Image Processing Reference
In-Depth Information
try to group similar objects together, the results will depend on the definition of
this similarity. Moreover, some clustering approaches such as partitioning
approaches require the experimenter to decide in advance the number of expected
clusters. In the context of fMRI data analysis, the elements to be classified are
the time series extracted from volume elements of the brain: an
-step-long time
series, describing the signal changes in a voxel, can be seen as an
n
n
-dimensional
vector, i.e., a point in
n
-dimensional space.
16.3.1
S
IMILARITY
The definition of similarity is a crucial concept, and different possible choices
can be made that can lead to different results. A simple definition of similarity
is the Euclidean distance in an
n
-dimensional space between two points (
x
,
y
)
given by
n
∑
d
( , ) ||
xy
=−=
x y
||
( ()
x
i
−
y
())
i
2
(16.1)
i
=
1
where
. The Mahalanobis distance is a gener-
alization of the Euclidean distance and can be written as
x
(
i
) is the
i
th element of vector
x
d
(,)
xy
=−
(
x y B
)
T
−
1
(
x y
−
)
(16.2)
where (
is chosen to be the identity matrix, then
the Euclidean distance is obtained. It is possible to choose
⋅
)
T
is the transpose operator. If
B
as a diagonal matrix
with the elements in the diagonal as the variances of each coordinate; multiplying
by the inverse of
B
is equivalent to weighting each coordinate by the inverse of
its variance, resulting in a normalization process. It is then possible to choose
B
BTT
=
T
(16.3)
In such a case the Mahalanobis distance is equivalent to the Euclidean after
the data have been transformed by
T
. This method was used in Reference 11, in
which the transformation
was used to find the correlation coefficient with the
stimulus reference function. This operation can be seen as focusing on the sim-
ilarity with the expected activation. In this work, interesting connections with
principal-component analysis preprocessing are outlined. Other metrics can be
defined such as the ones proposed for fMRI data in Reference 12, in which
decreasing functions of the Pearson's correlation coefficient are used. This metric
can be written as
T
d
(,)
xy
=
f cc
( (, )
xy
(16.4)
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