Prior Discrete Fourier Transform - Introduction to Imaging from Scattered Fields - page 202

Input Low pass PDFT

Figure c.1

Example of low pass filtered images versus a PDFT image.

where F n is determined by the reconstruction algorithm. For instance, to apply
an inverse DFT it is assumed that ˆ FF

n = . The PDFT is an attempt to find the
estimate that is in accordance with the prior information and has a minimum
deviation from the real object. Formally, this can be expressed as

1

·

∫

2

χ= f () r − f () r d

L

r → inimum

pr ()

Substituting for ˆ () f r the F n is the free parameter, and we find the minimum

as a solution of a system of linear equations

N

∑ ·
1

F () k = F P ( k − k )

n n n m

m =

with P ( k ) being the Fourier transformation of the prior. The parameter F n ,
which allows calculating the image estimate from an inverse DFT, is obtained
as the solution of this system of equations. To solve the system of equations
the P -matrix, P n,m = P ( k n - k m ), has to be inverted, which is typically ill-condi-
tioned. To overcome this problem a Tikhonov-Miller regularization is intro-
duced by multiplying the diagonal of the P -matrix with a value (1 + ε), ε being
much smaller than 1. This, in effect, accounts for a homogeneous background
noise in the image estimate, and the size of ε necessary to obtain a decent
image reflects the signal-to-noise ratio.
We have used the PDFT algorithm primarily to improve the Ewald sphere
data coverage as compared with a discrete inverse Fourier transform. We
would like to emphasize that the PDFT algorithm can be used to evaluate the
quality of data based on knowledge of the target geometry. Then the regular-
ization parameter in combination with the energy of the PDFT reconstruction
can be used to quantify the signal-to-noise ratio and detect the presence of
multiple reflection artifacts contributing to the data.
Figure C.2 illustrates the performance of the PDFT algorithm. It is evident
that the PDFT provides considerable benefit for nonuniformly oversampled
data. This is precisely the situation for monostatic backscatter data where only
a small number of views is obtained, but with a large number of time frequen-
cies. In this case, the incorporation of prior knowledge frees up the degrees of
freedom to improve the image quality and the resolution. It should be noted
that the quality of the PDFT reconstruction in Figure C.2 strongly depends