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Mixed Spectrum Analysis on fMRI Time-SeriesArun Kumar*, Lin Feng, Senior Member, IEEE, and Jagath C Rajapakse, Fellow, IEEE

Abstract—Temporal autocorrelation present in functional mag-netic resonance image (fMRI) data poses challenges to its analysis.The existing approaches handling autocorrelation in fMRI time-se-ries often presume a specific model of autocorrelation such as anauto-regressive model. The main limitation here is that the corre-lation structure of voxels is generally unknown and varies in dif-ferent brain regions because of different levels of neurogenic noisesand pulsatile effects. Enforcing a universal model on all brain re-gions leads to bias and loss of efficiency in the analysis. In thispaper, we propose the mixed spectrum analysis of the voxel time-series to separate the discrete component corresponding to inputstimuli and the continuous component carrying temporal autocor-relation. A mixed spectral analysis technique based on M-spec-tral estimator is proposed, which effectively removes autocorrela-tion effects from voxel time-series and identify significant peaks ofthe spectrum. As the proposed method does not assume any priormodel for the autocorrelation effect in voxel time-series, varyingcorrelation structure among the brain regions does not affect itsperformance. We have modified the standard M-spectral methodfor an application on a spatial set of time-series by incorporatingthe contextual information related to the continuous spectrum ofneighborhood voxels, thus reducing considerably the computationcost. Likelihood of the activation is predicted by comparing theamplitude of discrete component at stimulus frequency of voxelsacross the brain by using normal distribution and modeling spa-tial correlations among the likelihood with a conditional randomfield. We also demonstrate the application of the proposed methodin detecting other desired frequencies.

Index Terms—Conditional random field, functional MRI, mixedspectrum analysis.

I. INTRODUCTION

F UNCTIONAL magnetic resonance imaging (fMRI) hasbecome increasingly popular for studying and visualiza-

tion of human brain function in vivo. To improve signal-to-noiseratio (SNR) in fMRI, a large number of images are sequentiallyacquired in a single experiment, either in a block-based or in anevent-related manner [14], [7], [2]. Brain activation is detected

Manuscript received November 19, 2015; revised January 10, 2016; acceptedJanuary 13, 2016. This work was partly supported by AcRF Tier-1 grantRG19/15 to J. C. Rajapakse by the Ministry of Education, Singapore . Asteriskindicates corresponding author.*A. Kumar is with the School of Electrical and Electronic Engineering, Sin-

gapore Polytechnic, 139651 Singapore (e-mail: [email protected]).L. Feng is with Foshan University, Foshan, China [POSTALCODE? DEPT?], and also with Nanyang Technological Univer-sity, 639798 Singapore [DEPT?] (e-mail: [email protected]).J. C. Rajapakse is with the School of Computer Engineering, Nanyang Tech-

nological University, 639798 Singapore (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TMI.2016.2520024

by identifying voxels that have hemodynamic responses corre-lated with the input stimuli in task related experiments. This isusually performed by regression analysis assuming a generallinear model (GLM) and then followed by testing for signifi-cance of activation [8], [14]. The presence of temporal autocor-relation underestimates the variance of regression coefficientsin the GLM of voxel time-series and leads to errors in the esti-mated test statistics. Though temporal autocorrelation is less ofa problem for inter-group studies such as random effect analysis,it poses greater challenges to intra-subject activation detection.Two types of strategies namely, prewhitening [36] and col-

oring [9], have mainly been proposed to account for temporalautocorrelation of fMRI time-series. Prewhitening refers to datareduction techniques in time-domain whereas coloring refersto techniques involving filtering. The pseudo-generalized leastsquares (PGLS) method has been proposed to fit regressionmodels of time-series and then temporal correlations of resid-uals are fitted with a first-order auto-regressive (AR) model[3]. In the restricted maximum likelihood (ReML) approach,temporal autocorrelations were spatially smoothed to reducetheir variability and then fitted with the Yule-Walker equationsto obtain AR( ) parameters [36], [34]. This method was foundto have adverse effect on the classification of voxels with eitherapproximately white or negative autocorrelation, which wasovercome by a non-regularized AR(2) model [17]. Variationsof models, such as AR(1) model with an additionalwhite noise component [27] and ARMA model [20] have beensuggested. Coloring strategies shape the intrinsic autocorre-lation of data, usually with band pass filters consisting of alow-pass smoothing filter and a high-pass filter specified by theminimum period [9], [35]. Filtering imposes an autocorrelationstructure on the error terms. An AR(1) model is then fitted tothe altered correlations, which coefficients are assumed to beinvariant over the brain [9].In frequency domain approaches, detection of brain activa-

tion is usually performed by evaluating the null distribution ofno activation of the spectrum. Zarahn et al. (1997) modeledthe spectral power of the time-series with the inverse frequency( ) model for fitting the temporal autocorrelation [37]. Thenull distribution has also been derived from the smoothed peri-odogram on log scale excluding stimulus frequency [22], by av-eraging the squared residuals around each of fundamental andharmonic stimulus frequencies [16], and by subtracting the adja-cent epoch time courses [24]. In the above mentioned frequencydomain approaches, smoothing of the periodograms may retainsome of the effects of the stimuli and subtraction of adjacentepochsmay introduce spurious high frequency effects in the nulldistribution.A major drawback common to the above methods of han-

dling temporal autocorrelation is that an autocorrelation model

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2 IEEE TRANSACTIONS ON MEDICAL IMAGING

has to be assumed a priori. Thereafter, an iterative scheme ofremoving autocorrelation has to be used (for example, in ReMLor PGLS). For the detection of brain activation, PGLS techniquewith AR(1) model and the inverse frequency model are foundto be inadequate to model both the intermediate and long-rangeautocorrelations [9], [6], [23]. Coloring methods are found tobe less attractive unless the specified deconvolution matrix ex-actly matches the actual covariance structure. The prewhiteningmethods are found to be less accurate when the autocorrelationstructure and hemodynamic response are invariant across brainregions [36]. An alternative approach to avoid using a specifictemporal autocorrelation model is to remove the temporal au-tocorrelation effect by specifically modeling the known noisesources causing the alleged temporal autocorrelation, such as,hardware related low-frequency drift, residual movement ef-fects, and aliased physiological noises [21]. The major draw-back of this method is its high dependence on the sources ofnoise being characterized, and requires cleaning of data to getrid of unmodeled sources of temporal correlation [23].Spatio-temporal techniques that concurrently deal with

spatial and temporal correlations have also been suggestedfor fMRI data analysis. Woolrich et al. used temporally fixedspatial AR(1) model together with a spatially varying temporalAR model for short term correlated processes in additionto a high-pass filter in pre-processing stage to remove largescale temporal variations [33]. Descombes et al. proposed a3D spatio-temporal MRF model with two spatial dimensionsand one temporal dimension [5]. Gossl et al. used a Bayesianframework in which the spatial and temporal random fieldpriors are combined via the Kronecker product of their re-spective precision matrices [11]. One major drawback of thesetechniques is the use of deterministic model for drift withthe assumption that the remaining noise is white [33]. TheBayesian spatio-temporal technique suggested by Bowman [1]lacks an optimized model fitting procedure [19].In this paper, we present a novel approach using the mixed

