Prior Discrete Fourier Transform - Introduction to Imaging from Scattered Fields - page 203

k y

(a) (b)

k x

(c) (d)

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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 ,