A multidimensional similarity measure for bilateral adaptive filtering of fMRI data

A multidimensional similarity measure for bilateral adaptive filtering of fMRI data
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  A Multidimensional Similarity Measure for Bilateral AdaptiveFiltering of fMRI Data J. Rydell, H. Knutsson and M. BorgaDepartment of Biomedical Engineering, Link¨oping university, SwedenCenter for Medical Image Science and Visualization (CMIV), Link¨oping university, Sweden Abstract  In analysisof fMRI data, it is commonto average neigh-boring voxels in order to obtain robust estimates of thecorrelations between voxel time-series and the modelof the signal expected to be present in activated re-gions. We have previously proposed a method whereonly voxels with similar correlation coefficients are av-eraged. In this paper we extend this idea, and present a novel method for analysis of fMRI data. In the pro- posed method, only voxels with similar correlation co-efficients and similar time-series are averaged. The proposed method is compared to our previous method andto two well-knownfiltering strategies, andis shownto have superior ability to discriminate between activeand inactive voxels. 1 Introduction Analysis of functional MRI data deals with the prob-lem of detecting very weak signals in very noisy data.The common solution to this problem is to averagethe time series from neighboring pixels or voxels, andthereby enhance the signal to noise ratio [3]. In prac-tice, this is done by convolving the slices (or volumes)with a fixed low-pass filter kernel, e.g. a gaussian. Theprice to pay for this kind of noise reduction is loss of spatial resolution. Loss of spatial resolutionmeans thatthe shape of activated regions cannot be accurately de-termined and, perhaps worse, that small activated re-gions may remain undetected.Inordertomaintainhighspatialresolution,thespa-tial low-pass filtering can be made  adaptive  and  non-isotropic . This means that for each voxel, the size andshape of the local region, in which the averaging isperformed, is data dependent. A method for adaptivespatial filtering based on canonical correlation analysis(CCA)haspreviouslybeensuggested[2]. Thatmethodchooses the size and shape of the local averaging re-gion, i.e. the resulting adaptive filter, such that the correlation  between the averaged time series and themodel of the blood oxygen level dependent (BOLD)response model is maximized. This makes the methodvery sensitive. Indeed, the method is so sensitive thatrestrictions have to be imposed on the number of pa-rameters in the adaptive filter and their ranges in orderto maintain a reasonable selectivity. If given too muchfreedom,the methodmay find false signals in the noisesince the filter is optimized for making the filter outputas similar to the BOLD response model as possible.Another problem with this method is that when the fil-ter is centered in a non-activated voxel but close to anactivated region, the filter will try to ”reach in” to theactivated region in order to pick up as much activationas possible. This will make the resulting regions la-beled as active become larger than they should be, i.e.a growing of activated regions will occur.Wehavepreviouslyproposedanalternativemethodfor adaptive filtering [6]. That method is based on av-eraging of voxels which have similar correlation withthe BOLD model, and has the advantagethat edges be-tween active and inactive regions are preserved. Weherepresentanextensionofthisfilteringscheme,wherevoxels to be averaged are not only required to havesimilar correlation with the BOLD model, but shouldalso have similar time-series. We also show that thismodification providesa significant improvementof thedetection performance. 2 Theory When ordinary low-pass filtering is used for noise re-duction,voxels that are spatially close to each otheraretreatedassamplesfromonedistribution,andaweightedaverage of the voxels in a neighborhood is used as anestimate of the true signal value in the center of that re-gion. The weights are predetermined and based on thedistance from the center of the neighborhood. Close toedges in an image, the voxel values are actually sam-ples from two or more distributions, and using pre-determined weights for averaging causes blurring of the edges. Bilateral filtering [5, 7] extends low-passfiltering by also considering the distance between thevalue of a certain voxel and that of the center voxel,thereby creating a different filter kernel in