spectrum analysis to handle temporal and spatial correlationsof fMRI data. The spectrum of fMRI time-series is modeled asa sum of a discrete spectrum resulted due to task-related ac-tivation and a continuous spectrum carrying the autocorrela-tion of the signals. In other words, continuous spectrum refersto the spectrum of colored noise and discrete spectrum refersto the spectrum of periodic signal components. Our proposedmethod is based on M spectral estimator (MSPEC) [13] with amodification to improve computational efficiency and is calledC-MSPEC (Contextual-MSPEC). It uses robust regression tech-niques to reduce the influence of outliers in periodogram whencalculating the continuous spectrum and neighborhood corre-lations to improve efficiency. The advantage of C-MSPEC isthat it does not require explicit knowledge of autocorrelationfunction or the hemodynamic response function [18]. There-fore, it is capable of handling complex autocorrelations such asthose found in high-field fMRI [10]. However, our method isrestricted only to fMRI data gathered under periodic stimuli.There are two major contributions of our presented work,

firstly we remove the autocorrelation effect in voxel time-serieswithout assuming any specific autocorrelation model. This con-tribution has resulted in an improvement in activation detection.

Secondly, we have modified the standardMSPEC to incorporatespatial constraints. This method has previously been employedonly to calculate the continuous spectrum of individual time-se-ries. Our modification has resulted in a substantial decrease inthe number of iterations required to calculate the continuousspectrum of voxel time-series. The contextual information hasthus been used in our proposed method at two points, one whencalculating the continuous spectrum and the other when identi-fying the activated voxels.In Section II, we describe how fMRI time-series are modeled

with a mixed spectrum and the C-MSPEC method is usedto identify significant frequency components in each voxeltime-series. Brain activations are detected by using the nulldistribution of brain activation and modeling spatial correla-tion with conditional random fields (CRF). In Section III, wepresent the results of detecting activation on synthetic data aswell as on real fMRI gathered in a memory retrieval task. Wealso demonstrate the application of the proposed method indetecting other frequencies of interest other than the stimulusfrequency. Section IV concludes and gives future directions ofthis research.

II. METHODS

A. Modeling fMRI Time-Series With a Mixed Spectrum

An fMRI experiment consists of taking a series of three-di-mensional (3D) brain scans while the subject is resting or per-forming a functional task. Let the spatiotemporal fMRI imagebe where denotes the 3D spatial do-main of brain voxels, represents the scanning times, and therange of image intensities. Let the scanning times be indexed by

where denotes the total number of brain scans,representing the total scan time. Changes of blood-oxygenated-level-dependent (BOLD) signal over an fMRI experiment de-fine the hemodynamic time-series response at a brain voxel. Letthe fMRI time-series at voxel be and

(1)

where signal has a discrete spectrum, has a contin-uous spectrum and represents random noise. In this repre-sentation, the fMRI time-series signal generated by the periodicstimuli is modeled by and contribution by the autocorre-lated noise by .

B. M Spectrum Estimation

Let the spectrums for and be represented byand , respectively, with denoting the frequency. For anfMRI time series without the signal component having a discretespectrum, , a simple periodogram smoother(that is, a weighted moving average of periodogram ordinates)can be used to estimate the continuous spectrum :

(2)

where represents Parzen window of size andand represents the periodogram of the

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KUMAR et al.: MIXED SPECTRUM ANALYSIS ON FMRI TIME-SERIES 3

voxel time-series. To include the effect of the discrete spectrum,let . This leads to

(3)

where .The function , resulting in simple

periodogram smoothing, is modified to restrict the influence ofoutliers and is given by [18]:

ifififif

(4)

where and values are chosen within the range of the requiredcapped region and results in zero restriction (i.e., nocapping). The estimated value of is thus definedas the solution [18] of

(5)

With function, proof of (5) is given in Appendix A. Variableis related to as given below [18]:

if(6)

Equation (5) is solved for the value usually by using a nu-merical root solving method such as regula falsi method. Thesemethods start with two points: the maximum and min-imum values of spectral amplitudes values of time-se-ries. Some restrictions in the form of small value is requiredto ensure the opposite signs of respective values (lines 7–10of Algorithm1).

C. Contextual M Spectrum Estimator (C-MSPEC)

In the standardM spectrum estimator, the two valuesand are taken as the initial values and updated itera-tively with the following equations:

(7)

ifif

(8)

where is the iteration counts. This algorithm finds the con-vergence of the sequence to the solution . To im-prove the convergence, we propose a modified M spectral es-timator (C-MSPEC) algorithm that incorporates the spectrumamplitudes of the neighbors into the algorithm.Since the neighbor voxels in the brain have similar contin-

uous spectrum, we propose to initialize the value in (5) atfour points (lines 11–20 of Algorithm 1): two of the points are as

those mentioned in the standard algorithm and the other two areselected by the maximum and minimum values of continuousspectrum of the neighborhood voxels at the same frequency(lines 11–13 of Algorithm 1). From these four points, two pointsare selected to have values nearest to zero and of oppositesigns (lines 14–20 of Algorithm 1).The modified algorithm is given in Algorithm 1. Our algo-

rithm will result in a substantial reduction in the number of it-erations required to arrive at the solution for for a largenumber (not all) of voxels. Voxels that have no processed neigh-bors such as the first voxel follows the standard algorithm.The discrete spectrum amplitude value of brain voxelat stimulus frequency are calculated as,

. The likelihood of activation in brain voxels,, is calculated

assuming that values across the brain voxels pooled to-gether are normally distributed in the absence of any activation.

and denote the mean and standard deviation ofthe distribution, respectively.

D. Modeling Contextual Information of Activation

In the brain, neighboring voxels are likely to have the samefunction, so spatial correlations exist among the activated voxels

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and need to be taken into account when determining the acti-vation patterns of the brain. The aim is to find the optimal ac-tivation pattern by using the maximum a posteriori (MAP)estimator:

(9)

In order to estimate the brain activation pattern, the pair isassumed to form a conditional random field (CRF) [15]. Then,

(10)

where represents the neighborhood of voxel .From the fundamental theorem of random fields [12], the

above distribution is given by a Gibbs distribution:

(11)where represents the single-voxel potential at voxeland the potential of voxel interaction with the

neighborhood voxels [32].The single-voxel potential in (11) is decomposed as

(12)

which consists of representing the priorinformation of activations and repre-senting the data-dependent single-voxel potential. Let the priorpotential be a constant over the brain. Thedata-dependent term is calculated using the C-MSPECalgorithm.The pairwise potential term in (11) is then given by

(13)

where represents the data-independent joint potentialand represents the data-dependent joint potential.For spatial interactions, we use twenty six nearest neighbors in3D and assume that the potentials are constant over the image.The data-independent interaction term is set to encourage

contiguous voxels of the same class:

(14)

and the data-dependent term

(15)

where and represents the probability of voxeltime-series, given the state of activation of the voxels and re-spectively, and are determined using C-MSPEC estimator. And

and are positive constants.From (9)–(11), the MAP estimate of the activation pattern is

given by

(16)

However, computing the optimal activity pattern by using theabove equation and potential values is a computationally inten-sive task. We used the mean field approximation to efficiently

compute the optimal solution. The derivation of the mean fieldapproximation to the above estimate is given in Appendix B.

