Biomedical Engineering Reference
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because the signal-to-noise ratio is too low (in which case the low dimensional
manifold is hidden by the noise) or because signals are complex, which increases
the dimensionality of the required embedding space. In such cases, it may be
preferable to use model-driven approaches which leverage a priori on the structure
of the relevant information present in the signal. In this section, we will review a
class of iterative methods for data approximation, called Matching Pursuit, and we
will present its extension to multitrial data. Matching pursuit methods have been
used in the field of epilepsy, and sleep EEG, providing interesting information on
oscillatory phenomena present in the data. This section relates research that has been
originally presented in [
3
].
7.3.1
Matching Pursuit
The
matching pursuit
technique was introduced in [
17
]. It relies on a
dictionary
of
atoms
(a list of prototypical signals that are considered relevant). The algorithm
iteratively decomposes the signal into a sequence of atoms. At each iteration, it
seeks the atom of the dictionary that best matches the signal. The atom contribution
is then subtracted from the signal before starting the next iteration of the algorithm.
The method is repeated until the remaining residual is considered negligible.
Consequently, the algorithm mainly relies on: (1) a choice of a dictionary, (2) a
procedure to combine dictionary elements into a signal and a means to extract the
most significant atom out of a signal.
In our case, the dictionary
D
is a set of known predefined functions
ψ
p
i
(
t
)
,i
=
1
...M
(the atoms), where
p
i
is a set of parameters defining the function
ψ
p
i
.Given
D
of a given brain activity, classical matching
pursuit (independently used on each recording) aims at decomposing the signals
s
k
(
and multiple recordings (trials)
s
k
(
t
)
t
)
into a low number
P
of components chosen in
D
:
P
s
k
(
t
)=
a
ik
ψ
p
i
(
k
)
(
t
)+
n
k
(
t
)
,
(7.8)
i
=1
where
n
k
(
t
)
is the noise on trial
k
and
a
ik
is the amplitude associated to the atom
ψ
p
i
(
should encompass any activity
that is not present in all trials, but when the decomposition is done independently for
each trial, there is no way to guarantee this and thus noise represents any residual
signal that cannot be modelled with
P
atoms
ψ
p
i
(
k
)
.
Contrarily to usual bases in functional spaces, dictionaries
t
)
in
s
k
(
t
)
. Note that in theory, the noise
n
k
(
t
)
are most often over-
complete (the predefined functions are highly redundant so that decompositions
are not unique), but should have a good descriptive power, i.e., a good ability
to represent any allowed input signal with a low number of components
P
leading to sparse representations. Another important property of dictionaries is their
interpretability, i.e., the possibility to attach some meaningful semantics to each of
their atoms.
D
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