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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