III. RESULTS

A. Activation DetectionThe proposed method is demonstrated on synthetic data as

well as on real fMRI data. Synthetic data was used to quantita-tively evaluate the performance and robustness of the methodon different correlation structures. Functional MRI data gath-ered from a memory retrieval task was used in the testing. Forthe CRF, the prior probability was used for acti-vated voxels as the usual range of activated voxels is 2–5% inreal fMRI images. Small changes in do not affect the resultsnotably. Values of , around 1 were empirically found togive optimal accuracy with the synthetic images. These param-eter values were used for detection of activation from the realimages as well.1) Synthetic Data: Two-dimensional synthetic fMRI

datasets of 64 64 pixels per scan with 16 images per epochand 2 s interscan interval were generated. For simulating thedata, a stimulus cycle containing 4 s of stimulations and 28 s ofrest time was used. fMRI time-series response of the activatedpixels was obtained by convolving the stimulus with a hemo-dynamic response function defined by a mixture of twogamma functions [31]:

(17)

where , delay of response , delay of under-shoot , dispersion of response , dispersion ofundershoot , ratio of response to undershoot ,the length of kernel , represents the gamma func-tion and .In order to test the robustness of the method, synthetic data

was added with temporal autocorrelation at different complexi-ties. By using auto-regressive models at various ordersof : , we generated AR(1) modelwith , AR(2) with , and AR(3)with . The synthetic data wasgenerated with different AR models affecting specific regions:in Fig. 1(a), white/grey regions of circles were added withAR(3) autocorrelation, of diagonals with AR(2) and of outsidethe circle with AR(1) autocorrelation. The experiments wereperformed at signal-to-noise ratio (SNR) of 0.9, 1.2 and 1.9.SNR is defined as where is the amplitude of stimulussignal and is the standard deviation of autocorrelated noisedefined as , beingthe standard deviation of uncorrelated noise. The non-activatedpixels were left unchanged at zero amplitude [38]. Spatiallycorrelated Gaussian noise was added to all the brain pixels.The results of C-MSPEC method were compared with

the popular SPM method that implements a GLM on fMRItime-series [31] and with the independent component analysis

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Fig. 1. (a) Synthetic image: black portions are non-activated and grey/white portions are activated with different temporal autocorrelations (AR1-AR3), and theROC curves for synthetic data at (b) , (c) , and (d) .

(ICA) method [30]. When generating the results for SPMpackage, we have used time and dispersion derivatives toaccount for the small variations in hemodynamic responses andto correct temporal autocorrelation, assuming AR(1) model.Spatial smoothing with a Gaussian kernel was performed asa part of preprocessing for both SPM and ICA. With the ICAmethod, we chose the ICA component that corresponds tothe input stimuli. To evaluate the performance, the receiveroperating characteristic (ROC) curves were plotted by varyingthe threshold probability value for detecting activation. Theresults are given in Fig. 1. As is clear from the figure, ICAmethod did not perform very well and consistently. Note thatwe consider only the specificity range of 0.95–1.00 in the ROCcurves for computing the area under the partial ROC curves(AUC) values as the threshold point is very likely to be in thisrange. As is evident from ROC curves and AUC values, for twoof the three datasets, the C-MSPEC method has better results.The detected activation by the different method are given in

Fig. 2. The AUC values are shown in Table I. For the datasetgenerated at , the results were similar for both SPMand MSPEC method. This is understandable, as for this lowSNR data, the same HRF is used for both generation of dataand detection in GLM, whereas the proposed method does notmake use of the HRF model in detection.

We have used a number of parameters such as and inCRF, window size and in the formulation of the C-MSPECalgorithm. In order to understand the effect of these vari-ables on our analysis, additional experiments were performed.Fig. 3 shows the effect of various ( ) values.We have used three synthetic datasets at different SNR valuesfor this analysis. Also shown in this figure are the results for

and , that corresponds to the results withoutCRF modeling. As seen, values around 1, gave good AUCvalues. In our analysis of synthetic and real datasets, we haveused . Similarly, to study the effects of variouswindow sizes, we used synthetic data and plottedthe ROC curves at various window sizes (without the CRFstep) as shown in Fig. 4. As seen, beyond the window size of 7,there is not much difference in the results. In our synthetic andreal datasets, we have used the window size of 7. The effect ofsmall changes in values was also found to be negligible asshown in Fig. 5. We have set as used in [18].2) Memory Retrieval Task: In a memory retrieval task, sub-

jects learned three different sets (sizes 4, 6, and 8) of letters priorto the actual experiment with a corresponding cue for each set.During each trial of the experiment, a cue for the set and a letterwere presented and the subject decided whether the letter corre-sponds to the indicated set. A delay of 2.0 s was maintained be-tween the presentation of the cue and the probe. Fourteen slices

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Fig. 2. Synthetic data results: Activation patterns realized with SPM (top row),ICA (middle row), and C-MSPEC method (bottom row) at (a) , (b)

, and (c) .

TABLE ITHE AUC VALUES OF PARTIAL ROC CURVES (SPECIFICITY RANGE:

0.95–1.00) COMPARING C-MSPEC, SPM, AND ICA RESULTS.

Fig. 3. Area under the ROC curves (AUC values) at differentvalues for C-MSPEC method.

were acquired with 19.2 cm FOV, 7 mm thickness, and 64 64matrix using single shot gradient echo-planar imaging (EPI) se-quences ( and ). Ten cycles of datawere used for analysis. Details of the experiment are availablein [28].

Fig. 4. ROC curves at different window sizes of C-MSPEC method.

Fig. 5. ROC curves at different values of C-MSPEC method.

Brain processes involved in the experiment include encodingof the cue and probe, retrieval of information from the sec-ondary memory, scanning of the primary memory, response se-lection, and response execution [28]. Brain activation was de-tected using the C-MSPEC method and the SPM method [31].Fig. 6 shows activation detected on three representative slicesby the two methods. The cue and the probe both generated ac-tivation in the brain at same frequency but with an interval of2.0 s apart. The temporal differences in stimulus presentationappear as differences of the phases in frequency domain. In theC-MSPEC method, brain regions corresponding to the probeand cuewere thus separated based on the phase of voxel time-se-ries. In the SPM method, activations corresponding to cue andprobe were found by properly designing the design matrix, sep-arately taking into account the cue and the probe stimuli.Both of the methods effectively detected activation in the ex-

pected cortical areas of inferior precentral sulcus (pCS) and pos-terior middle frontal gyrus (MFG) following cue and in regionsof posterior parietal cortex following the probe [26]. Inferiorprecentral sulcus and posterior middle frontal gyrus are gen-erally involved in the retrieval of information from secondarymemory and also in maintaining this information for searchfollowing the presentation of the cue, hence are also activatedduring the probe.

