Image Processing Reference

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Four

Inverse Scattering Fundamentals

4.1 CAtegoRIzAtIon oF InveRSe SCAtteRIng pRobleMS

In general, inverse scattering problems can be placed in one of the two

categories:

1. Weak scattering

2. Nonweak or strong scattering

Weak scattering occurs when the incident wave is only “scattered” once,

and this incident wave basically undergoes very little perturbation as it trav-

els through or interacts with the target. This is significant in that in most of

these cases the wave inside the target can generally be approximated as the

incident wave, which allows the problem to be linearized in order to find a

solution. This approximation is what is exploited in the approach to the solu-

tion used in the Born and Rytov approximation methods (Avish and Slaney,

1988; Lin and Fiddy, 1990). In these approaches, by linearizing the problem,

one is able to establish a Fourier relationship between the measured scattered

field data and the target or scattering function. In principle, these methods are

only supposed to work well with weak scatterers due to the dependence on

this approximation. More precisely, the Rytov approximation requires targets

whose permittivity or index varies only very slowly on the scale of the wave-

length. This concept of weak scattering will be examined, and in some sense

challenged, in this research to understand the extent to which these methods

actually work. In the case of strong scatterers, the field inside the target is scat-

tered multiple times and can incur significant perturbation that introduces

nonlinearities to the integral equation of scattering (Chew, 1995). This negates

the linear approach to this problem, making it quite difficult to solve if and

when the above-mentioned methods do not perform well. A solution for the

problem involving strong scattering is highly desired, as most targets of inter-

est in real life would fall into this category.

4.2 InveRSe SCAtteRIng In two dIMenSIonS

A common setup for a 2-D inverse scattering experiment is shown in Figure

4.1. Typically, there is some fixed number of receivers or receiver locations

set up in some configuration around the center of the target, usually equally

spaced. An incident plane wave then illuminates the target at some known

angle ϕ
inc
with respect to the
x
-axis and the scattered field is measured at an

angle of ϕ
s
in the far field Now, a target or scattering object represented by
V
(
r
)

is placed in a homogeneous background, which has a permittivity of ε
0
, where

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