Alternate Inverse Methods - Introduction to Imaging from Scattered Fields - page 100

then the Fourier transform of f ( r ) is written as

∞ ∞

= () =

∫∫

FF k f () r e d 2 r

− i n kr

(7.7)

n n

−∞ −∞

for n = 1, 2, 3, …, N . The PDFT estimator is given by

M

∑

f = p () r ae

i m kr

(7. 8)

PDFT m

m = 1

where p ( r ) is the nonnegative prior weighting function containing informa-
tion about the object. The advantage of the PDFT algorithm is that the data
need not to be uniformly sampled, which is why this approach can be used to
interpolate and extrapolate both nonuniformly sampled and incomplete data
sets. The term PDFT comes from the fact that it is the product of a prior p ( r )
and the discrete Fourier transform. The coefficients a m for m = 1, 2, 3, …, M are
determined by solving a system of linear equations

M

∑ 1

F = a P ( kk − )

(7.9)

n mn m

m =

In matrix notation, we can write the above equation as

f a = (7.10)

where

f = [ FF ( ,( ), k k … ,( )] F k

T

(7.11)

1 2 M

a = … [, ,, ] aa a M

T

(7.1 2)

12

and where T denotes the transpose of a matrix and P is the M × M square matrix
and is the Fourier transform of prior p ( r ) . Thus, using the coefficients calcu-
lated by solving the above set of equations, the PDFT provides a data-consistent
image estimate by minimizing the weighted error, which is defined as

1

∫

ξ=

() ()

f r − f () r d

2

r ( inimum)

(7.13)

p r

PDFT

The above integral is taken over the support of the prior function p ( r ), which
contains the information about the true or estimated support of the object. The
computation of the P -matrix can be made even if the prior is not available in
a closed form but is estimated as a surrounding shape, as the matrix elements
can be readily computed. There are also challenges associated with Fourier data
being noisy, and a method of regularization usually needs to be employed. If the