B. Efficiency of C-MSPEC MethodIn order to test the effectiveness of our proposed modifica-

tions in the MSPEC algorithm, we compare the performanceof the C-MSPEC algorithm (Algorithm 1) with the standard

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Fig. 6. The detected activation of memory task by (a) the SPM method for thecue condition, (b) the SPM method for the probe condition, (c) the C-MSPECmethod for the cue condition, and (d) the C-MSPEC method for the probe con-dition.

MSPEC algorithm (Algorithm 1 without lines 11–20). Therewas no significant difference in the activation detected but thecomputational time was improved significantly with the modi-fied initialization. The average computation time on one subjectbrain with the C-MSPEC was 139.8 s compared with 159.0 sfor the standard MSPEC on a ACPI x64-based PC with an Inteli7 processor and 8 GB RAM. Fig. 7 shows the histogram of thenumber of iterations required for the detection of activation onbrain voxels on a representative scan. As seen, our modificationin the MSPEC algorithm significantly reduces the number of it-eration required for the detection of brain activation.

C. Detection of Other Prominent Frequency Components

We demonstrate an application of our technique on detectingother confounding frequencies of the brain. The C-MSPEC

Fig. 7. The number of iterations required to calculate continuous spectrum inthe memory-task data on one subject by the C-MSPEC method and the MSPECmethod.

Fig. 8. Significant frequency components present in the inferior brain slicesof a representative subject performing the memory task. Dashed lines show thelocation of stimulus frequency and its harmonics that have been removed.

method was applied on the voxel time-series of inferior slicesof the brain, which are likely to have significant physiolog-ical signals during the memory-retrieval task. In this case,frequency components other than the stimulus frequency andits harmonics were analyzed using the C-MSPEC method.Fig. 8 shows the histograms of the various frequency com-ponents that are significant on a representative subject. Notethat the stimulus related frequencies and their harmonics havebeen removed in the figure in order to clearly show the otherfrequency components. The most prominent frequency, that is,most statistically significant frequency, which was present inmost number of voxels corresponds to 0.1176 Hz. In order tounderstand the significance of this frequency component, weapplied spatial ICA [30] on these slices.Two independent components (ICs) were selected based on

the knowledge of anatomical regions that are likely to havestrong presence of physiological signals [4]. Fig. 9(a) shows theslices and time and frequency domain signals of these compo-nents. Both the components showed a prominent presence in themain trunk of middle cerebral artery: one component (Fig. 9(a))was significant in the branch of middle cerebral artery and theother (Fig. 9(c)) in posterior cerebral artery. Based on the FFTanalysis for component 1 and 2, frequency 0.1176 Hz was foundto be most significant as shown in Fig. 9(b) and (d). This is thesame frequency component that was also found to be signif-icantly present in the inferior location brain slices during theanalysis with the C-MSPEC method. Hence we can concludethat the proposed method is also able to detect the other fre-quencies such as physiological signals.

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Fig. 9. Spatial ICA components related to physiological signals: (a) and (c) show the location of these signals, and (b) and (d) show temporal variations of thesecomponents (top) and the FFT analysis results (bottom), The dashed lines indicate the stimulus and its harmonic frequencies.

IV. CONCLUSION

We presented the mixed spectrum analysis of fMRI data andits use in handling temporal autocorrelation. The frequencyspectrum of fMRI time-series was modeled with a mixedspectrum: a component having a discrete spectrum relating totask-related stimuli and a component having a continuous spec-trum comprising of temporal autocorrelation. With this model,the C-MSPEC method was proposed to find the amplitudeof discrete component of the stimulus frequency at the brainvoxels and the likelihood of activation at a particular brainvoxel. As we expect the neighborhood voxels to have similarcontinuous spectrum, the C-MSPEC method used the spectraof the neighbors when calculating continuous spectrum. Thisextension significantly improves the computational time of theMSPEC method.Spatial correlation among the likelihood of activated brain

voxels wasmodeled with a CRF. Compared to Gaussian randomfields [31] and MRF [29], [25], CRF accounts for data-depen-dent nature of neighborhood interactions better and significantly

improves the detection of activation [32]. The likelihood of ac-tivation of voxel given by the C-MSPEC method and the priorprovided by CRF allowed us to find MAP estimation of brainactivation.The main advantage of C-MSPEC method is that it is not

necessary to assume an a priori model of temporal autocor-relation or any form of hemodynamic response function. Themethod handles the autocorrelation in a unique way by sepa-rating it from the discrete spectra related to input stimuli. In theexperiments with synthetic data, the C-MSPEC method outper-formed the SPM method and ICA method in most of the cases.With real fMRI data gathered from a memory-related task data,the results from the C-MSPEC method were visually similar toSPMmethod; and the activations were more focal and localizedto the cortical areas.The C-MSPECmethod is only applicable for periodic stimuli.

It is also possible to look for any other significant confoundingfrequencies with our approach. We demonstrated this with anapplication of the C-MSPEC method in detecting the physio-logical signals present in real fMRI. It is worthwhile exploring

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more applications of the mixed spectral analysis and C-MSPECestimator for task-related and resting state fMRI data.

APPENDIX APROOF OF (5)

Considering (5) and (4) and using only the uncapped part of(4), we will prove that it leads to reasonable estimate of .It turns out to be similar to the way periodogram smoothing isused to detect the . The capped part in (4) is basically toremove the effect of outliers in periodogram, which makes thisalgorithm more effective than periodogram smoothing.Substituting the uncapped part of (4) in (5),

(18)

(19)

Since the window is considered such that,

(20)

Equating , to calculate the roots of the equa-tion, we get

(21)

(22)

(23)

That is, if we use only the uncapped part of (4), it leads tocalculation of using periodogram smoothing, which is wellknown to be a reasonable estimate of continuous spectrum inthe absence of outliers.

APPENDIX BMEAN FIELD APPROXIMATION OF THE MAP ESTIMATE

The CRF model to compute the posterior probabilities ofvoxel labels, as described in (11)–(15), is effectively imple-mented using the variational mean field approach. The poste-rior probability is approximated by a much simpler distribu-tion through the minimization of KL-di-

vergence to determine the approximate distribution [25]. Theterm denotes the approximated probability that thevoxel contains an activated brain signal.The K-L divergence is defined as

(24)

where the summation is taken over the presence and absenceof stimulated voxel time-series: . Substituting thevalues from (11)–(15),

(25)

which needs to be minimized based on the constraint. Using Lagrange multipliers, the above

optimization problem can be written as

(26)

Differentiating with respect to and by setting thederivatives equal to zero:

(27)This finally leads to the iterative update rule:

(28)where the normalization constant ensures that

and

(29)(30)

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[11] C. Gossl, D. P. Auer, and L. Fahrmeir, “Bayesian spatiotemporal infer-ence in functional magnetic resonance imaging,” Biometrics, vol. 57,no. 2, pp. 554–562, 2001.

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[18] T.-H. Li, Time Series with Mixed Spectra. Boca Raton, FL: CRCPress, 2014.

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[27] P. L. Purdon and R. M. Weisskoff, “Effect of temporal autocorrelationdue to physiological noise and stimulus paradigm on voxel-level false-positive rates in fMRI,”Human Brain Map., vol. 6, no. 4, pp. 239–249,1998.

