Born Approximation Observations - Introduction to Imaging from Scattered Fields - page 80

is evidence in the images that suggests that the requirement of minimum
degrees of freedom on the number of receivers is valid as well. This comple-
ments the series of images produced in the previous section.

6.4 IMAgIng RelAtIonShIp between boRn AppRoxIMAtIon
And MIe Q FAC t oR
It is obvious from the data for various cylinders in Tables 6.2 through 6.4 that
there appears to be a cyclic pattern to the quality of the Born reconstructed
images with increasing scattering strength. As it was demonstrated in the
previous two sections that the minimum degrees of freedom have been satis-
fied, this pattern of behavior must have another explanation. In considering
cylindrical targets, it is evident that this circle or cylinder can be analyzed or
considered as a 2-D Lorenz-Mie scatterer, as discussed in Section 3.4. This
being the case, it would logically follow that, as this cylinder is taken through
the different scenarios of varying size and permittivity, the incident wave
inside the target should be cycling through various points of resonance. As
discussed previously, for Lorenz-Mie scatterers, this is characterized by the
following Q factor equation which can be regarded as a measure of the scatter-
ing cross section of the cylinder, increasing at resonant frequencies:

4 4

Q =− + 2 sin p ( 1 − cos) p

(6.3)

p

p

2

where p is defined as

4 π rn ( − 1 )

p = (6.4)

λ

These two equations are defined almost entirely in terms of physical param-
eters of the target and the incident wave. These parameters are the radius r ,
the wavelength of the incident field λ, and the index of refraction n which is
also defined as

n =εµ (6.5)

rr

where the relative permeability is generally equal to 1. Tables 6.11 through
6.13 show families of images that demonstrate the effects of Q on Born recon-
structions as a function of the permittivity.
From a close examination of the images in these tables, it is quite clear that
there does seem to be a cyclic pattern in the quality of the reconstructions as
a function of increasing permittivity. Furthermore, examining the graphs in
Figures 6.3 through 6.5, it is evident that the performance of the reconstructed
images does seem to correlate to the predicted resonances of the target and
thus the “good” and “bad” reconstructions can therefore be predicted with
some certainty using this information. This then is highly suggestive as a
reasonable explanation for why strongly scattering cylinders, near a resonant
scattering condition at which the scattering cross section is larger, appear to