Image Processing Reference

In-Depth Information

the nature of scattered field data used to validate imaging algorithms and sug-

gest methods to generate such data. This of course is only necessary in the

absence of real measured data from known targets, but despite the best efforts

of many, real data sets are still few and far between. Data provided since the

early 1990s by the US Air Force and the Institut Fresnel (Belkebir and Saillard,

2001, 2005) have done a tremendous service in providing high quality data

from known objects, which provides a means for comparing different inverse

scattering techniques and thereby improves them. For a scatterer of compact

support (with
d
as the size of its largest dimension), the qualitative statement

is usually made that the (first) Born approximation is valid only when the

scatterer is “small” on the scale of the incident wavelength; this is discussed

in more detail in Chapter 4.

The Born series solution to the integral equation of scattering is an infinite

series which is traditionally defined as only valid when the criterion
kV
m
d
< 1

is met. Here,
k
is the wavenumber
k
= 2π/λ where λ is a measure of the wave-

length
inside
the scattering object. This is obviously difficult to determine for

an unknown object's constitutive parameter
V
(
r
) that is being imaged.
V
m
is

some measure of the maximum or mean value of
V
(
r
) which is also unknown;

consequently there is a temptation to apply the first Born approximation. This

requires that
kV
m
d
≪ 1 and, as we shall see, makes recovering an image com-

putationally straightforward. Indeed it reduces the inverse scattering prob-

lem to one of a limited-data Fourier estimation problem. This is a problem

on which there is much written, and it provides a comfort zone in which to

work and process scattered field data, in the (vain) hope that images obtained

when
kV
m
d
is not less than 1 still convey some meaningful information. The

criterion for the validity of the Rytov approximation is equally vague, relying

on the qualitative statement that spatial fluctuations in
V
be slow on the scale

of the wavelength, but that the magnitude of the fluctuations of
V
need not

necessarily be small or of low contrast. In other words, this physical inter-

pretation of the validity of the Rytov approximation is based on the require-

ment that the absolute value of the rate of change of the complex phase of the

scattered field within
V
be small compared with
k
∇
V
, where ∇ is the gradient

operator. If this assumption is reasonable, one can formulate the inverse Rytov

method as a limited-data Fourier estimation problem as well.

An interesting and important question to ask is what errors are introduced

if one does adopt the Born or Rytov approximation. This is a very reasonable

and insightful step to take and doing so has revealed classes of objects for

which one can expect the approximations to do poorly or fail altogether. There

is also much to be said for bringing to the inverse scattering problem a wealth

of signal and image-processing knowledge that has been established over the

years for dealing with limited data, especially limited Fourier data. By more

carefully formulating the inverse problem in terms of these approximations

and having a description for the errors and artifacts the “first Born approxi-

mate” image might possess, one can develop methods to postprocess those

images to try to recover
V
(
r
). This is the approach we have adopted and will

describe in more detail in a later chapter.

These inverse scattering algorithms that have been developed over the

years, often referred to as diffraction tomography algorithms, fall into two

classes. Devaney (1983) and Pan and Kak (1983) have modified the filtered

back-projection algorithm used in conventional tomography to give a filtered

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