Image Processing Reference

In-Depth Information

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Figure 9.10

FoamDielInt: (a) Actual object, (b) Born reconstruction.

FoamDielInt at 6 GHz. Figure 9.11 shows the implementation of the method to

estimate the FoamDielInt object. The available data was made causal by mov-

ing it into one quadrant, and then a reference point was added such that the

phase of
V
Ψ lies between +π and −π.

A Gaussian filter was then applied to the cepstral domain, and was then

reduced in diameter until no discernable wave-like features associated with

Ψ remained in the final image, which is obtained as a further inverse Fourier

transform of the filtered cepstrum and exponentiation. Figure 9.11e shows

the reconstruction obtained by applying cepstral filtering The ratio of the

contrast of the reconstructed cylinders matches the ratio of the permittivi-

ties in the original object. Figure 9.11c shows a log-plot of the cepstrum of

V
Ψ after the reference point has been added. The log-plot is used to view the

low energy features in the cepstral domain. The success of cepstral filtering

relies on
V
Ψ being minimum phase, which is dependent upon the amplitude

and location of the “artificial” reference point. Figure 9.12 shows the varia-

tion in the quality of reconstruction as we change the reference point ampli-

tude. Figure 9.12c shows that the quality of reconstruction improves as we

get closer to satisfying the minimum phase condition. FoamDielInt was also

computed for various other frequencies; the results are shown in Figure 9.13.

One of several factors that will affect the quality of the reconstruction (apart

from the difficulties associated with inverting multiply scattered data) is the

data determined image point-spread function. Figure 9.14 shows the point

spread function for different frequencies calculated from the inverse Fourier

transformation of the set of delta functions which mark the locations of the

sampling points in
k
-space, as provided by Institut Fresnel. These indicate the

extent of features that one can hope to resolve in the final reconstructions of the

scattering object
V
(
r
) from the measured scattered data in an ideal situation.

At lower frequencies, that is, 3 GHz, the locus of data in
k
-space is over a

much smaller radius Ewald circle than for the 10 GHz case. As a consequence,

the main lobe of the point-spread function is much larger at lower frequen-

cies of illumination, and therefore, one can expect to see a lower resolution

reconstruction. However, low frequency illumination also implies a better

image estimate of
V
(
r
) based on the conditions for the validity of the first Born

approximation, since a condition for the Born approximation to be valid is

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