each neigh-borhood. This approach causes voxels from the otherside of an edge to be treated as outliers, and thus theireffect on the estimate of the true signal value is re-duced or eliminated. An example of using low-passfiltering and bilateral filtering, respectively, of a noisyone-dimensional signal is shown in figure 1. The sig-  (a) Noisy data (b) After low-passfiltering(c) After bilateralfiltering Figure 1: Noisy data before and after low-pass and bi-lateral filtering.nal is a step function with additive gaussian noise, andit is obvious that low-pass filtering causes blurring of the edge while it is preserved by bilateral filtering.The bilateral filter kernel in each neighborhoodcanbe expressed as a product of two filter kernels: the spa-tial filter  F  s  and the range filter  F  r . The spatial filter isbased on spatial distance, and corresponds to the filterkernel used in low-pass filtering, while the range fil-ter is based on the difference in image intensity. Thatis, given an image  I  ( x,y ) , the bilateral filter kernel F  (∆ x, ∆ y )  at image coordinates  ( x,y )  can be written F  (∆ x, ∆ y ) =  F  s (∆ x, ∆ y )  ·  F  r (∆ x, ∆ y )  (1)where  F  s (∆ x, ∆ y )  is an ordinary spatial filter kernel g (∆ x, ∆ y )  and the range filter is defined as F  r (∆ x, ∆ y ) =  h ( I  ( x +∆ x,y  +∆ y ) − I  ( x,y ))  (2)A common choice of the filter kernels  g  and  h  is gaus-sian functions. 3 Method Godtliebsen et al [4] have proposed using bilateral fil-tering of the raw fMRI data, with a time dimension inaddition to the spatial and range dimensions describedabove. Our previous method is similar to bilateral fil-tering, but instead of basing the range filter on differ-ences in image intensity, we base it on the differencein correlationbetweenindividualvoxel time-series andthe BOLD model. Furthermore, instead of using thecorrelation coefficients directly, we use a mapping of the correlation. The reason for using this mapping isthat the correlation coefficients are not readily compa-rable on a linear scale. The mapping is defined as Λ( x,y ) = log   11  −  ρ ( x,y ) 2   (3)where ρ ( x,y )  isthecorrelationbetweenthetime-seriesat coordinates  ( x,y )  and the BOLD model. Under cer-tain conditions this measure, which is the logarithm of Wilks’ lambda, is equivalent to mutual information.Hereweproposeanextensionofthepreviousmethod,where we use  two  range filters. One of these ( F  r 1 ) isidentical to the range filter described above, while theother( F  r 2 )isbasedonthesimilaritybetweentheinten-sity time-series themselves. That is, two spatially closevoxelsare averagedif theirindividualcorrelationswiththe BOLD model are similar  and   their time-series re-semble each other.Often, the BOLD model used in fMRI data analy-sis is a linear subspace model, i.e. a model with two ormore temporal basis signals. The correlation betweena time-series and the model is then defined as the high-est correlation between the signal and any linear com-bination of the model basis signals. The model basiscan, for example, be generated by performing prin-cipal component analysis of a large number of simu-lated BOLD responses, generated by Buxton’s balloonmodel [1]. We propose that such a subspace modelis used, and use the angle between the projections of two time-series onto the model subspace as a measureof similarity between the two time-series. (In exper-iments with more than one stimuli, a linear subspacemodel can instead be based on the expected responsesfrom each of the stimuli.) By measuring the angle inthe signal subspace, large random variations that aredue to the high noise levels in the data are disregarded.If the time-series were directlycomparedto each other,any similarity would remain undetected because of thenoise.Simplycombiningthespatialandrangefilterswould,at each coordinate  ( x,y ) , yield a filter F  (∆ x, ∆ y ) =  F  s (∆ x, ∆ y ) · F  r 1 (∆ x, ∆ y ) · F  r 2 (∆ x, ∆ y ) (4)which averagesovervoxelsthat are close to each other,where the correlation with the BOLD model is similar,and where the projection of the signal onto the BOLDmodel basis functions is similar. However, this is gen-eralized slightly by introducing the parameters  α ,  β  and  γ   as follows: F  (∆ x, ∆ y ) =  (5) F  s (∆ x, ∆ y ) α ·  F  r 1 (∆ x, ∆ y ) β ·  F  r 2 (∆ x, ∆ y ) γ  These parameters can be used to tune the relative im-portance of the different filters. The parameters caneven be variable, to accommodatedifferent weightingsof the filter kernels in different