Advanced Topics in Inverse Imaging - Introduction to Imaging from Scattered Fields - page 176

The question of the quality or the appearance of each reconstructed image is
altogether another issue.
The two factors of the number of degrees of freedom and the effects of reso-
nance in determining the quality of a Born reconstructed image are extremely
important. This now gives a new and possibly more meaningful set of criteria
for determining or predicting when it is satisfactory to implement and expect
to be able to successfully interpret an image based on the Born approxima-
tion and on methods related to Born approximation. The previous criterion
of | kV ( r ) a | ≪ 1 only claims to predict whether a target is a weak scatterer
or not, which can be terribly subjective and inconsistent in predicting the
performance of image reconstruction. The new criterion gives a much more
definitive predictor since the minimum number of degrees of freedom can
be calculated precisely if n max is known, and the resonances might be deter-
mined or even identified for any given target.
Finally, several new techniques and algorithms were demonstrated and
examined that so far have proven to be very effective. The first technique
that was examined was that of the implementation of the code applying the
cepstrum filtering method on an individual source basis followed by a com-
bination of these processed data in either the image domain or the cepstrum
domain. This is in lieu of an earlier approach that simply applies the ceptrum
filtering method to the Born approximation composite image obtained from
the combined sources. As demonstrated previously, this modified method
shows a definite improvement in the appearance of the image in relation to
the boundaries of the target compared to previous methods. More specifically,
the procedure that combined the sources, after processing, in the cepstrum
domain showed the most promise in terms of more accurate image boundary
definitions The only difficulty with these methods initially was the quantita-
tive accuracy, in that the scale of magnitude for the improved reconstructed
images did not seem to be reasonable. This issue is addressed later.
One lingering question from the cepstral filtering method above was that
of the type of optimum filter to be used in the cepstrum domain. Through a
series of tests shown earlier, it was determined that the optimum filter to be
used is a Gaussian type filter centered at the origin using a σ value equal to
10 (for the specific conditions of the set of targets considered here). Another
striking discovery was that the absolute peak value of the filter plays a criti-
cal role in determining the magnitude of the reconstructed image produced
by the new cepstrum method described above. In particular, it was shown
that the magnitude of the output of the method that processes each source
individually and recombined them in the cepstrum domain was inversely
proportional to the magnitude of the peak of the Gaussian filter After further
examination, it was shown that the peak value of the Gaussian filter needed
to be scaled by a factor of 2/(3 N s ), where N s is the number of sources, to let the
magnitude of the reconstructed images be of the same range of the magnitude
of the Born reconstructed image for the same data. It is not known or under-
stood at this time why this scaling factor is needed. It could be a result of
applying 3-D techniques to a 2-D problem or some variance of this. In addition
to these experiments on the filter characteristics in cepstrum space, a new
approach was derived and evaluated for potential benefits as well. That new
approach involved subtracting a weighted cepstrum of the incident field dur-
ing the processing of the sources in cepstrum space. As explained earlier, this