise-free referenceimage. It shows both the magnitudes and locations ofthe errors in the reconstruction. For the in vivo data sets,the difference map and g-factor map were used for evalu-ation. The difference map depicts the difference in mag-nitudes between the reconstructions before and aftermotion correction. It can be used to qualitatively evalu-ate the reduction of motion artifacts and the preservationof diagnostically useful information. The g-factor mapsof convolution kernels were calculated using Eq. 5 inBreuer et al. (16) to demonstrate the impact on SNR dueto convolution. The values in the g-factor maps show the

FIG. 2. Motion correction with simulated phantom data. a: The reference image without motion artifacts or noise. e: Plot of the aver-aged convolution difference of each subset. The ‘‘convolution difference’’ means the magnitude of the difference between the synthetic

(by convolution) and the acquired (corresponding to (b)) k-space data sets after summation along channels and the FE direction. Thedashed horizontal line shows the mean of these four averaged differences. b: The motion-corrupted image. It is assumed that the verti-cal and horizontal axes of the object shown by the arrow extend by 2.5 and 5 pixels, respectively, during acquisition of the first subset.

c,d: The results obtained by COCOA and multiple average method, respectively. Arrows in (d) show the locations with residual motionartifact or GRAPPA reconstruction errors; (f)-(g) are the error maps of (b)�(d), which are the magnitude of the difference with the refer-

ence image (a). These error maps have been brightened 10 times for better visualization.

COCOA for Motion Compensation 161

corresponding noise amplification during reconstruction.Hence, smaller g-factor value means less impact onSNR.

All methods were implemented in the MATLAB pro-gramming environment (MathWorks Inc., Natick, MA).The MATLAB codes were processed on an HP xw4100workstation (Hewlett-Packard Company, Palo Alto, CA)with a 3.2-GHz Pentium 4 processor and 2-GB dual chan-nel random-access memory.

RESULTS

Figures 2 through 7 present the results of the proposedmethod. As pointed out in the ‘‘Theory’’ section, thecombination scheme is decided by two convolution dif-ference related values: var and err. Table 2 shows thesetwo values in all experiments. To demonstrate theimpact on SNR by the proposed method, Table 3 pro-vides the statistical analysis of g-factor values of eachexperiment.

Performance With Simulated Data and Comparisonto the Multiple-Average Method

Figure 2 shows the motion correction results with simu-lated data set 1. Figure 2e shows the plot of the averagedconvolution difference of each subset. Figure. 2b and fshows the artifacts due to nonrigid motion of an objectinside the phantom, while Fig. 2c and d shows thereconstructions by COCOA and multiple-averagemethod, respectively. Residual motion artifact can beobserved at the source of the motion in the result bymultiple-average method (Fig. 2d). In the correspondingerror map (Fig. 2h), errors due to GRAPPA reconstruc-tion can be clearly observed. The RMSE of Fig. 2b-d is1.7%, 0.3%, and 3%, respectively.

Figure 3 shows the change of noise level due to themotion-correction procedures. The simulated data set 2was used for this experiment. Figure 3a shows the noiseadded to the simulated data set. Figure 3b and c showsthe noise distribution after the motion-correction

Table 3Statistical Values of g-Factor Maps

Figure number Maximum Mean Standard deviation

3d 1.6 0.68 0.16

3e 5.0 1.24 0.624d 1.3 0.59 0.11

5d 2.7 1.09 0.316d 4.1 0.63 0.327e 2.8 1.34 0.26

Table 2Statistical Values for Decision of Combination Scheme

Figure number var err

2 65% 1.7%

3 2% 1.6%4 5% 84%

5 5% 110%6 3% 99%7 23% 36%

FIG. 3. Noise amplification due to the motion-correction procedure with the simulated data set 2. The different error maps are shown

for the (a) acquired data (showing basically the white noise), (b) for the COCOA reconstruction, and (c) result for the multiple averageapproach, respectively. These error maps are in the same scale. d,e: The g-factor maps for COCOA and the multiple average method.

