Image Processing Reference
In-Depth Information
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
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