Advanced Topics in Inverse Imaging - Introduction to Imaging from Scattered Fields - page 185

and of course far more effort has gone into developing methods that attempt to
directly address the strong scattering problem. We have only touched briefly
on most of these alternate methods here, because they require a deep theoreti-
cal treatment that is beyond the scope of this topic. Also, they often require
considerable prior knowledge and/or sophisticated pre- and postprocessing in
order to guarantee convergence to an unbiased solution. What we presented
here was an alternative and one-step nonlinear inverse scattering method,
which returns to the strategy of trying to linearize a blatantly nonlinear prob-
lem. By considering the integral equation of scattering for the strongly scatter-
ing situation, we model the Born approximation reconstruction as an image
corrupted by multiplicative band-limited noise (i.e., the different realizations
of the total field within the scattering volume). This allowed us to apply ceps-
tral filtering, that is, linear filtering techniques drawn from Fourier-based
theories to the logarithm of the recovered “image.” The steps involved in
processing this algorithm were described and several important conclusions
were drawn.
The first was that this method does indeed work quite effectively but only if
there were sufficient data to work with. Based on a more fundamental analysis
of a scattering experiment, one can identify the number of degrees of freedom
of that system, rather like viewing it as an information channel. In order to
recover an image in which one can have some confidence, the scattering geom-
etry and maximum values for the scattering parameters (i.e., size and index)
allow us to define a channel capacity or the number of degrees of freedom
associated with that imaging experiment. Only when a sufficient number of
data have been collected, for example, by either varying the source and/or
the receiver locations, can one expect to obtain a reliable image, the problem
being underdetermined otherwise. Being Fourier based, incorporation of prior
knowledge can alleviate limited data problems, but one must always be care-
ful about how this is done in order not to bias the recovered image. Also, we
noted that the precise form of the cepstral filter used is not that critical, and
more work could be done for any given problem to optimize this step, depend-
ing on the scattering object of interest.
It was also interesting to note that one cannot ignore resonant effects when
illuminating scattering objects. The size, shape, and material properties can
lead to energy storage in the scatterer, and there is always the fundamental
issue of addressing possible nonscattering structures leading to significant
ambiguities. We probed the simplest of resonant structures with this con-
cern in mind, namely, dielectric spherical scatterers, since there is an exact
solution available for spheres whose size is of the order the wavelength. The
solution is expressed in terms of an infinite series, but a truncated series can
provide very good estimates for the scattered field from various spherical
objects. We found that at resonances when the internal field in the scatterer
is at its maximum, the cepstral-based recovery of the image of the spherical
shape improved. This indicates that the more the complexity of the scattered
field inside the scattering object, the better this inversion method works. For
images of dielectric spheres based on this Mie scattering model, we found that
reconstructions (Born and cepstral) were better close to a Mie resonance, that
is, at a relatively high Q situation.
Also, in the presence of sufficient data as dictated by the number of degrees
of freedom, the recovered image of these objects reflected their scattering