Introduction to Inverse Scattering - Introduction to Imaging from Scattered Fields - page 10

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 x-ray 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 1-D 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 1-D
profile of the original target.
The much more complicated 2-D 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.