Biomedical Engineering Reference
In-Depth Information
ξ
=0
.
5
ξ
=1
.
5
ξ
=3
ξ
=9
ξ
=25
Fig. 7.6
Gabor atoms for different values of the oscillation parameter, obtained by varying
f
with
a constant
σ
(Eq.
7.9
). A low oscillation parameter produces a transient wave, and a high value a
sustained oscillation
7.3.2
A Dictionary Tuned for MEG and EEG
There are two types of activity that can appear in MEG or EEG measurements:
oscillatory activity such as gamma or alpha bursts, and transient, phasic activity. In
order to model both types of events, an over-complete dictionary of Gabor wavelets
is chosen. In the time domain, this dictionary is defined by:
e
2
iπf
(
t−t
0
)
e
−
(
t
−
t
0
)
2
1
ψ
t
0
,f,σ
(
t
)=
2
σ
2
.
(7.9)
1
4
(
πσ
2
)
An element of this dictionary is a Gaussian function centered at
t
0
, modulated by
a sinusoidal signal with frequency
f
. These atoms are thus both oscillatory (the
“evoked potential” type of activity being associated to low frequencies) and local-
ized (their temporal extent is bounded by the Gaussian with
σ
standard deviation).
The parameter
t
0
allows for time invariance (meaning that the representation of a
translated signal is the translation of the original signal representation). Each
ψ
t
0
,f,σ
is a wavelet centered around time
t
0
and oscillating at a frequency
f
. The duration
of the wavelet is given by
σ
but it is also interesting to specify it by its number of
oscillations which is given by
ξ
πfσ
. Figure
7.6
shows various such wavelets,
all centered around a same time instant, and for different values of
ξ
. Note that the
initial factor in (
7.9
) ensures that all atoms are normalized to have a unit energy (in
L
2
norm).
=2
Search WWH ::
Custom Search