Born Approximation Observations - Introduction to Imaging from Scattered Fields - page 65

Six

Born Approximation Observations

6.1 degReeS oF FReedoM
In the previous discussions, the scattering problem is viewed as an inverse
Fourier problem, which is a valid approach, but possibly not a complete
one. There is another way of viewing this problem, which may give more
insight into what is actually going on and what could possibly give a bet-
ter criteria for performance. If one were to view one isolated Ewald circle
map of the data from most penetrable targets, one would notice that most of
the information (that is, nonzero information) is located near the origin or
in the forward scattering section of the circle as shown in Figure 6.1. This
being the case, as most of the information is in the forward scattering mode
one could think of this as the information about the scatterer is “transmit-
ted” through the target. This could mean that another valid approach to this
problem would be to treat it as a transmission problem (Miller, 2007) where
one hase a source (the incident wave), a transmission medium (the target),
and a receiver (the receivers located in the forward half of the Ewald circle).
It is known from communication theory (Jones, 1988; Kasap, 2001) or analy-
sis that in order for certain types of signals to be successfully transmitted
over a given medium that a minimum number of modes or bandwidth must
be present to represent the necessary amount of information at the other
end of the medium. This is sometimes referred to as the minimum degrees
of freedom necessary for minimum image reconstruction. In this case, the
available degrees of freedom are merely a function of the physical charac-
teristics of the medium alone for a given wavelength (Jones, 1988; Kasap,
2001; Miller, 2007). Using this approach and applying it to this application
as shown and described in Miller (2007), Kasap (2001), Jones (1988), and
Ritter (2012), the general relationship that predicts the minimum degrees of
freedom in 3-D is

Bn
V

⋅
λ

max

(6.1)

N =

3 -D

3

where N 3-D is the minimum degrees of freedom required in 3-D, B v is the tar-
get volume, n max is the maximum index of refraction, and λ is the wavelength
which can be modified easily for 2-D as follows:

An
V

⋅
λ

max

(6.2)

N =

2 -D

2

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