Image Processing Reference
In-Depth Information
where r inc is the unit vector that specifies the direction of the incident field
An expression for the total field Ψ( r ) in terms of the coordinate system r can be
obtained by looking at the interaction of the incident field, Ψ inc ( r ), and the tar-
get V ( r ). This relationship can be expressed as an inhomogeneous Helmholtz
equation (Avish and Slaney, 1988) as follows:
(
∇+
2
k
2
)
Ψ
()
r
=−
k
2
V
() ()
r
Ψ
r
(4.5)
The total field in Equation 4.5 can now be expressed generally as the sum of
the incident field and the scattered field as follows:
ΨΨ Ψ
()
r
=
inc ()
r
+
()
r
(4.6)
s
where Ψ s is the scattered field All fields in Equation 4.6 are expressed in
terms of the position as defined by the point r . Now the total field, Ψ( r ), can be
expressed in terms of an inhomogeneous Fredholm integral equation of the
first kind (Morse and Feshbach, 1953) as follows:
kV
·
·
Ψ
(
rr
,
)
=
Ψ
()
r
2
(
r rr rr
)
Ψ
(
,
)
G
(
,
)
d
r
(4.7)
inc
inc
inc
0
D
where G 0 ( r , r ′) is (free space) Green's function which is the solution of the sca-
lar Helmholtz equation (Equation 4.4), and it satisfies the differential equation
(Morse and Feshbach, 1953)
(
∇+ =−
2
kG rr rr
2
)
(
,
)
δ
(
)
(4.8)
0
Green's function basically gives the field amplitude at any point r , generated
by any given point source located at r ′. Since the modeling space in this case
is homogenous and rotationally symmetrical, Green's function can be solved
in spherical coordinates with r ′ at the origin as follows:
(
∇+
2
kG
2
)
()
r
=− =−
δ
()
r
δ δδ
()()()
xyz
(4.9)
0
For homogenous, spherically symmetrical partial differential equations
(PDEs), the solution for Green's function in free space can be written as
Ce
r
ikr
Ue
r
ikr
(4.10)
G
0 ( r =
+
Utilizing the radiation boundary condition, which is simply that the
sources cannot be present at infinity, translates to this: that only the outgoing
wave solution(s) exist(s) and therefore U must be equal to 0 ( U = 0). This sim-
plifies Equation 4.10 as follows:
C
e ikr
(4.11)
G
0 ( r =
r

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