Image Processing Reference

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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.

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