Image Processing Reference
Introduction to Inverse Scattering
Considerable knowledge of the world around us is based on receiving and
interpreting electromagnetic and acoustic waves. We extend the bandwidths
and sensitivities of our senses by using instruments and collecting radiation
from sources and scatterers of radiation. Active illumination or insonification
of objects to probe and image their structures is an important tool in advancing
our knowledge. However, we need to have a good physical model that describes
the possible interactions of those waves with scattering objects. Constitutive
parameters (such as permittivity, permeability, refractive index, impedance,
etc.) that have spatially and temporally varying properties describe the scatter-
ing objects. Wave propagation and scattering characteristics are governed by
the fundamental relationships between these properties and their effects on
the components of the field, as governed, for example, by Maxwell's equations.
In either the electromagnetic case or the acoustical case, we need to derive a
wave equation, both in differential or integral form, with appropriate bound-
ary conditions or coefficients, and then analytically or numerically solve that
equation to find the field outside the object. This so-called “direct” problem,
which assumes that the object parameters are known and scattered fields are
to be determined, is itself a nontrivial exercise but well defined
The complementary or “inverse” problem is much more difficult and is the
focus of this topic. Making measurements of the scattered ield at various loca-
tions near or far from the object takes time and effort. One has to specify the
incident field properties such as wavelength, polarization, and direction, and
then, relative to these, measure the scattered field properties. The question
immediately arises as to how many measurements does one need in order to
recover the information one wants about the object being probed. Inverting
the governing wave equation is, from a purely mathematical perspective, the
so-called ill-posed problem. Such problems require that one formally estab-
lish the following:
1. Whether there is a solution at all.
2. Whether the solution, should it exist, is unique.
3. Whether a calculated solution is or is not ill conditioned.
In most practical situations, one only measures a finite number of data on the
scattered field, and uniqueness is impossible. One can fit an infinite number of
functions (i.e., images) to a finite data set. This lack of uniqueness requires that
we either explicitly or implicitly adopt a uniqueness criterion such as mini-
mum energy, maximum entropy or some other such criterion using which one
can define a unique solution and hope that it has a physical meaning.
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