Image Processing Reference

In-Depth Information

One

Introduction to Inverse Scattering

1.1 IntRoduCtIon

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.

3

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