Biomedical Engineering Reference
In-Depth Information
width, φ - phase).
The parameters γ can be sampled in various ways. The original idea of Mal-
lat [Mallat and Zhang, 1993] was to follow a dyadic scheme that mimics the over-
sampled discrete wavelets. For applications where the parameters are used to form
statistics of the atoms the dyadic sampling produced estimators that were biased by
the structure of the dictionary. Introduction of stochastic dictionaries relied on ran-
domization of the time-frequency coordinates and time width of atoms [Durka et al.,
2001a]. This allowed to obtain a bias free implementation. Further extensions of the
MP algorithm allow for analysis of multivariate signals i.e multichannel and multi-
trial decompositions [Durka et al., 2005b, Sieluzycki et al., 2009a] (Sect. 3.6.3).
The atomic decomposition of signal can be used to produce the time-frequency en-
ergy distribution. In this approach the best properties of energy distributions (WVD)
and atomic decompositions can be joined. The WVD of the whole decomposition is:
n = 0 R n x , g γ n
2 W g γ n (
W x
(
t
,
f
)=
t
,
f
)
n = 0
R n x
g γ n R m x
g γ m W g γ n g γ m (
+
,
,
t
,
f
)
(2.115)
m
=
0
,
m
=
n
where
Z
2 g γ n t
2 e i f τ d τ
g γ n t
τ
τ
W g γ n (
t
,
f
)=
+
(2.116)
is WVD of individual atoms. The double sum in equation (2.115) corresponds to the
crossterms, but since it is given explicitly, it can be omitted yielding the estimator of
the energy density distribution in the form:
M
n = 0 | R n x , g γ n |
E MP
x
2 W g γ n (
(
t
,
f
)=
t
,
f
)
(2.117)
This interpretation is valid since normalization of atoms (2.114):
Z +
Z +
2
W g
(
t
,
f
)
dt d f
=
g
=
1
(2.118)
leads to:
Z +
Z +
2
E MP
x
(
t
,
f
)
dt d f
=
x
(2.119)
This representation has implicitly no cross-terms and for Gabor atoms offers the
highest time-frequency resolution (Sect. 1.4.7).
MP dictionary can be adjusted to be coherent with particular structures in the
signal. The dictionary containing asymmetric functions was designed and proved to
be useful in description of components with different time courses of the rising and
decaying parts [Jedrzejczak et al., 2009] (Sect. 4.5.2).
 
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