Image Processing Reference
InDepth Information
4.4 Rytov AppRoxIMAtIon
While the Born approximation is one of the more commonly used methods for
linearizing an inverse scattering problem, there are other methods available.
One of the more common alternative methods is the Rytov approximation.
In this, the total field is represented in terms of a complex phase (Avish and
Slaney, 1988; Ishimaru, 1978) shown in Equation 4.40:
·
Ψ
(
rr
,
)
=
Ψ
( )
r
e
i
Φ
()
r
(4.40)
s
inc
inc
where Φ(
r
) is the complex phase function and is defined as
ΦΦ Φ
()
r
=
()
r
+
()
r
(4.41)
inc
s
In Equation 4.41, the Φ
s
(
r
) term is the phase function of the scattered field
If this new representation of the total field in Equation 4.40 is now substituted
back into the general inhomogeneous Helmholtz equation (Equation 4.5) and
the defined identity functions are used, then this yields the following:
∇
2
[
ΨΦ ΨΦ Ψ
() ()
r
r
]
= ∇
[
2
()
r
]
()
r
+ ∇
2
()
r
⋅∇+ ∇
Φ
()
r
[
2
Φ
(
r
)
]Ψ
inc
( )
r
(4.42)
inc
s
inc
s
inc
s
s
Simplifying, the inhomogeneous Helmholtz equation then becomes (Lin
and Fiddy, 1990)
{
}
(
∇+
2
k
2
)[
ΨΦ
( )()]
r
r
=
i kV
2
() [
r
− ∇
ΦΨ
()
r
]
2
()
r
(4.43)
inc
s
s
inc
As in Chapter 2, we utilize the freespace Green's function in Equation 4.43;
the complex phase function can then be written as
ik
2
∫
Φ
()
r
=
V(
r
′
)
Ψ
() (
r
′
G
rr r
,
′
)
d
′
s
Ψ
()
r
inc
0
inc
D
i
∫
−
[
∇
Φ
(
r
)
]
2
Ψ′
(
r
)(
G
rr r
,
′
)
d
′
(4.44)
Ψ
()
r
s
inc
0
inc
D
Interestingly, in Equation 4.44, the second term can be approximated to
zero if the following criteria are met (Lin and Fiddy, 1990)
( )
kV
2
r
∇Φ
( [
() )
r
] 
2
(4.45)
s
�
2

‚
∇
Φ
π
()
r
�
s

ε
()
r
−
1

(4.46)
2
The implication or impact of these inequalities is that in order for these cri
teria to be met, the incident wavelength must be very small in comparison to
the mean size of the target or scattering object, or that the spatial rate of phase
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