Image Processing Reference

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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

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