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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