Image Processing Reference
In-Depth Information
In electromagnetic propagation and scattering problems, we often assume
the source is actually a dipole antenna or emitter. This is a good approxima-
tion for sources when not too close to them. Considering a dipole located at
r
′, it has a dipole moment we can denote by
q
. The corresponding current
density is
Jr qr r
s
()
=−
iωδ
(
−
′
)
(2.24)
The electric field can then be calculated by introducing this expression into
Equation 2.23 for
E
, which gives
Er
(, )
ω ω
=
Grrq
( ,,)
ω
(2.25)
2
0
If
E
0
= 0 (or subtracted out), then Equation 2.25 represents the scattered field
2.2.1 Solving differential equations
Green's function of the wave equation is a solution for the special case when
the source term is a delta function in space and/or time.
∇−
∂
∂
1
2
2
�
2
(, ,, )
rr
′
tt
′
=− −
δ
(
rr
′
)(
δ
tt
′
)
(2.26)
cr
2
Green's function is therefore a field radiated by a point source. We can gen-
eralize this to arbitrary differential operators. For example, let
L
x
denote an
arbitrary linear differential operators in
n
variables
x
(i.e.,
x
1
,
x
2,
…,
x
n
) of
order α. In its most general form this would be
∂
∂
++
∂
∂
aa
x
+
+
a
a
+
0
1
2
n
∂
x
∂
x
1
2
n
∂
∂
2
∂∂
++
∂
∂
2
2
L
=
a
+
a
a
(2.27)
x
11
12
xx
nn
x
2
∂
x
2
12
1
n
∂
∂
α
α
∂
∂
α
α
+
a
++
a
11
nn
x
x
1
n
Using this operator, we now consider the inhomogeneous differential
equation
Lx
ϕ
()
=−
ρ
()
x
(2.28)
x
where ρ(
x
) is a source term (as we might expect in Laplace's equation or the
inhomogeneous Helmholtz equation). Then we want to find φ accounting for
whatever boundary conditions are imposed. We replace ρ(
x
) by a point source
function and then generalize it into distributed source function. This is then the
“impulse response” of the system that in optics is a point spread function (PSF)
LGxy
(
−=−−
)
δ
(
xy
)
(2.29)
x
where
G
(
x,y
) is Green's function.
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