Image Processing Reference
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so that if we now utilize the fact that as
r
approaches zero, 4π
C
= −1, which can
be used to solve for
C
as follows:
1
4π
C
=
(4.20)
This value of
C
can now be substituted back into Equation 4.11, which yields
e
ikr
(4.21)
G
(
r
=
0
4
π
r
This can now be generalized by shifting the source back to
r
′ which gives
Green's function in the general form for this application as
e
ik
rr
−
′
G
(
rr
′
)
=
(4.22)
0
4
π
rr
−
′
Equation 4.22 gives Green's function in the form necessary to solve the
inhomogeneous Helmholtz equations. In this topic, all of the experiments are
done in 2D; therefore, the 2D form of Green's function can be expressed in
terms of a zeroorder Hankel function of the first kind (Chew, 1995; Darling,
1984) as follows:
−
i
Hk
(4.23)
G
(, )
rr
′
=
()
1
[

rr
−
′
]
4
0
Now, an asymptotic assumption can be applied in that as
r
→ ∞, the zero
order Hankel function of the first kind becomes
41
8
1
Hk
()
[

rr
−=
+
′
]
e
ikr
(
π
/
4
)
e
− −−
ikr
·

rr
′

(4.24)
0
i
π
kr
where
ˆ
/  Substituting Equation 4.24 back into Equation 4.23, the
approximated 2D Green's function now becomes
r
′ =
r r
.
1
8
·
(4.25)
G
(, )
rr
′
≅−
e
ikr
(
+
π
/
4
)
e
− −−
ikr

rr
′

π
kr
Finally, a general equation for the scattered field can be obtained by sub
stituting Equation 4.25 into Equations 4.6 and 4.7, which gives the following:
1
8
∫
··
·
·
Ψ
(
rr
,
)
=
e
ikr
(
+
π
/
42
)
k
Ve
(
r
′
)
−−−
ikr
rr
′
Ψ
(
r r
′
,
)
d
r
′
(4.26)
s
inc
inc
π
kr
D
where
ˆ
= φ φ is the unit vector that describes the direction of
the incident plane wave and
ˆ
r
inc
(cos
,sin
)
inc
inc
r
= φ φ is the unit vector that describes
the direction of the scattered wave. This equation is of the form of a Fredholm
(cos ,sin
)
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