Advanced Cepstral Filtering - Introduction to Imaging from Scattered Fields - page 151

It appears in the filtering processes where the sources are independently fil-
tered, showing that this size inflation is decreased. Moreover, it appears that
the images from the process of where the independent source data were com-
bined in the cepstrum domain seem to perform slightly better in this regard.
Additionally, the independent source filtered images exhibit a scaling issue
as compared to the Born image and the cepstrum-filtered Born images. The
image with the data combined in cepstrum space has a magnitude or scale
that decreases as the number of sources is increased, while the image with the
data combined in image space has a magnitude that increases as the number
of sources is increased. It will be shown in a later section that a scaling factor,
dependent on the number of sources, can be applied to adjust or improve these
inappropriately scaled values. Taking all this into account, and analyzing the
images in Table 10.3, it appears that in general, the method of processing the
independent source data separately followed by combining these data sets
in the cepstrum domain appears to perform better than the other methods
examined.

10.2 eFFeCtS oF ModIFIed FIlteRS In CepStRAl doMAIn
As discussed earlier, it is known and understood that in the 2-D cepstral filter-
ing method, it is necessary to use some type of low pass filter to eliminate or
attenuate the Ψ component in the data; there is very little material available
to assist in determining the optimal parameters for this filter This is com-
pounded even further in that there is little information for one to intuitively
know where the “signal” data and “noise” are located in the cepstrum space
since the cepstrum domain does not have a linear relationship to the spectral
domain. This being the case, some experimentation may be necessary to gain
a better understanding of this domain to help in creating the optimum filter
for better image reconstruction. It was demonstrated in Shahid (2009) that
a Gaussian type filter performs reasonably well in filtering in the cepstrum
domain, and this was characterized by the following equation:

1
2

Filter = e

−+ [ xy

2 2

]/ 2 σ

2

(10.1)

πσ

The σ term in Equation 10.1 dictates how wide the filter will be, which we
will refer to as its bandwidth. Conventional filter theory would suggest that it
is desirable to have the filter bandwidth to be as wide as possible to allow as
much of the desirable signal spectrum to pass as possible, while at the same
time narrow enough to significantly attenuate the undesirable signal (or noise)
as much as possible. This assumption is based on the noise having mostly
higher spatial frequencies than those associated with the original target, V ( r ),
which need not be necessarily the case, especially in the cepstrum domain.
Therefore, it seems a logical place to start to vary the width or σ term and
observe the effects, if any, on the performance of the various filtering methods
described above. Table 10.4 contains a family of images that show how each
of the cepstrum filtering methods mentioned in the previous section performs
using a Gaussian filter described by Equation 10.1 with gradually increasing
width or σ term.