Image Processing Reference

In-Depth Information

Figure 9.17
FoamDielInt at 6 GHz: (a) Summation of cepstral reconstruction of 4, 5 and 6 GHz with PDFT and

(b) summation of cepstral reconstruction of 4, 5 and 6 GHz with DFT.

provide an additional mechanism for extracting spatial frequencies of
V
alone

from the cepstrum. A linear combination of estimates for
V
acquired in this

way will further improve the signal-to-noise ratio (SNR) of the estimate for

V
while suppressing any residual components from the Fourier transform in

each of these images. An example is shown below in Figure 9.17. It is also evi-

dent, as expected, that a Gaussian cepstral filter, being apodized, is preferable

to a circular or square hard-cut filter

The summation of cepstral reconstructions for different frequencies when

used with the PDFT has shown significant improvement over the summation

of cepstral reconstructions with DFT. To compare Figures 9.17a and 9.16f, one

sees that the summation of cepstral reconstructions for different frequencies

has shown improvement in shape estimation over a single frequency cepstral

reconstruction using the
P
, however summed up cepstral reconstructions still

need improvement to recover the permittivity distribution of the object,
V
.

One could feasibly employ a two-step process in which one would use sum-

mation for shape retrieval and single frequency reconstruction for quantita-

tive recovery for a more balanced approach.

9.2.2 Foamdielext

The FoamDielExt consists of a cylinder with of relative permittivity ε
r
≈ 1.45

and an external cylinder with relative permittivity ε
r
≈ 3.0. The Born recon-

struction and cepstral reconstruction at illumination frequency of 6 GHz are

shown in Figure 9.18.

It is again observed that the background permittivity level being incon-

sistent from the actual object's background. The PDFT implementation of

FoamDielExt with rectangular prior is shown in Figure 9.19.

In Figure 9.19b the background permittivity from the PDFT estimate is

improved significantly as compared to the DFT in Figure 9.18b by the use of a

rectangular prior function. The extrapolated values contribute to the improve-

ment of the resolution of the object estimate. Including more prior information

in PDFT, such as object feature information or relative permittivity differ-

ences, should lead to further improvements in the reconstruction.

9.2.3 Foamtwinext

FoamTwinExt is one of the most complex objects made available by Institut

Fresnel. It is the combination of FoamDielInt and FoamDielExt objects that

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