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178 9 Directional Filtering Transform
parameters related to the minor and major axes, respectively. The larger Ω
α
is, the
faster the correlation orthogonal to the edge direction decays; while the smaller Ω
β
is, the slower the correlation along the edge direction declines. By Fourier transform,
we get the PSD of ˜x as
˜
X
˜x ˜x
(Ω) =
2πσ
2
Ω
α
Ω
β
1 + Ω
T
¯
Γ Ω
−
3
2
, (9.12)
where Ω = (Ω
1
,Ω
2
)
T
and
¯
Γ =
cosϕ sinϕ
−sinϕ cosϕ
Ω
−2
α
0
0 Ω
−2
β
cosϕ sinϕ
−sinϕ cosϕ
T
.
For the discrete signal x[t], its PSD is
X
xx
e
jω
=
B
∑
i=−B
B
∑
j=−B
˜
X
˜x ˜x
((ω
1
−2iπ) f
1
,(ω
2
−2 jπ) f
2
), (9.13)
where B is a band-limit parameter. f
1
and f
2
are the sampling frequencies in the
horizontal and vertical ...