Image Processing Reference

In-Depth Information

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

Search WWH ::

Custom Search