Prior Discrete Fourier Transform - Introduction to Imaging from Scattered Fields - page 208

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