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[30] FMRLAB Software. [Online]. Available: http://sccn.ucsd.edu/fmrlab[31] SPM Software. [Online]. Available: http://www.fil.ion.ucl.ac.uk/spm[32] Y. Wang and J. C. Rajapakse, “Contextual modeling of functional MR

images with conditional random fields,” IEEE Trans. Med. Imag., vol.25, no. 6, pp. 804–812, Jun. 2006.

[33] M. W. Woolrich, M. Jenkinson, J. M. Brady, and S. M. Smith, “FullyBayesian spatio-temporal modeling of fMRI data,” IEEE Trans. Med.Imag., vol. 23, no. 2, pp. 213–231, Feb. 2004.

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IEEE TRANSACTIONS ON MEDICAL IMAGING 1

Mixed Spectrum Analysis on fMRI Time-SeriesArun Kumar*, Lin Feng, Senior Member, IEEE, and Jagath C Rajapakse, Fellow, IEEE

Abstract—Temporal autocorrelation present in functional mag-netic resonance image (fMRI) data poses challenges to its analysis.The existing approaches handling autocorrelation in fMRI time-se-ries often presume a specific model of autocorrelation such as anauto-regressive model. The main limitation here is that the corre-lation structure of voxels is generally unknown and varies in dif-ferent brain regions because of different levels of neurogenic noisesand pulsatile effects. Enforcing a universal model on all brain re-gions leads to bias and loss of efficiency in the analysis. In thispaper, we propose the mixed spectrum analysis of the voxel time-series to separate the discrete component corresponding to inputstimuli and the continuous component carrying temporal autocor-relation. A mixed spectral analysis technique based on M-spec-tral estimator is proposed, which effectively removes autocorrela-tion effects from voxel time-series and identify significant peaks ofthe spectrum. As the proposed method does not assume any priormodel for the autocorrelation effect in voxel time-series, varyingcorrelation structure among the brain regions does not affect itsperformance. We have modified the standard M-spectral methodfor an application on a spatial set of time-series by incorporatingthe contextual information related to the continuous spectrum ofneighborhood voxels, thus reducing considerably the computationcost. Likelihood of the activation is predicted by comparing theamplitude of discrete component at stimulus frequency of voxelsacross the brain by using normal distribution and modeling spa-tial correlations among the likelihood with a conditional randomfield. We also demonstrate the application of the proposed methodin detecting other desired frequencies.

Index Terms—Conditional random field, functional MRI, mixedspectrum analysis.

I. INTRODUCTION

F UNCTIONAL magnetic resonance imaging (fMRI) hasbecome increasingly popular for studying and visualiza-

tion of human brain function in vivo. To improve signal-to-noiseratio (SNR) in fMRI, a large number of images are sequentiallyacquired in a single experiment, either in a block-based or in anevent-related manner [14], [7], [2]. Brain activation is detected

Manuscript received November 19, 2015; revised January 10, 2016; acceptedJanuary 13, 2016. This work was partly supported by AcRF Tier-1 grantRG19/15 to J. C. Rajapakse by the Ministry of Education, Singapore . Asteriskindicates corresponding author.*A. Kumar is with the School of Electrical and Electronic Engineering, Sin-

gapore Polytechnic, 139651 Singapore (e-mail: [email protected]).L. Feng is with Foshan University, Foshan, China [POSTALCODE? DEPT?], and also with Nanyang Technological Univer-sity, 639798 Singapore [DEPT?] (e-mail: [email protected]).J. C. Rajapakse is with the School of Computer Engineering, Nanyang Tech-

nological University, 639798 Singapore (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TMI.2016.2520024

by identifying voxels that have hemodynamic responses corre-lated with the input stimuli in task related experiments. This isusually performed by regression analysis assuming a generallinear model (GLM) and then followed by testing for signifi-cance of activation [8], [14]. The presence of temporal autocor-relation underestimates the variance of regression coefficientsin the GLM of voxel time-series and leads to errors in the esti-mated test statistics. Though temporal autocorrelation is less ofa problem for inter-group studies such as random effect analysis,it poses greater challenges to intra-subject activation detection.Two types of strategies namely, prewhitening [36] and col-

oring [9], have mainly been proposed to account for temporalautocorrelation of fMRI time-series. Prewhitening refers to datareduction techniques in time-domain whereas coloring refersto techniques involving filtering. The pseudo-generalized leastsquares (PGLS) method has been proposed to fit regressionmodels of time-series and then temporal correlations of resid-uals are fitted with a first-order auto-regressive (AR) model[3]. In the restricted maximum likelihood (ReML) approach,temporal autocorrelations were spatially smoothed to reducetheir variability and then fitted with the Yule-Walker equationsto obtain AR( ) parameters [36], [34]. This method was foundto have adverse effect on the classification of voxels with eitherapproximately white or negative autocorrelation, which wasovercome by a non-regularized AR(2) model [17]. Variationsof models, such as AR(1) model with an additionalwhite noise component [27] and ARMA model [20] have beensuggested. Coloring strategies shape the intrinsic autocorre-lation of data, usually with band pass filters consisting of alow-pass smoothing filter and a high-pass filter specified by theminimum period [9], [35]. Filtering imposes an autocorrelationstructure on the error terms. An AR(1) model is then fitted tothe altered correlations, which coefficients are assumed to beinvariant over the brain [9].In frequency domain approaches, detection of brain activa-

tion is usually performed by evaluating the null distribution ofno activation of the spectrum. Zarahn et al. (1997) modeledthe spectral power of the time-series with the inverse frequency( ) model for fitting the temporal autocorrelation [37]. Thenull distribution has also been derived from the smoothed peri-odogram on log scale excluding stimulus frequency [22], by av-eraging the squared residuals around each of fundamental andharmonic stimulus frequencies [16], and by subtracting the adja-cent epoch time courses [24]. In the above mentioned frequencydomain approaches, smoothing of the periodograms may retainsome of the effects of the stimuli and subtraction of adjacentepochsmay introduce spurious high frequency effects in the nulldistribution.A major drawback common to the above methods of han-

dling temporal autocorrelation is that an autocorrelation model

0278-0062 © 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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has to be assumed a priori. Thereafter, an iterative scheme ofremoving autocorrelation has to be used (for example, in ReMLor PGLS). For the detection of brain activation, PGLS techniquewith AR(1) model and the inverse frequency model are foundto be inadequate to model both the intermediate and long-rangeautocorrelations [9], [6], [23]. Coloring methods are found tobe less attractive unless the specified deconvolution matrix ex-actly matches the actual covariance structure. The prewhiteningmethods are found to be less accurate when the autocorrelationstructure and hemodynamic response are invariant across brainregions [36]. An alternative approach to avoid using a specifictemporal autocorrelation model is to remove the temporal au-tocorrelation effect by specifically modeling the known noisesources causing the alleged temporal autocorrelation, such as,hardware related low-frequency drift, residual movement ef-fects, and aliased physiological noises [21]. The major draw-back of this method is its high dependence on the sources ofnoise being characterized, and requires cleaning of data to getrid of unmodeled sources of temporal correlation [23].Spatio-temporal techniques that concurrently deal with

