Image Processing Reference
In-Depth Information
1
0.8
0.6
0.4
0.2
0.05
π
0.95
π
0
0
0.5
1
1.5
2
2.5
3
Fig. 9.13. The graphs of a Gabor filter bank with minimum and maximum angular frequencies
0
.
1
π
and 0
.
9
π
, respectively. The filter corresponding to Fig. 9.12 is shown in
magenta
, where
the
black
constant illustrates the characteristic function of the corresponding frequency cell
need to have filters that cover the positive
ω
-axis in the Gabor filter bank because we
assumed that the function
f
is real-valued so that there is no additional information
in the negative axis, compared to the positive axis due to
F
(
ω
)=
F
∗
(
ω
).Given
the attenuation at the cell boundaries, for a 1D Gabor filter bank we thus need the
minimum frequency, the maximum frequency, shown as vertical arrows, as well as
the total number of filters to determine the filter bank. The tip of a green arrow and
the closest yellow arrow is half the width of the angular frequency cells.
One can extend these results into 2D and higher dimension in a straightforward
manner. The window function is an
N
D Gaussian, which has certain advantages,
e.g., direction isotropy in image analysis applications. The resulting Gabor filters are
given in the spatial domain by
−
1
(2
πσ
2
)
N
−
2
2
σ
2
r
T
T
g
n
(
r
)=
w
(
t
) exp(
−
i
ω
n
r
)=
exp(
) exp(
−
i
ω
n
r
)
(9.40)
2
and in the frequency domain, by
−
ω
−
ω
n
2
2
σ
ω
G
n
(
ω
)=
W
(
ω
−
ω
n
)=exp(
)
(9.41)
where
σ
ω
=1
/σ
. Without loss of generality, we assumed that the frequency cells
are axis-parallel hypercubes with equal edge sizes in all directions and that the cell
centers are at
ω
n
. We illustrate the frequency cells of one such Gabor filter bank in
Fig. 10.28 along with the isocurves of the filters (green). The DC extracting Gabor
