good detection performance for activated regions of differ- ent shapes can be obtained. An alternative adaptive filtering method, based on rotating and scaling a ...
DIMENSIONALITY AND DEGREES OF FREEDOM IN FMRI DATA ANALYSIS - A COMPARATIVE STUDY Joakim Rydell, Magnus Borga, Peter Lundberg† and Hans Knutsson Medical Informatics, Department of Biomedical Engineering † Department of Radiation Physics Link¨oping University, Sweden http://www.imt.liu.se/mi/
ABSTRACT Two- and three-dimensional isotropic and anisotropic spatial filters for adaptive fMRI data analysis are compared in terms of activation detection sensitivity and specificity. Evaluations using both real and artificial data are presented. It is shown that three-dimensional anisotropic filters provide superior activation detection performance.
This paper presents a comparison between using steerable filters and using a set of two isotropic filters of different spatial extent for CCA-based fMRI data analysis. The filter sets are compared in two-dimensional and three-dimensional analysis.
2. THEORY 1. INTRODUCTION Canonical correlation analysis (CCA) has previously [3, 6, 2] been shown to work well for detecting neural activity in fMRI data. In contrast to the general linear model (GLM), CCA finds a combination of outputs from several spatial filters that is optimal in the sense that it correlates maximally with the specified model of the blood oxygen level dependent (BOLD) signal. Different sets of spatial filters have been suggested, ranging from simple voxel-based filters to the steerable filters introduced by Knutsson et al [5]. By adaptively combining the steerable filters using CCA, good detection performance for activated regions of different shapes can be obtained. An alternative adaptive filtering method, based on rotating and scaling a spatial smoothing filter, has been proposed by Shafie et al [7]. Earlier, fMRI data has mostly been considered one slice at a time, and therefore two-dimensional filters have been used. In two dimensions, four spatial filters (one isotropic and three oriented filters), whose outputs are linearly combined by CCA, are used to create steerable filters. By exploiting the correlation between adjacent slices, even better activation detection performance can be expected. In three dimensions, however, one isotropic and six oriented filters are needed to be able to form filters in any direction. The outputs of seven filters can be combined in a very large number of different ways, and the risk of falsely declaring voxels to be active may increase with the number of degrees of freedom if the possible combinations are not properly constrained.
Canonical correlation analysis was introduced by Hotelling [4] as a method for finding the maximum correlation between linear combinations of two sets of variables. That is, given two multivariate variables x = (x1 , x2 , ..., xm )T and y = (y1 , y2 , ..., yn )T , CCA finds two projection vectors wx and wy such that the correlation between the projections x = wxT x and y = wyT y is maximized. This maximum correlation is called the canonical correlation. When CCA is used for fMRI data analysis, x represents the outputs of the spatial filters and y represents the basis vectors for the subspace of BOLD signals considered to match the paradigm. Because of linearity, filtering the data spatially and then combining the filter outputs (as done by CCA) is equivalent to first combining the filters and then using the resulting filter on the data. That is, N ¡X i=1
N X ¢ Fi ∗ It = (Fi ∗ It ) i=1
where Fi is the i:th filter kernel and It is the (two- or threedimensional) data at time t. To reduce the risk of overestimating the correlation in voxels with no paradigm-related activation, restricted CCA [1] is used to enforce certain restrictions on the vectors wx and wy . This restricts the possible filter shapes and BOLD signals to plausible choices [2].
(a) Isotropic filters
Fig. 1. Activated regions in artificial data. (b) Anisotropic filters
3. MATERIALS AND METHODS Fig. 2. Two-dimensional filters. 3.1. Data To evaluate the different filter types, both real and artificial data was used. The real data consists of a finger tapping task, and the artificial data was generated by embedding a square wave in nulldata, i.e. data from a scanning session with no paradigm. The signal to noise ratio (SNR) of the artificial data is 0.5. Both data sets where acquired in a resolution of 64 × 64 × 32 isotropic voxels and sampled at 120 timepoints. The repetition time (TR) was 4 seconds, giving a scan time of eight minutes. The period of the paradigm (and the square wave) was 160 seconds, or 40 timepoints. The activated regions in the artificial data are shown in figure 1. 3.2. Filter sets To obtain the two-dimensional isotropic filters, a large number of gaussian low pass filters of different sizes were generated. The mean filter and the first principal component of the filter set were then used in the analysis. By combining these filters, approximately gaussian filters of different sizes can be produced. Figure 2a shows the shapes of the two-dimensional isotropic filters. In three dimensions, filters with the same radial function put too much weight far from the center. To obtain comparable filters in three dimensions, low pass filters with a radial function of e− σr22 r 6= 0 r F (r) = 1 r=0 were generated with different values of σ, and the mean filter and the first principal filter were used in the analysis. The anisotropic filters were constructed as described in [2], i.e. by creating an isotropic low pass filter and dividing it into one smaller isotropic and several oriented parts using different weight functions. In two dimensions, a gaussian filter was used as initial isotropic filter. Again, using a
three-dimensional gaussian filter puts too much weight far from the center, and thus an isotropic filter with the radial function described above was used in the generation of the three-dimensional anisotropic filters. The shapes of the twodimensional anisotropic filters are shown in figure 2b. The filters shown in the figure are larger than those actually used. 3.3. BOLD models When analyzing the artificial data, it is known that the signal to be found is the same square wave that was used to generate the data. Thus, that square wave should be used as BOLD model for the artificial data. In the analysis of the real data, however, the exact shape of the signal to find is unknown. Following the same pattern as in the generation of the isotropic filters, a large number of plausible BOLD signals were generated and the mean signal and first principal signal were used as basis signals. Linear combinations of these basis signals are assumed to span the subspace of all plausible BOLD signal shapes. 3.4. Pre-processing of data When the data is acquired, different slices of the volumes are sampled at different times. Since the slices are not acquired sequentially, but interleaved (the order in which they are sampled is s1 , s3 , ..., s31 , s2 , s4 , ..., s32 ) the time between the sampling of adjacent slices is approximately TR/2 = 2 seconds. This delay reduces the SNR improvement in three-dimensional filtering. Also, the BOLD signal in most slices is shifted relative to the paradigm, which reduces the detection sensitivity. To compensate for the sampling delay, the voxel timecourses in each slice were interpolated (using cubic interpolation) before the analysis. To improve activity detection performance, linear and quadratic drifts were removed from each voxel timecourse
(a) Isotropic, 2D
(b) Anisotropic, 2D
(c) Isotropic, 3D
(d) Anisotropic, 3D
Fig. 3. Detected activations in artificial data. before the data was spatially filtered. No spatial registration of the data was performed.