spatial and temporal correlations have also been suggestedfor fMRI data analysis. Woolrich et al. used temporally fixedspatial AR(1) model together with a spatially varying temporalAR model for short term correlated processes in additionto a high-pass filter in pre-processing stage to remove largescale temporal variations [33]. Descombes et al. proposed a3D spatio-temporal MRF model with two spatial dimensionsand one temporal dimension [5]. Gossl et al. used a Bayesianframework in which the spatial and temporal random fieldpriors are combined via the Kronecker product of their re-spective precision matrices [11]. One major drawback of thesetechniques is the use of deterministic model for drift withthe assumption that the remaining noise is white [33]. TheBayesian spatio-temporal technique suggested by Bowman [1]lacks an optimized model fitting procedure [19].In this paper, we present a novel approach using the mixed

spectrum analysis to handle temporal and spatial correlationsof fMRI data. The spectrum of fMRI time-series is modeled asa sum of a discrete spectrum resulted due to task-related ac-tivation and a continuous spectrum carrying the autocorrela-tion of the signals. In other words, continuous spectrum refersto the spectrum of colored noise and discrete spectrum refersto the spectrum of periodic signal components. Our proposedmethod is based on M spectral estimator (MSPEC) [13] with amodification to improve computational efficiency and is calledC-MSPEC (Contextual-MSPEC). It uses robust regression tech-niques to reduce the influence of outliers in periodogram whencalculating the continuous spectrum and neighborhood corre-lations to improve efficiency. The advantage of C-MSPEC isthat it does not require explicit knowledge of autocorrelationfunction or the hemodynamic response function [18]. There-fore, it is capable of handling complex autocorrelations such asthose found in high-field fMRI [10]. However, our method isrestricted only to fMRI data gathered under periodic stimuli.There are two major contributions of our presented work,

firstly we remove the autocorrelation effect in voxel time-serieswithout assuming any specific autocorrelation model. This con-tribution has resulted in an improvement in activation detection.

Secondly, we have modified the standardMSPEC to incorporatespatial constraints. This method has previously been employedonly to calculate the continuous spectrum of individual time-se-ries. Our modification has resulted in a substantial decrease inthe number of iterations required to calculate the continuousspectrum of voxel time-series. The contextual information hasthus been used in our proposed method at two points, one whencalculating the continuous spectrum and the other when identi-fying the activated voxels.In Section II, we describe how fMRI time-series are modeled

with a mixed spectrum and the C-MSPEC method is usedto identify significant frequency components in each voxeltime-series. Brain activations are detected by using the nulldistribution of brain activation and modeling spatial correla-tion with conditional random fields (CRF). In Section III, wepresent the results of detecting activation on synthetic data aswell as on real fMRI gathered in a memory retrieval task. Wealso demonstrate the application of the proposed method indetecting other frequencies of interest other than the stimulusfrequency. Section IV concludes and gives future directions ofthis research.

II. METHODS

A. Modeling fMRI Time-Series With a Mixed Spectrum

An fMRI experiment consists of taking a series of three-di-mensional (3D) brain scans while the subject is resting or per-forming a functional task. Let the spatiotemporal fMRI imagebe where denotes the 3D spatial do-main of brain voxels, represents the scanning times, and therange of image intensities. Let the scanning times be indexed by

where denotes the total number of brain scans,representing the total scan time. Changes of blood-oxygenated-level-dependent (BOLD) signal over an fMRI experiment de-fine the hemodynamic time-series response at a brain voxel. Letthe fMRI time-series at voxel be and

(1)

where signal has a discrete spectrum, has a contin-uous spectrum and represents random noise. In this repre-sentation, the fMRI time-series signal generated by the periodicstimuli is modeled by and contribution by the autocorre-lated noise by .

B. M Spectrum Estimation

Let the spectrums for and be represented byand , respectively, with denoting the frequency. For anfMRI time series without the signal component having a discretespectrum, , a simple periodogram smoother(that is, a weighted moving average of periodogram ordinates)can be used to estimate the continuous spectrum :

(2)

where represents Parzen window of size andand represents the periodogram of the

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voxel time-series. To include the effect of the discrete spectrum,let . This leads to

(3)

where .The function , resulting in simple

periodogram smoothing, is modified to restrict the influence ofoutliers and is given by [18]:

ifififif

(4)

where and values are chosen within the range of the requiredcapped region and results in zero restriction (i.e., nocapping). The estimated value of is thus definedas the solution [18] of

(5)

With function, proof of (5) is given in Appendix A. Variableis related to as given below [18]:

if(6)

Equation (5) is solved for the value usually by using a nu-merical root solving method such as regula falsi method. Thesemethods start with two points: the maximum and min-imum values of spectral amplitudes values of time-se-ries. Some restrictions in the form of small value is requiredto ensure the opposite signs of respective values (lines 7–10of Algorithm1).

C. Contextual M Spectrum Estimator (C-MSPEC)

In the standardM spectrum estimator, the two valuesand are taken as the initial values and updated itera-tively with the following equations:

(7)

ifif

(8)

where is the iteration counts. This algorithm finds the con-vergence of the sequence to the solution . To im-prove the convergence, we propose a modified M spectral es-timator (C-MSPEC) algorithm that incorporates the spectrumamplitudes of the neighbors into the algorithm.Since the neighbor voxels in the brain have similar contin-

uous spectrum, we propose to initialize the value in (5) atfour points (lines 11–20 of Algorithm 1): two of the points are as

those mentioned in the standard algorithm and the other two areselected by the maximum and minimum values of continuousspectrum of the neighborhood voxels at the same frequency(lines 11–13 of Algorithm 1). From these four points, two pointsare selected to have values nearest to zero and of oppositesigns (lines 14–20 of Algorithm 1).The modified algorithm is given in Algorithm 1. Our algo-

rithm will result in a substantial reduction in the number of it-erations required to arrive at the solution for for a largenumber (not all) of voxels. Voxels that have no processed neigh-bors such as the first voxel follows the standard algorithm.The discrete spectrum amplitude value of brain voxelat stimulus frequency are calculated as,

. The likelihood of activation in brain voxels,, is calculated

assuming that values across the brain voxels pooled to-gether are normally distributed in the absence of any activation.

and denote the mean and standard deviation ofthe distribution, respectively.

D. Modeling Contextual Information of Activation

In the brain, neighboring voxels are likely to have the samefunction, so spatial correlations exist among the activated voxels

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and need to be taken into account when determining the acti-vation patterns of the brain. The aim is to find the optimal ac-tivation pattern by using the maximum a posteriori (MAP)estimator:

(9)

In order to estimate the brain activation pattern, the pair isassumed to form a conditional random field (CRF) [15]. Then,

(10)

where represents the neighborhood of voxel .From the fundamental theorem of random fields [12], the

above distribution is given by a Gibbs distribution:

(11)where represents the single-voxel potential at voxeland the potential of voxel interaction with the

neighborhood voxels [32].The single-voxel potential in (11) is decomposed as

(12)

which consists of representing the priorinformation of activations and repre-senting the data-dependent single-voxel potential. Let the priorpotential be a constant over the brain. Thedata-dependent term is calculated using the C-MSPECalgorithm.The pairwise potential term in (11) is then given by

(13)

where represents the data-independent joint potentialand represents the data-dependent joint potential.For spatial interactions, we use twenty six nearest neighbors in3D and assume that the potentials are constant over the image.The data-independent interaction term is set to encourage

contiguous voxels of the same class:

(14)

and the data-dependent term

(15)

where and represents the probability of voxeltime-series, given the state of activation of the voxels and re-spectively, and are determined using C-MSPEC estimator. And

and are positive constants.From (9)–(11), the MAP estimate of the activation pattern is

given by

(16)

However, computing the optimal activity pattern by using theabove equation and potential values is a computationally inten-sive task. We used the mean field approximation to efficiently

compute the optimal solution. The derivation of the mean fieldapproximation to the above estimate is given in Appendix B.