neighborhoods. Thismakes the proposedmethod verygeneral. We do, how-ever, here propose specific choices of the parameters.As was mentioned in the last section, it is commonto choose  F  s  and  F  r  to be gaussian functions. Accord-ingly, we suggest that all of   F  s ,  F  r 1  and  F  r 2  are se-lected as such. Thus, F  s (∆ x, ∆ y ) = exp  − d s (∆ x, ∆ y ) 2 2 σ 2 s   (6) F  r 1 (∆ x, ∆ y ) = exp  − d r 1 (∆ x, ∆ y ) 2 2 σ 2 r 1   (7) F  r 2 (∆ x, ∆ y ) = exp  − d r 2 (∆ x, ∆ y ) 2 2 σ 2 r 2   (8)  where the distance measures are defined as d s (∆ x, ∆ y ) =   ∆ x 2 + ∆ y 2 (9) d r 1 (∆ x, ∆ y ) =  (10) Λ( x,y )  −  Λ( x  + ∆ x,y  + ∆ y ) d r 2 (∆ x, ∆ y ) =  (11) arccos( ˆw ( x,y )  ·  ˆw ( x  + ∆ x,y  + ∆ y )) The different  σ :s are the standard deviations of the re-spective gaussian functions and  ˆw ( x,y )  is the projec-tion direction in the subspace model for the time-seriesat coordinates  ( x,y ) .The values of the exponents  α ,  β   and  γ   should bein the range from  0  to  1 , where  0  means that the filterhas no effect and  1  means that the filter has full ef-fect. This implies that setting  α  =  β   = 1  and  γ   = 0 yields our previous method as a special case. We pro-pose that these parameters are used as weights for thedifferent filters according to the certainties of their re-spective distance measures. The exact spatial distanceis always known, and thus its certainty  α  = 1 . Thereis no good certainty estimate for the correlation, andthus we also propose that  β   is constant, for example β   = 1 . However, the certainty of the projection ontothesubspacemodelisrelatedtoourestimate ofthecor-relation. The higher the correlation estimate, the morecertain the projection direction is. Thus we select γ  (∆ x, ∆ y ) =  4   | ρ ( x,y ) ρ ( x  + ∆ x,y  + ∆ y ) |  (12)i.e. the square root of the geometric mean of the cor-relations in the two pixels under consideration. Thechoice of the square root is not of crucial importance,but it appears to provide a better weighting than usingthe geometric mean directly. Then, in regions wherethe correlation is high, the filter based on time-seriessimilarity will be important, while in other regions itwill have little or no effect. This is an advantage inboth active and inactive regions. In inactive regions,the correlation is low and the similarity between thetime-series is random. By ignoring the second rangefilter ( F  r 2 ) in these regions, the final filter will aver-age over larger areas, thus reducing the probability of finding spurious correlations in the noise. In these re-gions,  F  r 1  precludes filters that would pick up signalfrom activated voxels. In active regions, on the otherhand, the correlations are higher and thus  F  r 2  has ef-fect. This decreases the risk of extending the effectivefilter beyond the active region.An example of the different filter kernels is shownin figure 2. Figure 2a shows where activity has beenembedded in the noise in an artificial data set. In fig-ure 2b, the spatial filter  F  s  is shown. Figures 2c andd show the range filters  F  r 1  and  F  r 2  when they are lo-cated in the dashed square in figure 2a. In this case,the center pixel is located in an activated region. It isclear that the two range filters complement each other,excluding pixels outside of the activated region fromthe averaging. In figure 2e the resulting filter obtained (a) Activatedlocations(b) Spatial filter F  s (c) Range filter F  r 1 (d) Range filter F  r 2 (e) Resulting filter F  (f) Resulting filterwith inactivecenter pixel Figure 2: Example of filter kernels based on the differ-ent distance measures, and final filter combined usingequation 5. The resulting filter in figure e is used forweighting the time-series in the region surrounded bythe dashed line in figure a.by combining  F  s ,  F  r 1  and  F  r 2  according to equation5 is shown. The coefficients in this filter are used asweights for averaging the time-series in the marked re-gion. As can be seen in the figure, the filter has almostzero weight for inactive pixels but large weights forspatially close pixels with activation similar to that of thecenterpixel. Ifthecenterpixelhadbeenlocatedbe-side the active region, a filter with large weights for in-active pixels and small weights for active pixels wouldinstead have been obtained. Such a resulting filter,where the center pixel is just outside of the active partof the marked regionin figure 2a, is shown in figure 2f.When the filters  F  (∆ x, ∆ y )  have been