Applications to Real Measured Data - Introduction to Imaging from Scattered Fields - page 140

before guidelines were known, of the kind we have provided here, relating to
degrees of freedom.
In Chapter 6.1 it was stated in Equation 6.1 that

Bn
V

⋅
λ

N =

max

3 -D

3

where
N 3-D is the minimum degrees of freedom required in 3 dimensions
B v is the target volume
n max is the maximum index of refraction
λ the wavelength

which could be modified as in Equation 6.2 for 2-D as

An
V

⋅
λ 2

N =

max

2-D

where the target volume is replaced with the target area A v .
From the many simulations presented here, we can conclude how the num-
ber of degrees of freedom provides a metric for the number of data points
necessary to recover a reasonable image. In the figures shown in Chapter 6,
the number of degrees of freedom, N , is given and then the images are shown
as function of ε( r ) and the number of data used, expressed both as a function
of N and as an absolute number given the objects' dimensions and permit-
tivities. We noted that there are obviously many factors that are affecting the
quality of these reconstructions, not the least being the fact that even using a
cepstral filter, though the filter shape has not been optimized. Nevertheless, a
reasonable sense of how many measurements must be made can be estimated
and at least 4 N 2 appears to be necessary. One can consider this to represent the
fact that there needs to be at least 2 N source locations and at least 2 N receiver
locations. However, in practice, we have found that assigning more degrees of
freedom to the receiver locations rather than the source locations appears to
be slightly more beneficial We cannot rule out that a statement such as this
might be object-dependent.
Take the IPS008 target which is discussed in Section 9.1.1 above, which
was considered a strongly scattering object since k | V | a ≈ 87 where k , the wave
number, is calculated as k = 2π/λ = 2π/0.03 = 209.5 m −1 , V is the scattering
strength or average permittivity and “ a ” is the dimension of the largest feature
of object. The 2-D number of degrees of freedom can be estimated to be π(5λ) 2
3 1/2 /λ 2 ~ 45. The value of 4 N 2 ~ 8000. The measured Ipswich data consisted of
36 illumination directions, at equal angular separations of 10° and 180 com-
plex scattered field measurements for each view angle using a frequency of
10 GHz and hence the total number of data available is of the order 6500 which
we argue could well be insufficient This being the case, then even a perfect
inverse scattering algorithm would not be able to generate an image that is a
complete rendition of the original object.
Let us consider the measured data provided by Institut Fresnel, where the
models are defined in Figure 9.9. The FoamDielInt consists of two cylinders: a