Image Processing Reference

In-Depth Information

“foam” of relative permittivity ε
r
≈ 1.45 and inside the “foam” there is another

circular dielectric of relative permittivity ε
r
≈ 3.0. From Chapter 5, Table 5.1

shows the reconstruction from the inverse Fourier transform of scattered

field data, that is, first Born reconstruction of FoamDielInt. The Born recon-

struction is computed at 6 GHz operating frequency for which the scatter-

ing strength of the object is |
kVa
| ≈ 22. The object represents a fairly strong

scatterer, |
kVa
| ≫ 1, as a result of which we see that the first Born recon-

struction is not all that good. For FoamDielInt and FoamDielExt, the emitting

antenna was placed at eight different locations which were 45° apart whereas,

for FoamTwinDiel and FoamMetExt, which are more complicated objects, the

emitting antenna was positioned at 18 locations with 20° angular intervals.

The receiving antenna collected complex data at 1° intervals. The scattering

experiment was conducted using nine operating frequencies, which range

from 2 GHz to 10 GHz.

The number of degrees of freedom for one of these objects is approximately

given by 11 × 11 cm
2
× 1.8/λ
2
, and the wavelength varies from 15 cm to 3 cm

depending on the frequency used. This gives 1 <
N
< 25 assuming no metal is

included. However, it is the highest frequency case that will dictate whether

the number of measurements taken is sufficient At lower frequencies, the

data collection might well be adequate but the resolution limit using 2 GHz

illumination is approximately the size of the entire scattering object, revealing

only whether it is there or not; not very useful in our opinion. If we consider

N
= 25, then 4
N
2
= 2500. Assuming 18 antenna positions and measurements of

the scattered field for each and every degree, the total number of data avail-

able are 6300. This should be sufficient for the nonmetallic objects. For the

choice of only eight incident field directions, 2800 is just about sufficient

Consequently, we would argue that we can have some confidence in algo-

rithms that recover good reconstructions from the Institut Fresnel data of the

sets of nonmetallic objects.

Finally, in Section 6.4, we discussed imaging resonant objects and noted

the improved appearance of the reconstruction of cylinders when illuminated

at frequencies close to their Mie resonant frequencies. Figure 9.26 shows on

the left a reconstruction of a cylinder with large permittivity, the black circle

(a)

(b)

×10
-0

0

0

0.03

12

11

10

9

8

7

6

5

4

3

50

50

0.025

100

0.02

100

0.015

150

150

0.01

200

200

0.005

250
0

250
0

50

100

150

200

250

50 00

150 00

250

Figure 9.26
(a) An image of a cylinder with large permittivity making the first Born approximation invalid.

(b) More uniform reconstruction of the cross section of a cylinder for a weaker scatterer with
ε
r
=
1.1. The left side

image, as expected, has approximately the correct diameter.

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