Image Processing Reference

In-Depth Information

Five

Data Processing

5.1 dAtA InveRSIon In
k
-SpACe: A FouRIeR peRSpeCtIve

Basically, there are two methods or algorithms that are commonly used in

performing Fourier-based inversion of scattering field data. They are

1. Fourier-based interpolation

2. Filtered back propagation

Other methods, which are not addressed in this topic, are more complex

and computationally intensive methods such as the modified gradient tech-

niques that are iterative in nature and do not necessarily converge for strong

scatterers. Detailed discussions of these methods can be found in Avish and

Slaney (1988). The basic methodology used in the Fourier interpolation method

is that the scattered field data are placed on semicircular arcs in 2-D
k
-space or

the Fourier domain where the loci of points are defined by the Ewald circles

or, in 3-D, Ewald spheres (Wolf, 1969). The Ewald circles arise naturally from

Equation 4.26 when adopting the first Born approximation. They are tangent to

the origin of 2-D
k
-space with a radius of
k
, where
k
represents the magnitude

of the scattered field's wavenumber. The transmitted data, that is, forward scat-

tered data lying on the part of the circle closest to the origin and the reflected

data, that is, backscattered data lying on the part of the circle farthest away

from the origin are depicted in Figure 5.1. The position of the circle is rela-

tive to the direction of the source of the incident wave that is illuminating the

target. As the incident wave is moved or rotated around the target and the

subsequent data gathered from the stationary receivers and mapped onto the

respective Ewald circle, additional Ewald circles of data are formed in
k
-space.

Ideally, as the source is rotated all the way around the target,
k
-space is filled

to some degree with data out to a “limiting circle” of radius 2|
k
| as shown

in Figure 5.2. This method can be used to develop an estimate of the Fourier

transform of the target for the given incident frequency, but theoretically this

is only a representative of an image in the weakly scattering limit known as

the first Born approximation. For more general scattering targets, these Fourier

data must be interpreted in terms of the integral equation given in Equation

4.26 indicating the dependence on both the scattering distribution and the total

field that resides inside the scattering volume. Other incident frequencies can

be used, which will in turn vary the radius of the corresponding Ewald circle

to help fill
k
-space. The radius of the Ewald circle is directly proportional to

the frequency of the incident illuminating source as illustrated in Figure 5.3.

In the filtered back propagation method mentioned above, the scattered field

data are “propagated” backwards into the object domain using an appropriate

57

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