with higher accuracy by the anisotropic filters than by the isotropic filters. ROC curves for the different filter sets are shown in figure 4a. It is clear from these curves that the anisotropic filters perform better than the isotropic filters, and that the best detection performance is obtained by using the three-dimensional anisotropic filters. However, these ROC curves are based on the ability of the different filters to detect activation in the artificial data shown in figure 1, which contains a significant amount of anisotropic activated regions. This may overemphasize the importance of anisotropic filters. For comparison, a second artificial data set was created, consisting mostly of isotropic activated regions. As the ROC curves in figure 4b show, the performance of the three-dimensional anisotropic filters was superior also for this data set. This is explained by the possibility for anisotropic filters to average over large planar regions close to edges of activated regions. When using isotropic filtering close to edges of activation, the resulting filter will either be very small or average over a region that is only partly activated. The superior performance of the three-dimensional anisotropic filters is caused by their ability to form planes in any orientation, while the two-dimensional filters are restricted to planes within each slice. 1
4. RESULTS
1 Isotropic, 2D Isotropic, 3D Anisotropic, 2D Anisotropic, 3D
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4.1. Artificial data Figures 3a-d show the active regions detected by the different filter sets in the artificial data. The correlation threshold used for each filter was selected to minimize spurious activation while maintaining the sensitivity. For all the filters, this optimal threshold was found to be approximately 0.4. It is evident from the figures that three-dimensional filtering provides better detection performance for activated regions that range over several slices and affect few voxels within each slice. An example of this is the vertical line. This is true both for the isotropic and the anisotropic filter sets. It can also be seen that fine structures are detected
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An important difference between the real and the artificial data sets is that the correct classification (active or inactive) for each voxel is known for the artificial data. This makes an objective evaluation of activity detection performance possible. Because of this, the results are presented differently for the two sets of data. For both data sets, activation images showing typical detection performance are presented, but for the artificial data, receiver operating characteristic (ROC) curves showing the specificity (ability to correctly classify inactive voxels) and sensitivity (ability to correctly classify active voxels) are also included.
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(a) Artificial data shown in figure 1.
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(b) Artificial data isotropic activation.
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Fig. 4. ROC curves for artificial data. The large number of degrees of freedom in threedimensional anisotropic filtering does not introduce a problem with specificity. This is explained by the use of restricted CCA to impose constraints on the shape of the filters. 4.2. Real data In figures 5a-d, detected activation in the real data is shown overlaid on anatomical images. The correlation thresholds used in these figures are approximately 0.55. As in the artificial data, three-dimensional filters seem to be better at
(a) Isotropic, 2D
(b) Anisotropic, 2D
(c) Isotropic, 3D
(d) Anisotropic, 3D
Fig. 5. Detected activations in real data. detecting activation ranging over several slices. The difference between isotropic and anisotropic filtering in two dimensions is hardly noticeable, while the difference in threedimensional filtering is significant. Since the true activation is unknown, it is a difficult task to use real data to decide which filter set provides the best detection performance. 5. CONCLUSIONS The activation detection performance of CCA-based analysis of fMRI data using two- and three-dimensional isotropic and anisotropic spatial filters has been evaluated. It has been shown that anisotropic filters provide better performance than isotropic filters. It has also been shown that the large number of degrees of freedom in three-dimensional anisotropic filtering does not introduce specificity problems and that three-dimensional filtering is preferable to two-dimensional filtering. 6. ACKNOWLEDGEMENTS The authors wish to thank Dr. Ola Friman for valuable assistance. The financial support from the Swedish Research Council is gratefully acknowledged.
7. REFERENCES [1] S. Das and P.K. Sen. Restricted canonical correlations. Linear Algebra and its Applications, 210:29–47, 1994. [2] O. Friman, M. Borga, P. Lundberg, and H. Knutsson. Adaptive analysis of fMRI data. NeuroImage, 19(3):837–845, 2003. [3] O. Friman, J. Carlsson, P. Lundberg, M. Borga, and H. Knutsson. Detection of neural activity in functional MRI using canonical correlation analysis. Magnetic Resonance in Medicine, 45(2):323–330, February 2001. [4] H. Hotelling. Relations between two sets of variates. Biometrika, 28:321–377, 1936. [5] H. Knutsson, R. Wilson, and G. H. Granlund. Anisotropic non-stationary image estimation and its applications — Part I: Restoration of noisy images. IEEE Transactions on Communications, 31(3):388–397, March 1983. [6] R. R. Nandy, C. G. Green, and D. Cordes. Canonical correlation analysis and modified ROC methods for fMRI techniques. In Proceedings of the ISMRM Annual Meeting (ISMRM’02), Hawaii, USA, May 2002. ISMRM. [7] K. Shafie, B. Sigal, D. Siegmund, and K.J. Worsley. Rotation space random fields with an application to fMRI data. The Annals of Statistics, 31, 2003.
