Image Processing Reference

In-Depth Information

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

=

201

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