Introduction to Inverse Scattering - Introduction to Imaging from Scattered Fields - page 8

of measurements of scattered field data. Future sections will go into this in
more detail. When the object in question is strongly scattering, this is known
well to be a very difficult problem to solve. While advances have been made
primarily using iterative techniques, long computational times are neces-
sary, and convergence is not guaranteed, especially if only incomplete and
noisy data are measured. Most computationally feasible algorithms assume a
weakly scattering condition in order to linearize the inversion procedure and
can be broadly classified as diffraction tomography techniques. The primary
motive for writing this topic is to introduce a method, which remains rela-
tively fast and simple to implement and which builds on these known inverse
scattering algorithms (despite their limitations) and more advanced signal
processing concepts. We have developed a method, which is simple to imple-
ment, and which can provide a good estimate of the image of the scatterer. It is
a diffraction tomography technique that has built in to it a nonlinear filtering
step, which directly addresses the limitations of assuming kV m d ≪ 1 when it
is clearly not correct. The success of the improved method inevitably depends
on the quality and extent of the data, and it is useful if one has data from
several illuminating wavelengths assuming that the constituent parameters
representing the object do not change significantly over the frequency range
employed. As we shall see, the quality of the reconstructed image depends on
a number of factors. The new method is computationally fast and at the very
least is useful in providing a quick look at the object features and serves as an
initial guess for a more rigorous inversion scheme. Such a method based on
diffraction tomography can obviously be used for both imaging and structure
synthesis applications. If one measures the scattered field and can compute an
image of the scattering object, then one can also specify a scattered field and
thereby compute and define a scattering object. If one can make this object
such that it will behave as expected, then this rich field of object synthesis
or target camouflaging goes hand in hand with the imaging emphasis of this
topic. In the case of imaging from scattered ield data, it is important that the
content of the reconstructed image can be relied upon in order to discern
features that might have some practical significance The structure synthesis
problem is not so constrained. In principle, any structure that provides the
desired scattering characteristics is a solution to the synthesis problem. The
only concern might be the ease with which the predicted scattering structure
can be fabricated, that is, in terms of its material properties and structural
features.
What is diffraction tomography? This family of inverse scattering tech-
niques is formulated as a Fourier inversion procedure. Complex scattered field
data are collected at a number of scattering angles in the (near or) far field
for each angle of illumination as shown in Figure 1.1. As will be detailed in
Chapter 4, the scattered far field data under these conditions, based on the
first-order Born approximation, are mapped onto Ewald spheres in a space
we will refer to as k -space which is the Fourier space associated with the real
space in which the scattering object exists. With sufficient k -space coverage
from measurements of scattered fields at different angles of incidence and dif-
ferent scattering angles, the inverse Fourier transform provides information
about the scattering object (Figure 1.1). Traditionally, should kV m d ≪ 1 then
this information will be directly proportional to the object's permittivity or
refractive index profile, relative to that of the background in which it resides.