Image Processing Reference

In-Depth Information

We suppose that we have
p
(
x
) ≥ 0 as our prior estimate of the overall shape

of the function
f
(
x
); let
P
(ω) be the Fourier transform of
p
(
x
). Let us take as

the special functions the set
H
m
(ω), with
m
= 1, …,
N
, where
H
m
(ω) is the

function whose inverse Fourier transform is
h
m
(
x
) =
p
(
x
)exp(
ix
ω
m
); that is,

H
m
(ω) =
P
(ω − ω
m
). Now we apply our estimation procedure to each of these

H
m
(ω).

We fix a value of
m
and apply the estimation procedure in the previous

equation to estimate
h
m
(
x
). We then have

N

∑
1

h

()

x

=

a H

()

ω

m

nm n

n

=

for each
m
= 1, …,
N
. Putting in what
H
m
and
h
m
are, we obtain the system of

equations

N

∑
1

px

()exp(

ix

ω

)

=

aP

()

x

(

ω

−

ω

)

m

n

n

m

n

=

for
m
= 1, …,
N
.

Let
D
be the square matrix with entries
P
(ω
n
− ω
m
), let
a
(
x
) be the column vec-

tor with entries
a
n
(
x
), and let
e
(
x
) be the column vector with entries exp(
ix
ω
m
).

We can then write the previous equation as
p
(
x
)
e
(
x
) =
Da
(
x
). Solving for
a
(
x
),

we get

ax

()

=

pxDex

()

()

−1

We now apply these coefficients to the original Fourier transform estima-

tion problem. Let
d
be the column vector with entries
F
(ω
n
). Our estimate of

f
(
x
) is then

ˆ
()

fx

=

ax dpxexDd

()

=

()()

T

TT

−

which is the PDFT.

Morphing the prior

The resolution of the PDFT estimate systematically improves as the prior
p
(
r
)

improves. When no assumptions are warranted regarding the prior function,

one can then choose a generic prior and then reduce its size and shape and

monitor the energy of the associated PDFT estimate. It can be shown that as

the prior shrinks to a size smaller than the target shape, the energy of the PDFT

estimate dramatically increases. A systematic approach to this “morphing” of

p
(
r
) allows target information to be deduced, either an improved image of the

target or a defining signature shape (Figure C.5). This is illustrated below.

By identifying a region in an image that looks like a likely target, one

can extract this array of pixels (a large enough area around the target which

includes the significant portion of the SAR point spread function), and then

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