Resolution and Degrees of Freedom - Introduction to Imaging from Scattered Fields - page 215

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!