created ateach coordinate  ( x,y ) , they are used to filter the rawdata in each timepoint. After this, each time-series inthe resulting data is analyzed separately to detect acti-vation.It is important to notice that this is different fromcalculating the correlation in each pixel and then per-forming bilateral filtering of the correlation map. 4 Results and discussion The proposed method has been evaluated on both realandsyntheticdata. Figures3b-eshowcorrelationmapsgeneratedby analyzingsimulateddata usingfixed low-pass filtering, adaptive filtering using CCA, adaptivefiltering using our previous method and adaptive filter-ing using the proposedmethod,respectively. The areaswhere BOLD-like signals were embedded in the noiseare shown in figure 3a. The signal to noise ratio of thesimulated data is approximately 5 – 10 %. Brighter re-gions in figure 3a have higher SNR. The noise is gaus-sian, with spatial autocorrelation similar to that found  (a) Locations withsimulatedactivation(b) Fixedlow-passfiltering(c) Adaptivefiltering basedon CCA(d) Our previousmethod(e) The proposedmethod Figure 3: Locations with simulated activity and activeregions detected using the different analysis methods.in real fMRI data.In figure 4, receiveroperating characteristic (ROC)curves,showingthesensitivity(abilitytocorrectlyclas-sify active voxels) versus the specificity (ability to cor-rectly classify inactive voxels) of the different meth-ods, are displayed.It is evident from the ROC curves that the meth-ods based on bilateral filtering have superior ability todiscriminate between active and inactive voxels in thesimulated data. This is also supported by the corre-lation maps in figures 3d-e, which show sharper edgesbetweenactiveandinactiveregionsthanthecorrelationmaps generated by the CCA method and the methodbasedona fixedfilter. This edge-preservingpropertyisclearly an advantage of these methods. While the vis-ible difference between the correlation map from ourprevious method and that from the proposed method isnot very large, the ROC curves clearly show that theproposed inclusion of a second range filter, based ontime-series similarity, provides a further enhancementof the detection performance. This is to be expected,since the new range filter reduces the risk of creatingtoo large filters.Figure5showsactivationdetectedinrealdatafroma finger tapping task, overlaid on an anatomical imageof the brain. The activation in the motor cortex is con-sistent with the task. 5 Conclusion A new method for adaptive filtering of fMRI data hasbeen presented and evaluated. The method, which isbased on bilateral filtering, extends our previous fMRIanalysis scheme. Experimentalresults have shown that 10 −4 10 −3 10 −2 10 −1 10 0 − specificity        S     e     n     s      i      t      i     v      i      t     y Adaptive (proposed)Adaptive (previous bilateral)Adaptive (CCA)Fixed low−pass filter Figure 4: ROC curves for the different analysis meth-ods. The proposed method provides the best detectionperformance.Figure 5: Activation detected using the proposedmethod on real data from a finger tapping experiment.As expected, the detected activation resides in the mo-tor cortex.thenewmethodprovidesimprovedactivationdetectionperformance. References [1] R.B. Buxton, E.C. Wong, and L.R. Frank. Dynamicsof blood flow and oxygenation changes during brain ac-tivation: the Balloon model.  Magnetic Resonance in Medicine , 39(6):855–864, 1998.[2] O. Friman, M. Borga, P. Lundberg, and H. Knutsson.Adaptive analysis of fMRI data.  NeuroImage ,19(3):837–845, 2003.[3] K.J. Friston, P. Jezzard, and R. Turner. Analysis of func-tional MRI time-series.  Human Brain Mapping , 1:153–171, 1994.[4] F. Godtliebsen, C.-K. Chu, S. H. Sørbye, andG. Torheim. An estimator for functional data with appli-cation to MRI.  IEEE Transactions on Medical Imaging ,20(1):36–44, 2001.[5] F. Godtliebsen, E. Spjøtvoll, and J. S. Marron. A non-liear Gaussian filter applied to images with discontinu-ities.  Nonparametric Statistics , 8:21–43, 1997.[6] J. Rydell, H. Knutsson, and M. Borga. Correlation con-trolled adaptive filtering for fMRI data analysis. In  Proceedings of the 13th Nordic-Baltic conference onbiomedical engineering and medical physics (NBC’05) ,Ume˚a, Sweden, June 2005. NBC.[7] C. Tomasi and R. Manduchi. Bilateral filtering for grayand color images. In  IEEE International Conferenceon Computer Vision 98  , pages 839–846, Bombay, India,January 1998. IEEE.
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