Born Approximation Observations - Introduction to Imaging from Scattered Fields - page 67

unified degrees of freedom criterion. Experiments are carried out to test this,
and speculation is made as to the exact form of this alternate requirement.

6.2 RequIReMentS FoR degReeS oF FReedoM FoR SouRCeS
For the purpose of demonstrating the role of degrees of freedom and its useful-
ness in predicting performance when imaging from scattered fields, a series or
“families” of reconstructed images will now be presented. These reconstruc-
tions are presented in a systematic way for a group of selected targets, all illu-
minated by an increasing number of incident waves, all with a frequency of
5 GHz. The targets of choice for this demonstration are defined and described
in Table 6.1.
For each of these target parameters, the degrees of freedom and the weakly
scattering metric ( kVa ), discussed in Section 6.1, are calculated and shown
for reference. Also, to aid in the analysis of these images, the borders of each
image are shaded in either “red” or “green,” with “red” indicating that for the
given parameters and number of sources, the minimum degrees of freedom
are not satisfied, and the “green” indicating that for the given parameters and
number of sources, the minimum degrees of freedom are satisfied With this
color code the reader should be able to distinguish, as the number of sources
is increased (moving “down” the table), when the minimum criteria are met.
It should be noted that there are obviously other factors that are affecting
the reconstruction in these images, namely obvious resonances that occur for
different parameters that definitely affect some of the images as compared to
others. These phenomena will be examined in more detail in the following
sections. It should also be kept in mind that the measure of and meeting of the
number of degrees of freedom do not necessarily indicate that the image will
look exactly like the target because the challenges and issues involved with
the inverse scattering problem remains; it simply means that for the given
parameters, adequate sources have been used to “communicate” or transfer as
much information about the target as is possible under the given experimental
arrangement. So, with this in mind, one would expect that after the minimum
degrees of freedom have been met, and for all images with higher degrees of
freedom, the image of the “target” should not change significantly from one
image to the next. This is not to say that the entire image will not change as
there could be significant differences in the “noise” artifacts in the free space

table 6.1 Definition of Various Target Types and Scenarios
Used to Test the Degrees of Freedom Requirement along with
Corresponding Table Number of Results

table number shapes Dimensions Permittivity Range

5.2 1 Circle Radius = 1 λ 2-10

5.3 1 Circle Radius = 1 λ 11-19

5.4 1 Circle Radius = 2 λ 2-10

5.5 2 Circles 1.1-1.9

Radius = 1 λ

5.6 2 Circles Radius = 1 λ 2-10

5.7 1 Square Sides = 1 λ 1.1-1.9

5.8 2 Squares 1.1-1.9

Sides = 1 λ

5.9 2 Triangles Base/height = 2 λ /3 λ 1.1-1.9