Image Processing Reference
In-Depth Information
Fig. 9.12. The graphs of the real (
green, solid
) and the imaginary (
green, dashed
) parts of a
Gabor filter
,
g
(
t
), in the time domain. The window function
w
(
t
)=
|g
(
t
)
|
is drawn in
black
)=
F
(
mT, n
2
π
T
f
(
t − mT
)
w
(
t
) exp(
−in
2
π
T
t
)
dt
(9.36)
=2
π
n
2
π
T
W
(
ω
−
)
F
(
ω
) exp(
−
imT ω
)
dω
(9.37)
The filters of the filter bank are called the Gabor filters and they are given as follows
in the spatial domain:
in
2
π
T
1
(2
πσ
2
)
2
t
2
2
σ
2
in
2
π
T
g
n
(
t
)=
w
(
t
) exp(
−
t
)=
exp(
−
) exp(
−
t
)
(9.38)
with 2
σ>T
. In Fig. 9.12, the real and the imaginary part of one such Gabor filter,
g
n
(
t
) with
n
=2, is shown. The black curve shows
, which is the Gaussian
window. The Fourier transform of the filter is shown in magenta in Fig. 9.13. The
higher the value of
n
, the more the filter function oscillates in the Gaussian window.
In the Fourier domain, Gabor filters are translated Gaussians:
G
n
(
ω
)=
W
ω
|
g
n
(
t
)
|
=exp
n
2
T
)
2
σ
2
n
2
π
T
(
ω
−
−
−
(9.39)
Figure 9.13 illustrates
G
n
(
ω
) for
n
=1
7 for a set of Gabor filters uniformly
distributed in the angular frequency range [0
.
05
π,
0
.
95
π
]. The filters are centered
in the seven equally sized frequency cells and attenuate 50% at the cell boundaries.
Note that this amount of overlap is motivated by our finding in Sect. 9.3, according to
which a Gaussian approximates the characteristic function of its frequency cell best
in the
···
L
∞
norm when it attenuates at the characteristic function boundaries with
50%. The characteristic function of one frequency cell that is to be approximated by
its underlying Gaussian (magenta) is drawn in black in Fig. 9.13. Note that we only
