Image Processing Reference

In-Depth Information

of incorporating specific object characteristics into the choice of prior infor-

mation. It is immediately obvious that the PDFT algorithm introduces addi-

tional information through the choice of prior and the
P
-matrix. These are

composed of the weighting function
p
(
x
) and the regularization parameter, ε,

the latter in effect improving the estimation process by accounting for signal

noise. The trade-off between the amount of data and the use of prior informa-

tion can be further emphasized if we consider the trivial case |
p
(
x
)|
2
=
V
obj
(
x
)

(i.e., the prior is chosen to be the true object function). The reconstruction

is always perfectly accomplished from a single sample in the Fourier plane.

It is clear that the choice of prior may incorporate object properties to any

degree, and a weighting function
p
(
x
) already consisting of small features

to some degree facilitates the extrapolation of the spectrum in the Fourier

domain. Conversely, we may accomplish an improved regularization by using

a smoothly modulated weighting function.

Such estimators have indeed proved superior for resolving localized object

structures and may be interpreted in terms of having selected a more appro-

priate basis set for the given object being imaged. This is why some
a priori

knowledge of a support constraint for a compact scatterer is so powerful.

Effectively, the system degrees of freedom are put to much better use!

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