Biomedical Engineering Reference
In-Depth Information
theoretical and technical details are referred to the textbooks on Gabor analysis
[20], wavelet packets [21], and the original paper on brushlet [22].
6.2.3.1 Gabor Transform and Gabor Wavelets
In his early work, Gabor [23] suggested an expansion of a signal
s
(
t
) in terms of
time-frequency atoms
g
m
,
n
(
t
) defined as:
s
(
t
)
=
c
m
,
n
g
m
,
n
(
t
)
,
(6.20)
m
,
n
where
g
m
,
n
(
t
)
,
m
,
n
∈
Z
, are constructed with a window function
g
(
x
), combined
to a complex exponential:
g
m
,
n
(
t
)
=
g
(
t
−
na
)
e
i
2
π
mb t
.
(6.21)
Gabor also suggested that an appropriate choice for the window function
g
(
x
)
is the Gaussian function due to the fact that a Gaussian function has the theoret-
ically best joint spatial-frequency resolution (uncertainty principle). It is impor-
tant to note here that the Gabor elementary functions
g
m
,
n
(
t
) are not orthogonal
and therefore require a biorthogonal dual function
γ
(
x
) for reconstruction [24].
This dual window function is used for the computation of the expansion coeffi-
cients
c
m
,
n
as:
f
(
x
)
γ
(
x
−
na
)
e
−
i
2
π
mb x
dx
,
c
m
,
n
=
(6.22)
while the Gaussian window is used for the reconstruction.
The biorthogonality of the two window functions
γ
(
x
) and
g
(
x
) is expressed
as:
g
(
x
)
γ
(
x
−
na
)
e
−
i
2
π
mb x
dx
=
δ
m
δ
n
.
(6.23)
From Eq. (6.21), it is easy to see that all spatial-frequency atom
g
m
,
n
(
t
)
share the same spatial-frequency resolution defined by the Gaussian func-
tion
g
(
x
). As pointed out in the discussion on short-time Fourier transforms,
such design is suboptimal for the analysis of signals with different frequency
components.
A wavelet-type generalization of Gabor expansion can be constructed such
that different window functions are used instead of a single one [25] according
to their spatial-frequency location. Following the design of wavelets, a Gabor
