Image Processing Reference

In-Depth Information

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

65

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