162 Huang et al.

procedures. Significantly amplified noise can be ob-served in the result obtained by the multiple-averagemethod (Fig. 3c). RMSEs of the images corresponding toFig. 3a-c are 2.6%, 2.0%, and 5.2%, respectively. TheRMSE of the result by COCOA is the lowest. Figure 3dand e is the g-factor maps of COCOA and multiple-aver-age method. Due to difference on convolution kernels,COCOA produced much lower g-factor values.

Application on Artifacts Due to Swallowing or Breathing

Figure 4 shows the results for the spine data set 1, whichwas corrupted by swallowing, whereas Fig. 5 presentsresults for shoulder imaging. Before correction, obviousartifacts can be observed in Figs. 4a and 5a. After theproposed operation, the artifact level was dramaticallyreduced. The boundary definition, as shown by thearrow in Fig. 5a, was clearly improved after the correc-tion (Fig. 5b). From Figs. 4c and 5c, the removed motionartifacts can be observed. The image difference is insig-nificant at spatial locations that are free of motion orwhere the SNR is high, as shown by the black regions inFigs. 4c and 5c. Figures 4d and 5d are the correspondingg-factor maps.

Application With Partially Acquired Data

Spine data set 2 was artificially undersampled to showthe application of the proposed method with partiallyacquired data. Figure 6a was the reconstruction with fullk-space data. Due to swallowing, coughing, and randomneck motion, Fig. 6a has ghosts and blurred boundaries.Figure 6b illustrates the result of COCOA followed byGRAPPA. Compared with Fig. 6a, Fig. 6b has reducedthe artifact level and increased image sharpness. Theedge definition of the abnormal cervical vertebrae was

enhanced after motion correction. The difference map(Fig. 6c) shows the removed artifacts and enhancedboundaries. The corresponding g-factor map is presentedby Fig. 6d.

FIG. 5. Application to reduce artifacts due to breathing. a: The

motion-corrupted image. b: The image corrected by the proposedmethod. c: Magnitude of the difference between (a) and (b). c:The image has been brightened five times. d: g-factor map of the

proposed operation.

FIG. 4. Application to reduce artifacts due to swallowing. a: Themotion-corrupted image. b: The image corrected by the proposed

method. c: Magnitude of the difference between (a) and (b). c:The image has been brightened five times. d: g-factor map of theproposed operation.

FIG. 6. Application to partially acquired data. a: The motion-cor-rupted image. b: The image corrected by the proposed methodwith partially acquired data. c: Magnitude of the difference

between (a) and (b). c: The image has been brightened five times.d: g-factor map of the proposed operation.

COCOA for Motion Compensation 163

Application With Linearly Acquired Data

Figure 7 shows results of the abdomen data set corruptedby blood flow. This data set was acquired with a linearacquisition scheme, which means that immediately adja-cent PE lines were acquired one after another. Figure 7fshows the convolution kernel used in this experiment.In Fig. 7a, multiple copies of vessel can be clearlyobserved due to flow artifacts. Figure 7b shows the resultafter the application of COCOA. It can be seen that theghosts were significantly reduced. The removed ghostsare shown in the differences image (Fig. 7d). When com-paring image quality before and after correction, it canbe seen that the change in SNR was insignificant. Peri-odic local maxima introduced by inflow can be seen onthe plot of convolution difference (Fig. 7c). By smoothingthe convolution difference, these local maxima can beeasily identified. The dotted line in Fig. 7c shows theplot of the smoothed convolution difference. At locationswith errors, the solid line has a higher value than thedotted line and the data from the synthetic k-space wereused. At other locations, the original acquired k-spacedata were used.

DISCUSSION

Advantages of the Proposed Method

There are three main advantages for COCOA: autocalibra-tion, robustness, and SNR preservation. First of all, thismethod does not require extra hardware or additionaldata. Even for undersampled data, the equally spacedpartial k-space data provide sufficient calibration signal.