III. RESULTS

A. Activation DetectionThe proposed method is demonstrated on synthetic data as

well as on real fMRI data. Synthetic data was used to quantita-tively evaluate the performance and robustness of the methodon different correlation structures. Functional MRI data gath-ered from a memory retrieval task was used in the testing. Forthe CRF, the prior probability was used for acti-vated voxels as the usual range of activated voxels is 2–5% inreal fMRI images. Small changes in do not affect the resultsnotably. Values of , around 1 were empirically found togive optimal accuracy with the synthetic images. These param-eter values were used for detection of activation from the realimages as well.1) Synthetic Data: Two-dimensional synthetic fMRI

datasets of 64 64 pixels per scan with 16 images per epochand 2 s interscan interval were generated. For simulating thedata, a stimulus cycle containing 4 s of stimulations and 28 s ofrest time was used. fMRI time-series response of the activatedpixels was obtained by convolving the stimulus with a hemo-dynamic response function defined by a mixture of twogamma functions [31]:

(17)

where , delay of response , delay of under-shoot , dispersion of response , dispersion ofundershoot , ratio of response to undershoot ,the length of kernel , represents the gamma func-tion and .In order to test the robustness of the method, synthetic data

was added with temporal autocorrelation at different complexi-ties. By using auto-regressive models at various ordersof : , we generated AR(1) modelwith , AR(2) with , and AR(3)with . The synthetic data wasgenerated with different AR models affecting specific regions:in Fig. 1(a), white/grey regions of circles were added withAR(3) autocorrelation, of diagonals with AR(2) and of outsidethe circle with AR(1) autocorrelation. The experiments wereperformed at signal-to-noise ratio (SNR) of 0.9, 1.2 and 1.9.SNR is defined as where is the amplitude of stimulussignal and is the standard deviation of autocorrelated noisedefined as , beingthe standard deviation of uncorrelated noise. The non-activatedpixels were left unchanged at zero amplitude [38]. Spatiallycorrelated Gaussian noise was added to all the brain pixels.The results of C-MSPEC method were compared with

the popular SPM method that implements a GLM on fMRItime-series [31] and with the independent component analysis

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Fig. 1. (a) Synthetic image: black portions are non-activated and grey/white portions are activated with different temporal autocorrelations (AR1-AR3), and theROC curves for synthetic data at (b) , (c) , and (d) .

(ICA) method [30]. When generating the results for SPMpackage, we have used time and dispersion derivatives toaccount for the small variations in hemodynamic responses andto correct temporal autocorrelation, assuming AR(1) model.Spatial smoothing with a Gaussian kernel was performed asa part of preprocessing for both SPM and ICA. With the ICAmethod, we chose the ICA component that corresponds tothe input stimuli. To evaluate the performance, the receiveroperating characteristic (ROC) curves were plotted by varyingthe threshold probability value for detecting activation. Theresults are given in Fig. 1. As is clear from the figure, ICAmethod did not perform very well and consistently. Note thatwe consider only the specificity range of 0.95–1.00 in the ROCcurves for computing the area under the partial ROC curves(AUC) values as the threshold point is very likely to be in thisrange. As is evident from ROC curves and AUC values, for twoof the three datasets, the C-MSPEC method has better results.The detected activation by the different method are given in

Fig. 2. The AUC values are shown in Table I. For the datasetgenerated at , the results were similar for both SPMand MSPEC method. This is understandable, as for this lowSNR data, the same HRF is used for both generation of dataand detection in GLM, whereas the proposed method does notmake use of the HRF model in detection.

We have used a number of parameters such as and inCRF, window size and in the formulation of the C-MSPECalgorithm. In order to understand the effect of these vari-ables on our analysis, additional experiments were performed.Fig. 3 shows the effect of various ( ) values.We have used three synthetic datasets at different SNR valuesfor this analysis. Also shown in this figure are the results for

and , that corresponds to the results withoutCRF modeling. As seen, values around 1, gave good AUCvalues. In our analysis of synthetic and real datasets, we haveused . Similarly, to study the effects of variouswindow sizes, we used synthetic data and plottedthe ROC curves at various window sizes (without the CRFstep) as shown in Fig. 4. As seen, beyond the window size of 7,there is not much difference in the results. In our synthetic andreal datasets, we have used the window size of 7. The effect ofsmall changes in values was also found to be negligible asshown in Fig. 5. We have set as used in [18].2) Memory Retrieval Task: In a memory retrieval task, sub-

jects learned three different sets (sizes 4, 6, and 8) of letters priorto the actual experiment with a corresponding cue for each set.During each trial of the experiment, a cue for the set and a letterwere presented and the subject decided whether the letter corre-sponds to the indicated set. A delay of 2.0 s was maintained be-tween the presentation of the cue and the probe. Fourteen slices

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Fig. 2. Synthetic data results: Activation patterns realized with SPM (top row),ICA (middle row), and C-MSPEC method (bottom row) at (a) , (b)

, and (c) .

TABLE ITHE AUC VALUES OF PARTIAL ROC CURVES (SPECIFICITY RANGE:

0.95–1.00) COMPARING C-MSPEC, SPM, AND ICA RESULTS.

Fig. 3. Area under the ROC curves (AUC values) at differentvalues for C-MSPEC method.

were acquired with 19.2 cm FOV, 7 mm thickness, and 64 64matrix using single shot gradient echo-planar imaging (EPI) se-quences ( and ). Ten cycles of datawere used for analysis. Details of the experiment are availablein [28].

Fig. 4. ROC curves at different window sizes of C-MSPEC method.

Fig. 5. ROC curves at different values of C-MSPEC method.

Brain processes involved in the experiment include encodingof the cue and probe, retrieval of information from the sec-ondary memory, scanning of the primary memory, response se-lection, and response execution [28]. Brain activation was de-tected using the C-MSPEC method and the SPM method [31].Fig. 6 shows activation detected on three representative slicesby the two methods. The cue and the probe both generated ac-tivation in the brain at same frequency but with an interval of2.0 s apart. The temporal differences in stimulus presentationappear as differences of the phases in frequency domain. In theC-MSPEC method, brain regions corresponding to the probeand cuewere thus separated based on the phase of voxel time-se-ries. In the SPM method, activations corresponding to cue andprobe were found by properly designing the design matrix, sep-arately taking into account the cue and the probe stimuli.Both of the methods effectively detected activation in the ex-

pected cortical areas of inferior precentral sulcus (pCS) and pos-terior middle frontal gyrus (MFG) following cue and in regionsof posterior parietal cortex following the probe [26]. Inferiorprecentral sulcus and posterior middle frontal gyrus are gen-erally involved in the retrieval of information from secondarymemory and also in maintaining this information for searchfollowing the presentation of the cue, hence are also activatedduring the probe.

