Image Processing Reference
In-Depth Information
It is clear from the images in Table 10.4 that there is no significant gain or
benefit to have a width for σ greater than about 10 for the particular family of
targets considered here. It should also be noted that there was no valid recon-
structed image output for the “combined in cepstrum space” filtering method
for values of σ of 3 and 10. It will be shown later that this is due to the peak
value of the filter, which can be addressed by scaling. However, it should be
noted that this form of Gaussian filter has a maximum value at the center that
varies inversely as σ varies to maintain a value of 1 for the total area under
the curve. The larger the value of σ is the smaller the peak of the filter It again
will be shown later that this plays a very significant role in the filtering pro-
cess in the cepstrum domain.
Now that it is known that the majority, if not all, of the “good” or desir-
able part of the spectrum lies within σ = 10 of the center of the filter, the next
question is whether there is any location within this region that has “bad”
or undesirable data. To investigate this, a series of modified Gaussian filters
are used to try and further characterize the mapping of data in the cepstrum
domain. To do this, a Gaussian filter with a σ value set to 30 was strategically
“notched” with gaps of varying widths and locations to see if there was any
location “under” the Gaussian filter that is more important than any other for
the “good” or the “bad” data.
The results of this filter experiment and its effects on the various types of
cepstrum filtering discussed previously are shown in Table 10.5. It is evident
again from this table and from Table 10.4 that the important signal data is
located within the σ = 10 range. There does not seem to be any benefit, in
general, in trying to include any data from the outer limits of the cepstrum
domain. It is demonstrated clearly from the last set of images in Table 10.5,
where the σ = 0 to σ = 5 range is filtered out that this is where the most impor-
tant information is since the reconstructions for this condition are very poor,
and the original target image is undetermined even with 36/360 degrees of
freedom for sources/receivers, respectively.
Now that the optimal value for σ has been investigated and evaluated for
the Gaussian filter in regard to the bandwidth, we will now investigate the
peak value. It is observed in Table 10.4 that the “combined cepstrum” method
does not produce any results for three of the scenarios presented. It was
assumed that this was due to the fact that the pertinent “harmonics” were
either outside the pass band, or they were excessively attenuated due to the
reduced peak that is inherent to the Gaussian function. By using some simple
experimentation on the peak value of the Gaussian by using simple scaling
techniques, it was discovered that the issue was not that the Gaussian filter
peak was too low, but that it was, in fact, too high. The fact is that there is an
apparent inverse relationship between the peak value of the Gaussian filter
and the overall range in magnitude of the resulting image using the “com-
bined cepstrum” method. Moreover, it was observed through simulations that
the minimum constraint for the peak of the filter to insure that the “combined
cepstrum” method produces a reconstructed image is that the peak must be
equal to or less than 1/ N s , where N s is the total number of sources. It was also
observed that, in general, to scale the magnitude of the reconstructed image
produced by the “combined cepstrum” method in order to be in the same
general range as the magnitude of the image produced using the Born recon-
struction, it was necessary for the filter to be scaled by an additional value
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