This is different from PPI techniques, which need anextra set of calibration signal. Moreover, this methoddoes not require a special acquisition trajectory.Although interleaved acquisition is preferred, the dataacquired in a regular Cartesian acquisition scheme aresufficient for the proposed method. Hence, the proposedmethod can be widely applied to any general-purposeimaging sequences.

Second, the method is robust. As shown in the‘‘Results’’ section, the proposed method is applicable forartifacts due to nonrigid motion (Fig. 2), swallowing(Fig. 4), breathing (Fig. 5), mixed motion (Fig. 6), andblood flow (Fig. 7). For images without motion artifacts,the proposed method will not degrade the image quality(Fig. 3). For images with serious artifacts (Fig. 6), theconvolution kernel calculated from severely corruptedcalibration signals can still efficiently reduce motionartifacts.

Third, this method can preserve SNR well whilereducing ghost artifacts. Figure 3 is used to show thisquantitatively. When the original image was onlydegraded by gaussian distributed noise (Fig. 3a), theRMSE of the result by COCOA (Fig. 3b) was 23% lowerthan the original noisy image. This can also be observedfrom the g-factor values due to the convolution (Fig. 3dand corresponding row in Table 3). This means the con-volution did not amplify the overall noise level butreduced it instead. This phenomenon is significantly dif-ferent from PPI.

The reduction of noise level can be attributed to tworeasons: First, the closest neighbors along the PE direc-tion were included in the kernel (Fig. 1a), and the coil

FIG. 7. Application to a data set corrupted by the periodic blood inflow. a,b: The images before and after correction. c: Plot of the con-volution difference that was used to decide the combination scheme. The dotted line is the smoothed version of the convolution differ-

ence. d: Magnitude of the difference between (a) and (b). It has been brightened five times. e: g-factor map of the proposed operation.f: The convolution kernel used in the proposed method. The notations are identical to the ones used in Fig. 1.

164 Huang et al.

(eight-channel head coil) provided independent sensitiv-ity maps. Therefore, the approximation using Eq. 1 isaccurate. Second, it has been shown and discussed in lit-erature (16,17) that a k-space-based method could haveg-factor values less than unity due to the implicit regula-rization during calibration. When full k-space data areavailable and the convolution kernel includes immediateadjacent neighbors, the regions with lower than unity g-factor values could be dominant; g-factor values ofexperiments with in vivo data sets further highlight thisobservation. When Fig. 1 was used to design the convo-lution kernel, the mean g-factor value was lower thanunity when the number of coil elements was 8 (Fig. 3d)or 16 (Figs. 4d and 6d). The mean was slightly higherthan 1 when the number of coil elements was 4 (Fig.5d). To further quantitatively show the reduction ofnoise at low g-factor regions, the standard deviation atthe left lower corner (shown by the white box in Fig. 4a)was calculated before and after the proposed operation.The standard deviation was reduced from 6.5 (Fig. 4a) to4.6 (Fig. 4b). Since this region had no motion artifacts inFig. 4a, the reduction was achieved due to the noise sup-pression, resulting in a noise level reduction of over30%. To sufficiently reduce flow artifacts, the immediateadjacent neighbors were not used in the linearlyacquired abdominal data set (Fig. 7f). Hence, the mean g-factor value became 1.34, which was higher than thosevalues observed in other experiments. However, from thecomparison of Fig. 7a and b and the difference of thesetwo images (Fig. 7d), it can be seen that the impact onSNR was limited.

