Introduction to Inverse Scattering - Introduction to Imaging from Scattered Fields - page 4

In some imaging applications, one cannot measure the scattered field itself,
for example at very high frequencies. Above ~1 THz we do not have detectors
fast enough to measure the fluctuating field and all we acquire is a time aver-
aged quantity. In electromagnetic problems we assume this is proportional to
the magnitude squared of the (electric) field, | E | 2 . As we shall see, the infor-
mation required to solve the inverse problem and calculate an image of the
object demands that we solve another problem, namely that of estimating
from | E |, the function E = | E | exp( i ϕ) or solve the so-called phase-retrieval
problem (ϕ denotes phase). This is also nontrivial and, without knowledge of
the phase, the information we can recover about the object is severely limited
and at best statistical in nature.
Most problematic is the inevitable presence of noise in our measured data.
Inverse procedures, as we shall see in the coming chapters, are always ill
conditioned. This means that one can expect small changes in the data as a
result of noise to lead to very large differences in our images. The instability
of inverse methods can be understood mathematically and remedied using
the so-called regularization techniques. The price to be paid to control ill
conditioning is a degradation of the image, for example, a loss of resolution.
However, since we cannot guarantee a unique solution in practice, we have
to accept further compromises in order to obtain an image we can have some
confidence in.
From a practical standpoint, we hope to collect the minimal amount of
data to provide the image quality needed for the task at hand. Maps of spa-
tially varying contrast might suffice while, for other purposes, for example,
in medical imaging, a quantitatively accurate map of a constitutive parameter
such as impedance might be essential. Overarching all of these issues is the
more important problem of the governing equation to be inverted being inher-
ently nonlinear in nature. The scattered field for all but the weakest scatter-
ing objects depends on the complexity of the scattering processes that occur
within the object itself. For inverse problems, for by very definition we do not
know the structure of the object, we cannot know a priori the extent of multi-
ple scattering that occurs within it. We can define what we mean by “weakly”
scattering, and that assumption, while rarely valid in practice, does lead to a
more tractable inversion method but one that still suffers from the questions
of uniqueness and ill conditioning mentioned above. For more interesting,
but strongly scattering objects, we need to address the nonlinear aspect of the
inverse problem. We will describe methods that do this but emphasize now
that there are, at the time of this writing, still no fast and reliable methods one
can take off the shelf and use. Indeed, despite many decades of effort, inverse
scattering methods remain very challenging and an active field of research.
Methods we describe here have a range of applicability that limits their use to
situations for which some prior knowledge about the object is available. This
is certainly possible in some applications such as imaging a limb or probing
a suitcase, and prior knowledge can play an important role in addressing the
uniqueness question, as we shall see.
1.2 InveRSe SCAtteRIng pRobleM oveRvIew
The wavelength of the radiation used with respect to the scale of the features
one wishes to image provides a convenient way to segregate inverse scattering