Image Processing Reference
In-Depth Information
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ł
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æ
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Fig. 9.11. Music scores are represented and interpreted as a sequence of time-limited musical
tones chosen from a finite set
As originally proven by Gabor [77], for a Gaussian window function
w
, it is possible
to discretize
F
(
t
0
,ω
0
) w.r.t.
t
0
and
ω
0
such that
f
as well as
F
can be recovered, up
to a lowpass filtering, from the samples. The samples corresponding to the same
t
0
constitute the discrete local spectrum. It is also possible to come to this conclusion
by using our results in the previous sections. From the definition, the local spectrum
around
t
0
equals:
F
(
t
0
,ω
0
)=
w
(
t
) exp(
itω
0
)
,f
(
t
−
t
0
)
=2
π
W
(
ω
−
ω
0
)
,F
(
ω
) exp(
−
it
0
ω
)
(9.33)
where we have used theorem 7.2 (Parseval-Plancherel). In consequence, we can
write
F
(
t
0
,ω
0
)=
f
(
t
−
t
0
)
w
(
t
) exp(
−
iω
0
t
)
dt
(9.34)
=2
π
W
(
ω
−
ω
0
)
F
(
ω
) exp(
−
it
0
ω
)
dω
(9.35)
For
t
0
=0, Eq. (9.35) represents a lowpass filtering of the spectrum,
F
, with the
filter
W
. Accordingly, the smoothed spectrum can be sampled with the discretiza-
tion step
Ω
=
2
T
where
T
is the effective width of the window,
w
(
t
), in the time
domain. However, the smoothed spectrum,
F
(0
,ω
), continuous in
ω
, represents the
Fourier transform of the local function around the origin, i.e.,
f
(
t
)
w
(
t
) in Eq. (9.34),
and it can be recovered from the samples,
F
(0
,n
2
T
) precisely because the func-
tion
f
(
t
)
w
(
t
) has limited extension due to windowing. The origin
t
=0is in no way
unique, because the function
F
(
ω
) exp(
−
it
0
ω
) appearing in Eq. (9.35) is the Fourier
transform of
f
(
t
t
0
), which is a shifted version of the function such that
t
0
is the
origin. In consequence, what we have in Eq. (9.35) is a smoothing of the Fourier
transform of the shifted function
f
(
t
−
t
0
) with the filter
W
(
ω
) in the Fourier do-
main. Because of this,
F
(
t
0
,ω
) can be sampled w.r.t.
ω
, i.e.,
F
(
t
0
,n
2
T
), where
T
corresponds to the effective extension of
w
(
t
) and recovered from the samples. Using
the same arguments, and using the symmetry of the Fourier transform, we conclude
that
F
(
t, ω
0
) can be sampled w.r.t.
t
, i.e.,
F
(
mT, ω
0
), which in turn can be recov-
ered from the samples of
F
(
mT, n
2
T
). The discrete local spectrum, also called the
Gabor spectrum
,or
Gabor decomposition
, is computed as a projection onto a filter
bank either in the spatial domain, as in Eq. (9.36), or in the Fourier domain, as in Eq.
(9.37).
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