Image Processing Reference
In-Depth Information
where Θ is the scattering angle as defined in Figure 3.3. Furthermore,
1
π
n
(
cos
Θ
)
=
P
(
cos
Θ
)
(3.34)
1
sin
Θ
d
τ
n
(
cos
Θ
)
=
P
(
cos
Θ
)
(3.35)
1
d
Θ
where
P
n
1
represents the associated Legendre polynomials of the first kind. In
addition, we have
ψαψα ψαψα
ψαξα ψαξα
′
(
m
)
()
−
m
(
m
)
′
()
a
=
n
n
n
n
(3.36)
n
′
(
m
)
()
−
m
(
m
)
′
()
n
n
n
n
mm
ψαψα ψαψα
ψαξα ψαξα
′
(
)
()
−
(
m
)
′
()
b
=
n
n
n
n
(3.37)
n
mm
′
(
)
()
−
(
m
)
′
()
n
n
n
n
The size parameter, α, is defined as follows:
=
2
π
am
α
0
(3.38)
λ
0
Finally, in Equations 3.36 and 3.37, the Ricatti-Bessel functions are
defined as
12
/
π
z
=
�
-
‚
ψ
()
z
J
()
z
(3.39)
n
2
n
+
12
/
12
/
π
z
=
�
-
‚
ξ
()
z
Hz
()
=
ψ
()
ziXz
+
()
(3.40)
n
2
n
+
12
/
n
n
12
/
π
2
z
=−
�
-
‚
Xz
()
Yz
()
(3.41)
n
n
+
12
/
where
Y
n
is a typical Bessel function of the second kind.
In addition to these relationships, there are other characteristics of Mie
scatterers that are of great interest in this topic, in particular the eficiency
factor of scattering, which can be approximated by (Walstra, 1964)
4
4
Q
=− +
2
sin
()
p
(
1
−
cos
(
p
))
(3.42)
p
p
2
where
4
π
rn
(
−
1
)
p
=
(3.43)
λ
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