Image Processing Reference
InDepth Information
k
y
k
(
ˆ
inc

ˆ
sct
)
Ewald
limiting circle

k
ˆ
sct
k
ˆ
inc
k
x
2
k
2
k
Ewald circle
Figure 1.2
Fourier space (
k
space) of the object as a result of interaction of different incident plane waves
with scattering object. The direction of the incident field
r
i
nc
and the direction
r
sct
of a particular plane wave com
ponent of the scattered field define a point at the Ewald circle. Changing the incident field directions
r
sct
fills the
interior of Ewald limiting circle.
From Equation 1.2, it can be seen that as
k
tends to infinity, corresponding
to an almost zero wavelength, the radius of these Ewald circles becomes infi
nite, and for each incident wave direction, the circles become lines tangent to
the
k
space origin. This limiting case, when the Born approximation is valid,
reduces to the Fourier Central Slice Theorem and the interpretation of projec
tion data, for example, as used in xray CAT scanning that was mentioned
earlier. This is a satisfying result but reinforces the fact that if
kV
m
d
≪ 1 is not
valid; because of multiple scattering, we should not expect to compute a use
ful image without taking further steps.
1.4 theoRetICAl ISSueS And ConCeRnS
In one dimension, the imaging from inverse scattering problem is well under
stood as discussed in detail in a paper by Zoughi (2000) and Devaney (1978). In
general, in these 1D problems, the scattered data to be inverted are treated as
reflection and transmission coefficients The inverse algorithm in these spe
cial cases is simply applied to this data to retrieve a reliable and unique 1D
profile of the original target.
The much more complicated 2D version of this problem has been under
examination for well over 100 years now. For the most part, there has been
some limited success in implementing useful algorithms to perform this
inversion for the very special case of weakly scattering targets. As already
mentioned, the majority of these algorithms are based on using a technique
to linearize this type of problem using either the Born approximation or the
Rytov approximation, both of which will be discussed in more detail later.
Historically, the success of these approaches or approximations is dependent
on the target being a weak scatterer and that they fail to perform well when
this is not the case. This historical viewpoint is due for a closer examination,
as we shall see later.
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