Image Processing Reference

In-Depth Information

k
y

(a)

(b)

k
x

(c)

(d)

0.12

140

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

120

0.1

100

0.08

80

60

0.06

40

0.04

20

0.02

0

-2 -1.5

-1 -0.5

0

0.5

1

1.5

-2 -1.5

-1 -0.5

0

0.5

1

1.5

Cross range
x
(m)

Cross range
x
(m)

Figure c.2
Demonstration of the PDFT algorithm from synthetic Fourier transform data. (a) Target (two cylin-

ders in free space), (b) map of data in
k
-space (wide angle ISAR configuration), (c) DFT estimate of target, and (d)

PDFT estimate.

on the signal-to-noise ratio. In other words, the PDFT is capable of providing

the optimum linear image estimate based on a given set of data points, prior

information about the target, and the quality of the data.

the power of the pdFt

From a mathematical perspective, the PDFT is thus a method for estimat-

ing an unknown vector or a function (an object such as a discrete image, for

example) from linear functional information about that vector that by itself is

insufficient to specify uniquely the vector in question. In such cases, it is cru-

cial to use prior information to single out one specific estimate from the many

possibilities. The approach we use with the PDFT is to select
a priori
informa-

tion about the target and incorporate it into the design the PDFT estimate. It

is a method for reconstructing (estimating) an object from many finite linear

functional values (linear “projections,” if you will). The PDFT is designed to

be applied to the underdetermined problem, to combat the degrading effects

of limited data and nonuniqueness of solutions through the inclusion of prior

information about the object to be reconstructed. In simplest terms, the data is

insufficient to determine a unique solution. The significance of this approach

is that it permits us to incorporate prior information about the object to be

recovered. We illustrate that point now using the case of reconstruction for

Fourier transform data.

Suppose now that our object
f
=
f
(
x
) is a function of a continuous real

variable
x
in
R
. We use
f
to denote either
V
u
,
V
, or a target region within
V
,

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