Prior Discrete Fourier Transform - Introduction to Imaging from Scattered Fields - page 201

Appendix C: Prior Discrete Fourier
Transform

bACkgRound And deFInItIon
Since our goal is to recover images of scattering objects with a size similar
to the wavelength of the illuminating wave, an image estimate based on an
inverse discrete Fourier transformation of the data points in k -space is not
sufficient to perform the task. Therefore, we have proposed basing the inverse
scattering step on a spectral estimation technique in order to obtain a suf-
ficient resolution and distinguish different types of objects. The method we
have been applying is called the prior discrete Fourier transform (PDFT) algo-
rithm. This algorithm had been in use for some time to postprocess back-
propagated fields It is the result of rather recent efforts to adapt the PDFT
algorithm as a general substitute for any linear backpropagation procedure.
The PDFT algorithm was developed to estimate signals from sparse dis-
crete measurements, simultaneously incorporating prior knowledge about the
object or the measurement system. The PDFT can be applied to a large variety
of problems. Typically, measurements provide partial knowledge about the far
field of the scattering amplitude. However, in addition, information is avail-
able about the typical lateral extension of the object or the area from which the
scattered field originates.
For example, in Figure C.1, a low pass filtered image (center) can be enhanced
by using the larger outer box in the middle image as the function p ( r ), to give
the image on the right. We assume that the data we get from the scattering
experiments is the knowledge of the Fourier space F ( k ) of some target f ( r ) at N
arbitrarily sampled spatial frequencies k n :

∞

∫ ∞

F () k = f () r exp ( − i kr ) d r

L

n n

−

with L referring to the dimension of the problem. We further assume that the
target is of compact support and that the object has a physical size, which is
described by a real-valued positive function p ( r ). This prior can be used to
incorporate information about the shape as well. In its simplest form p ( r ) can
be a top hat that circumscribes the object entirely. Then, an estimate of the
object can be obtained from an inverse DFT

N

ˆ

∑ 1

ˆ

f

PDFT ()

r = p () r

n Fi

exp ( kr )

n

n =

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