B. Efficiency of C-MSPEC MethodIn order to test the effectiveness of our proposed modifica-

tions in the MSPEC algorithm, we compare the performanceof the C-MSPEC algorithm (Algorithm 1) with the standard

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Fig. 6. The detected activation of memory task by (a) the SPM method for thecue condition, (b) the SPM method for the probe condition, (c) the C-MSPECmethod for the cue condition, and (d) the C-MSPEC method for the probe con-dition.

MSPEC algorithm (Algorithm 1 without lines 11–20). Therewas no significant difference in the activation detected but thecomputational time was improved significantly with the modi-fied initialization. The average computation time on one subjectbrain with the C-MSPEC was 139.8 s compared with 159.0 sfor the standard MSPEC on a ACPI x64-based PC with an Inteli7 processor and 8 GB RAM. Fig. 7 shows the histogram of thenumber of iterations required for the detection of activation onbrain voxels on a representative scan. As seen, our modificationin the MSPEC algorithm significantly reduces the number of it-eration required for the detection of brain activation.

C. Detection of Other Prominent Frequency Components

We demonstrate an application of our technique on detectingother confounding frequencies of the brain. The C-MSPEC

Fig. 7. The number of iterations required to calculate continuous spectrum inthe memory-task data on one subject by the C-MSPEC method and the MSPECmethod.

Fig. 8. Significant frequency components present in the inferior brain slicesof a representative subject performing the memory task. Dashed lines show thelocation of stimulus frequency and its harmonics that have been removed.

method was applied on the voxel time-series of inferior slicesof the brain, which are likely to have significant physiolog-ical signals during the memory-retrieval task. In this case,frequency components other than the stimulus frequency andits harmonics were analyzed using the C-MSPEC method.Fig. 8 shows the histograms of the various frequency com-ponents that are significant on a representative subject. Notethat the stimulus related frequencies and their harmonics havebeen removed in the figure in order to clearly show the otherfrequency components. The most prominent frequency, that is,most statistically significant frequency, which was present inmost number of voxels corresponds to 0.1176 Hz. In order tounderstand the significance of this frequency component, weapplied spatial ICA [30] on these slices.Two independent components (ICs) were selected based on

the knowledge of anatomical regions that are likely to havestrong presence of physiological signals [4]. Fig. 9(a) shows theslices and time and frequency domain signals of these compo-nents. Both the components showed a prominent presence in themain trunk of middle cerebral artery: one component (Fig. 9(a))was significant in the branch of middle cerebral artery and theother (Fig. 9(c)) in posterior cerebral artery. Based on the FFTanalysis for component 1 and 2, frequency 0.1176 Hz was foundto be most significant as shown in Fig. 9(b) and (d). This is thesame frequency component that was also found to be signif-icantly present in the inferior location brain slices during theanalysis with the C-MSPEC method. Hence we can concludethat the proposed method is also able to detect the other fre-quencies such as physiological signals.

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Fig. 9. Spatial ICA components related to physiological signals: (a) and (c) show the location of these signals, and (b) and (d) show temporal variations of thesecomponents (top) and the FFT analysis results (bottom), The dashed lines indicate the stimulus and its harmonic frequencies.

IV. CONCLUSION

We presented the mixed spectrum analysis of fMRI data andits use in handling temporal autocorrelation. The frequencyspectrum of fMRI time-series was modeled with a mixedspectrum: a component having a discrete spectrum relating totask-related stimuli and a component having a continuous spec-trum comprising of temporal autocorrelation. With this model,the C-MSPEC method was proposed to find the amplitudeof discrete component of the stimulus frequency at the brainvoxels and the likelihood of activation at a particular brainvoxel. As we expect the neighborhood voxels to have similarcontinuous spectrum, the C-MSPEC method used the spectraof the neighbors when calculating continuous spectrum. Thisextension significantly improves the computational time of theMSPEC method.Spatial correlation among the likelihood of activated brain

voxels wasmodeled with a CRF. Compared to Gaussian randomfields [31] and MRF [29], [25], CRF accounts for data-depen-dent nature of neighborhood interactions better and significantly

improves the detection of activation [32]. The likelihood of ac-tivation of voxel given by the C-MSPEC method and the priorprovided by CRF allowed us to find MAP estimation of brainactivation.The main advantage of C-MSPEC method is that it is not

necessary to assume an a priori model of temporal autocor-relation or any form of hemodynamic response function. Themethod handles the autocorrelation in a unique way by sepa-rating it from the discrete spectra related to input stimuli. In theexperiments with synthetic data, the C-MSPEC method outper-formed the SPM method and ICA method in most of the cases.With real fMRI data gathered from a memory-related task data,the results from the C-MSPEC method were visually similar toSPMmethod; and the activations were more focal and localizedto the cortical areas.The C-MSPECmethod is only applicable for periodic stimuli.

It is also possible to look for any other significant confoundingfrequencies with our approach. We demonstrated this with anapplication of the C-MSPEC method in detecting the physio-logical signals present in real fMRI. It is worthwhile exploring

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more applications of the mixed spectral analysis and C-MSPECestimator for task-related and resting state fMRI data.

APPENDIX APROOF OF (5)

Considering (5) and (4) and using only the uncapped part of(4), we will prove that it leads to reasonable estimate of .It turns out to be similar to the way periodogram smoothing isused to detect the . The capped part in (4) is basically toremove the effect of outliers in periodogram, which makes thisalgorithm more effective than periodogram smoothing.Substituting the uncapped part of (4) in (5),

(18)

(19)

Since the window is considered such that,

(20)

Equating , to calculate the roots of the equa-tion, we get

(21)

(22)

(23)

That is, if we use only the uncapped part of (4), it leads tocalculation of using periodogram smoothing, which is wellknown to be a reasonable estimate of continuous spectrum inthe absence of outliers.

APPENDIX BMEAN FIELD APPROXIMATION OF THE MAP ESTIMATE

The CRF model to compute the posterior probabilities ofvoxel labels, as described in (11)–(15), is effectively imple-mented using the variational mean field approach. The poste-rior probability is approximated by a much simpler distribu-tion through the minimization of KL-di-

vergence to determine the approximate distribution [25]. Theterm denotes the approximated probability that thevoxel contains an activated brain signal.The K-L divergence is defined as

(24)

where the summation is taken over the presence and absenceof stimulated voxel time-series: . Substituting thevalues from (11)–(15),

(25)

which needs to be minimized based on the constraint. Using Lagrange multipliers, the above

optimization problem can be written as

(26)

Differentiating with respect to and by setting thederivatives equal to zero:

(27)This finally leads to the iterative update rule:

(28)where the normalization constant ensures that

and

(29)(30)

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