Image Processing Reference
In-Depth Information
is real if and only if its Fourier transform is Hermitian. This means that in
order for
V
(
r
) to be real, it must be equal to the transpose of its complex con-
jugate. This is expressed as
i
{
}
=
·
·
·
·
·
·
(4.37)
−
f
(
kk
rr
,
)
−
f
BA
(
kk
rr
,
)
∗
Im
f
BA
(
kk
rr
,
)
BA
2
inc
inc
inc
which can also be written as
k
∫
·
2
·
d
Ω
f
(
kk
,
rr
)
=
0
BA
(4.38)
4
π
inc
Ω
∈
H
2
where Ω is defined as the solid angle over which
H
2
is integrated in either 2-D
or 3-D. Equation 4.38 clearly suggests that the target must be zero everywhere
for the Born approximation to be valid. This simply is not possible for a real
scattering object. Therefore, this indicates that there can be no exact solution
for real objects for the Born approximation. Physically,
V
(
r
) is always complex,
as dictated by dispersion relations that follow directly for causal systems.
This notwithstanding, for sufficiently small scattering objects with negligible
noise levels, a consistent stable estimate of
V
(
r
)Ψ/Ψ
inc
can be obtained by using
a regularized inversion of the Fourier data (Ritter, 2012), such as the discrete
(inverse) Fourier transform (DFT).
Taking another look at Equation 4.34, it is evident that the inverse Fourier
transform of the complex far-field scattering amplitude can provide a reason-
able representation of the scattering object. As already discussed in the previ-
ous section, this can be accomplished with one of the two methods, namely,
the Fourier transform-based interpolation and the filtered back propagation
method (Avish and Slaney, 1988). When the target or scattering object is con-
sidered to be a nonweak or a strong scatterer, theoretically the Born approxi-
mation is said to be invalid. When this is the case, the Fourier inversion of
many Ewald circles discussed in the next chapter results in an approximation
of
V
(
r
), which is expressed as
·
(
rr
rr
,
)
Ψ
Ψ
·
inc
V
(
rr
,
)
≈
V
( )
r
(4.39)
BA
inc
·
(
,
)
inc
inc
In this equation, the angle bracket symbols, 〈 〉, indicate an averaged depen-
dence on the direction of the incident plane wave. The form of this equation
suggests that the Ψ-averaged term is independent of the
V
(
r
) term, which,
if true, implies that it should be separable. It is unclear at this time if this
approximation, that is, that these terms are independent of each other,
holds for all classes of targets, but we can argue that for increasingly com-
plex targets, the averaged internal multiplied scattered fields will become
increasingly noise-like, while
V
(
r
) remains unchanged. Equation 4.39 is an
approximation because, in principle, the Fourier transformation can only
be applied for each
r
inc
= constant, and also, in practice, its accuracy will be
affected by the limited data (i.e.,
k
-space) covered, which is always the case
in this text.
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