Scattering Fundamentals - Introduction to Imaging from Scattered Fields - page 36

mentioned previously. When the wavelength is not relatively small, scattering
and diffraction effects become important. It is this specific case that we will
focus on here.
The problem of determining the image of an object from scattered field mea-
surements has received a lot of attention lately because the associated applica-
tions are so vast. For objects that are sufficiently weakly scattering, it is well
known that the inverse scattering problem can be linearized and thus becomes
readily soluble using the first Born or Rytov approximations (see later). Object
reconstruction can be achieved by extracting the Fourier data of the object from
the scattered field This is available over semicircular arcs in the Fourier plane
( k -space) of the object, which normally often requires interpolation and extrap-
olation to improve the image estimate. The more the measurements of the scat-
tered field that are made, the more the points obtained in k -space. For compact
objects, the Fourier transform is analytic, or more precisely, an entire function
of exponential type. It is this fundamental underlying property that leads to the
definition of sampling theorems and provides effective methods of extrapola-
tion and interpolation. In principle, knowing an analytic function over a small
but continuous interval of k -space should allow it to be extrapolated by analytic
continuation throughout k -space. Of course, the fact that the data are measured
at discrete points and are finite in number means that we have an infinite num-
ber of possible solutions. The analyticity does provide an overall constraint
and the compact support, even if it is only approximately known, provides con-
siderable prior knowledge that can be exploited. A support constraint not only
allows an (in principle) adequate sampling rate to be defined, but also allows
the Fourier basis being used to be modified to a more appropriate and more
effective basis to be defined We discuss these concepts more in Chapter 4.
Improved sampling strategies combined with more effective representations for
the object suggest that we might be able to get better images with fewer mea-
surements and there is some truth to this. The most recent implementation of
these ideas is described and explored in Chapters 6 and 10.

3.3.1 Scattering from obstacles
When a field excites a dipole, energy is transferred and seen as a loss or
“extinction” of the incident wave. Time-averaged scattered power is defined
as P scat ≡ σ scat I 0 with some 3-D scattered field pattern. There may also be some
absorption P abs ≡ σ abs I 0 and a total extinction cross section given by P ext ≡ σ ext I 0
where σ ext = σ scat + σ abs . The differential cross section is defined in Figure 3.1.

z

r sinθ dθ

d P
dΩ

dσ(θ , φ)
dΩ

≡ I 0

r dθ

r

θ

y

dσ Scattered power per unit solid angle at (θ,ϕ)

Incident power per unit area

=

dΩ

ϕ

x

Figure 3.1 Illustration of differential cross section.