Comparison With Existing Data Consistency–BasedMotion Compensation Techniques

The consistency of data, such as k-space data (3,12), sen-sitivity maps (11), and reconstructed images (4), hasbeen explicitly enforced in the techniques previouslyproposed. In COCOA, data consistency is also used todetect the locations of motion-corrupted k-space signalsand implicitly used for motion correction. It should benoted that COCOA needs motion detection only whenthe motion is sudden (Fig. 2e) or periodic (Fig. 7c). Inthese cases, the use of threshold works well. When errorsdue to motion are widely and uniformly distributed in k-space, some previously proposed techniques strongly de-pendent on motion detection (3) may fail. COCOA uses adifferent strategy in these cases: if the standard deviationof convolution difference (var) is low (Tables 1 and 2),the result of convolution (synthetic k-space) is used forthe final reconstruction. Figures 4 to 6 show that quiteoften motion artifacts result in low var, and in thesecases the convolution step alone can dramatically reducemotion artifacts. Hence, the proposed method does notheavily rely on the motion detection. But the multiple-average method (12) does not rely on motion detectioneither. Figures 2 and 3 compare this method with CO-COA. In Figs. 2d and h, significant residual motion arti-fact and GRAPPA-related artifact can be observed. TheRMSE of the multiple-average method (Fig. 2d) was 10times higher than that of the proposed method (Fig. 2c).When the original image contained only noise, the recon-

struction by GRAPPA with k-space subset produced ahigh g-factor. Even after averaging, the amplified noise(Fig. 3c) is 2.6 times higher than that of COCOA (Fig.3b). This is because the acceleration factor was 4 forreconstruction of each extra k-space copy for the multi-ple-average method in this example. On the contrary, thecorresponding acceleration factor for the convolutionkernel defined by Fig. 1a was only 1.3.

Compared with the method using the consistency ofthe reconstructed images (4), the convolution and combi-nation in k-space are much faster. As mentioned inAtkinson et al. (4), the processing time was from minutesto hours for each two-dimensional image. On the otherhand, the convolution scheme takes similar reconstruc-tion time to GRAPPA, which is from less than 1 sec toless than 1 min, based on the convolution kernel sizeand the image size. The combination step takes onlymilliseconds.

Potential Improvement of the Proposed Method

There are several potential approaches to furtherimprove the proposed algorithm. A detailed discussionof these approaches is beyond the scope of this paper.Hence, only the outline and a proof of principle are pro-vided. First, the definition of the convolution kernel canbe further improved. Methods for GRAPPA convolutionkernel optimization (18,19) can be adopted to furtherreduce the g-factor. Second, empiric thresholds are usedin Table 1. They are not optimal. More research on thecombination scheme can result in a further reduced over-all error. Third, iterative reconstruction schemes can beused to further reduce artifacts. During the iteration,both calibration signals and the shape/size of the convo-lution kernel can be updated. One specific example isprovided as follows. Based on the experimental results,the accuracy of detection is not sensitive to the errorlevel in the motion-corrupted data. But the synthetic k-space may contain considerable errors if the GRAPPAconvolution kernel is not accurate due to the errors inthe calibration signal. One solution is to recalculate theconvolution kernel with the updated data from the previ-ous iteration. Fourth, a k-space correlation-based motion-detection method can be adopted. In this work, the con-volution difference is used to detect the motion cor-rupted k-space lines. If the motion is rigid, convolutiondifference cannot explicitly provide the motion parame-ters, which can be detected by k-space correlation basedmethod (7) have the ability to detect the rigid motionparameters. This information can be used to correctoriginal k-space and/or reject some k-space lines withsignificant errors. A PPI or compressed sensing (20)reconstruction can be followed to approximate therejected k-space lines.

CONCLUSION

A robust and efficient novel method, COCOA, is intro-duced for motion compensation. The proposed algorithmis based on the error redistribution due to convolution.Experiments demonstrate that the proposed method cansignificantly reduce nonrigid motion artifacts while

COCOA for Motion Compensation 165

preserving SNR. Future work has to include kernel opti-mization, combination scheme optimization, and thecombination with other motion-compensationtechniques.

ACKNOWLEDGMENTS

The authors thank the anonymous reviewers for theirinvaluable feedback. We are grateful to Rodney Housenfor critically reading and editing this manuscript.

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