Image Processing Reference

In-Depth Information

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!

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