Image Processing Reference

In-Depth Information

table 11.2
Family of Reconstructed Images for a Circle with a

Radius of 1λ for Varying Permittivity Values and Varying

Number of Randomly Spaced
Data Points in
k
-Space

ε
r
=
1.1;
N
=
3.3

ε
r
=
1.5;
N
=
3.8

ε
r
=
1.9;
N
=
4.3

NN/2

8

8

15

NN

16

16

30

2NN

32

32

48

4NN

72

72

108

12,960

12,960

12,960

Next, we look at the results from the square targets shown in Table 11.3. It

again is clear that, for the cases of
N
2
/2 and
N
2
total data points, the degrees

of freedom are clearly not satisfied for any of the permittivity cases. Likewise,

as before, the images are starting to take shape for the 2
N
2
case for all permit-

tivity values, but still appear as if they have not quite satisfied the overall

number of degrees of freedom. Finally, again as before, once the number of

data points reaches the 4
N
2
case it is evident for all permittivity ranges that

the image is fairly well formed. This even strengthens the idea that there is

an overall degrees of freedom criterion and that this limit is in the neighbor-

hood of 4
N
2
. This needs more exhaustive testing before it can be declared as a

theorem, but it surely presents a strong case for the relationships suggested for

overall degrees of freedom for imaging while using these techniques.

11.3 ConCluSIon

The problem of imaging from scattered fields is at least 100 years old and con-

tinues to provide challenges, both theoretical and experimental, across liter-

ally a broad spectrum of probing waves. The interaction of waves with matter,

be they acoustic or electromagnetic, is complex; even the direct problem, that

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