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. 9.14. The Gabor filters in the Fourier domain plotted against the linear frequency scale.
The green arrows show the used minimum and the maximum frequency parameters
a coordinate transformation, such that a small interval is mapped to a larger interval
through the transformation. To be precise, we want a mapping such that when we di-
vide
ξ
equally, where each interval has the width
δξ
, the widths of the corresponding
intervals in
ω
should be larger and larger in such a way that
δω
grows proportionally
with
ω
, i.e.,
δω
=
Cωδξ
(9.44)
where
C
is an unknown constant. Effectively, this leads to a differential equation
when we let
δξ
, and thereby
δω
(which depends on it), be ever smaller.
dω
dξ
δω
=
Cωδξ
→
=
Cω
(9.45)
The differential equation with the boundary condition yields the unique solution
ω
=
u
(
ξ
)=
ω
min
exp
log
ω
max
ω
min
ξ
(9.46)
or
ω
max
ω
min
ξ
ω
=
ω
min
·
(9.47)
with the inverse
ξ
=
log
ω
max
ω
min
−
1
log
ω
ω
min
·
(9.48)
We divide
ξ
into
N
equal cells and place the same translated Gaussian in each
ξ
cell in analogy with Fig. 9.13 and Eq. (9.41):
