Introduction to Inverse Scattering - Introduction to Imaging from Scattered Fields - page 9

Receivers
(Typical)

Scattered
field

Ta rget

Incident
plane wave

Measurement
sphere

Figure 1.1 Typical experimental setup for diffraction tomography. The target or object is illuminated with a
monochromatic plane wave and the scattered waves, after the interaction of the incident plane wave with the scat-
tering object, are measured by receivers placed either in the near or far field around the object.

Of course the quality of that image will only be as good as the sampling of its
Fourier transform in k -space. It is the limited sampling that inevitably results
from experiments that leads to the need for considerable postprocessing and
the application of estimation methods to get improved images. If the inequal-
ity is not appropriate, a distorted image results which may convey nothing
useful whatsoever about the actual permittivity profile
Ideally, the diffraction tomography method (Ritter, 2012), when restricted
to weakly scattering targets, also assumes that the target is illuminated by a
known quasi-monochromatic incident wave or waves, and the scattered field(s)
is measured ALL around the target by a number of receivers. An illustration
of a typical general experimental setup for this method is shown in Figure
1.2. The corresponding equation, which will be derived later in Chapter 4, is
given by

1

∫

ˆ

ˆˆ

Ψ s BA ( rr , = ) e

ikr ( + π / ) 42

k Ve ( r ′ )

− ik ( rr r − ) g ′

d r

(1.2)

inc

inc

8 π kr

D

where Ψ s is the scattered field and Ψ s BA is the scattered field in the Born
approximation. It should be noted that V ( r ) is assumed to be zero outside of
the volume D . The relationship between the scattered field measurements and
the scattering object is given by Equation 1.2. It can be seen that the Fourier
variable in k -space is actually a locus of points mapping out a circle whose
radius depends on the k -value of the incident wave and whose center depends
on the direction of the incident plane wave. Deviating from incident plane
waves and incident quasi-monochromatic waves complicates the interpreta-
tion of the scattered field using this Fourier picture quite considerably!