18-08-2012, 05:08 PM
MULTISCALE ANALYSIS OF FMRI DATA WITH MIXTURE OF GAUSSIAN DENSITIES
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ABSTRACT
We describe in this work an exploratory analysis of fMRI data. We
regard the fMRI dataset as a set of spatiotemporal signals sx indexed
by their position x. The analysis is performed on the wavelet
packet coefficients of the fMRI signals sx, and we show that we
can characterize the coefficients in terms of a mixture model of
multivariate Gaussian distributions.
INTRODUCTION
Functional Magnetic Resonance Imaging (fMRI) can quantify hemodynamic
changes induced by neuronal activity. The goal of the
analysis is to detect the “activated” voxels x where the dynamic
changes in the fMRI signal sx(t), t = 0, · · · , T −1 can be considered
to be triggered by the (sensory or cognitive) stimulus. Statistical
techniques are commonly used for the detection of activated
voxels. Unfortunately, these techniques often rely on oversimplified
assumptions. We describe in this work an exploratory analysis
of the fMRI data. We regard the fMRI dataset as a set of spatiotemporal
signals sx, indexed by their position x. Our analysis is not
performed directly on the raw fMRI signal. Instead, the raw data
are projected on a set of basis functions conveniently chosen for
their ability to reveal the structure of the dataset. Several studies
indicate that one finds dynamic changes of the fMRI signal in time
and in frequency [1]. Wavelet packets are time-frequency “atoms”
that are localized in time and in frequency. We favor therefore the
use of wavelet packets to perform the analysis of fMRI data.
LOCAL STATISTICS OF THE COEFFICIENTS
In the previous section we studied the empirical distribution of the
wavelet packet coefficients computed over the entire brain. While
we expect the background signal to be constant (except for a possible
slowly varying drift), the strength of the signal sx may vary
from one activated voxel x to another. We can interpret the empirical
distribution of αx(γ0) in Fig. 4 as follows : the activated region
is composed of a small number of subregions wherein αx(γ) is
Gaussian distributed ; and the mean is different for each subregion.
The distribution of αx(γ0) can be thus described by a mixture of
Gaussian distributions. We can test this hypothesis by performing
a local analysis of the distribution of αx(γ). For each position of
the neighborhood N(x0) we test the hypothesis that the distribution
of the coefficients {αx(γ), x ∈ N(x0)} is Gaussian. Several
test for normality exist, and we use the W Shapiro-Wilk test [5]
because of its ability to detect non Gaussian distributions. As x0
is moved throughout the brain we collect many samples for theW
statistic. We can compute the empirical distribution of W (for all
values of x0), as is shown in Fig. 5 and Fig. 6. In order to quantify
non-normality, we compare this empirical distribution with the
distribution of W obtained under the normality assumption. The
comparison is performed using the χ2 distance.