Prior Discrete Fourier Transform - Introduction to Imaging from Scattered Fields - page 204

depending on the problem we are facing. The Fourier transform of f is F (ω), the
function of the continuous real variable ω in R given by

∫

F () ω = f x ()exp( − ixx ω ) d

where the integral is over the real space R . F here denotes the far field aris-
ing from the domain D enclosing V . Let our data be many finite values of the
Fourier transform of f ; that is, let the data be

∫

dF = () ω = fx ( exp( ixx ω ), dfor n =… 12 ,, , N

n n n

where ω n 's ( n = 1,2, …, N ) are N arbitrary points in R . Let p ( x ) ≥ 0 be a prior
estimate of the profile of the function f ( x ) to be reconstructed. Let H con-
sist of all linear combinations of functions of the form h m ( x ) = p ( x )exp( i ω m x ),
m = 1,2,…, N . We then have

∫ ∫

dhx = ()exp( − i ω xx ) d = hx ( )exp( − i ( ω − ω )d x ) x

n m m m n

Therefore, d mn = P (ω n − ω m ) for each m and n , where P is the Fourier
transform of p . The vector b ( x ) has entries p ( x )exp( i ω m x ). With
e ( x ) = (exp( i ω 1 x ) … exp( i ω N x )) T we have b ( x ) = p ( x ) e ( x ). Then the PDFT esti-
mate of f ( x ) is

ˆ ()

fx = ax dbxDdpxexc () = () = ()()

T T − T T

where c = D − T d .
Two important observations can now be made. The PDFT estimator is easily
regularized in the Tikhonov-Miller sense when data are noisy. Secondly, the
prior constraints employed can be very general. For example, one could use
the radar beam pattern as prior knowledge of where the scattering arises from,
if one is processing the radar returns directly. If an image has been formed,
for example, as a SAR image, then one can apply a window around the region
of interest and improve the resolution within that subdomain. The effective-
ness of this depends on the proportion of energy in the cluttered background
to that in the vicinity of the target. A smaller window resolves this problem
and actually assists with the resolution enhancement step, as is clear from the
theoretical section above. An example is shown here of real data inversion
imaging a calibration sphere and a smaller plastic landmine (Figure C.3).
We note that a tomographic reconstruction can alleviate problems of the
strong return, which comes from the surface of the ground, a particular prob-
lem for mine and bunker detection. The ease with which this can be done
depends on the point spread function or side lobes arising from the limited
data. The PDFT can improve the point spread function, but it can also assist
with this problem before the image formation step. Through appropriate
k -space gating one can remove the data most strongly associated with a sur-
face reflection prior to using the PDFT, as shown in Figure C.4.