IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 58

A Rationale for Backprojection in Spotlight Synthetic Aperture Radar Image Formation
Figure 17.
Centered impulse reconstructed from summing plane waves sampled
in the k-plane and numbering 62 in the radial direction and
1440 in angle.
represent the ground patch. Then, the approximate reconstruction
is
d xu;v ¼
1
4p2
X
n
X
m
ejkm;nxu;v
jjkm Dk Du
(35)
which serves the same conceptual purpose as (32) except
here the discretization is carried to the left-hand side, the
reconstruction plane, simply for expository purposes. The
visualization of this discretization plan is Figure 18.
Figure 17 shows d xu;v
Figure 18.
Discretization parameters for a full-circle polar grid in the
k-plane. Compare to Figure 14.
seen outside this radius in three of four corners of the
figure. If the ground patch image is considered to be the
entire square then the radial sampling interval in the
k-plane must be further diminished by a factor of 1=
p
ffiffiffi
2
the distance in the x-plane from the center to the nearest
edge divided by the distance from the center to a corner.
As a final example consider Figure 20, which uses the
constructed from plane waves
represented by a densely sampled k-plane with k varying
from 0 to 1:2p radians/m (0 to 0.6 cycles/m) in 62 samples
and u varying around the entire circle in 1440 samples,
which is enough to prevent angular aliasing in this example.
Here, the radial sampling rate in the k-plane is the
same as the sampling rate in either direction in Figure 15.
Rather than fully-realized aliased copies appearing periodically,
in this case the aliased energy appears as a ring of
radius 51.25 surrounding the impulse. Other alias rings
appear at multiples of51.25 and are not shown in the figure.
Once again, this undersampling is done intentionally for
illustrative purposes. This particular form of aliasing is
analyzed in terms of Hankel transforms and Bessel functions
in [44].
Figure 19 shows a reconstruction of four impulses
located in the x-plane at 0; 0ðÞ, 40; 10 ðÞ, and
20;30
ðÞ,35; 15
ðÞmade according to (31) and with a discretization
similar to (35). This is an example in showing the utility
of the off-center field and receiver functions such as (7)
and (10) and indicates how a general ground patch can be
imaged. Relative to Figure 17, the radial sampling interval
in the k-plane was roughly halved by increasing the number
of samples to 124 over the same range of 0 to 1:2p;
this is enough to prevent aliased energy from appearing
within a circle with a radius of about 51.25 as discussed in
connection with Figure 17, even for reflectors or impulses
placed at the edge of that circle. Aliasing artifacts can be
58
same scatterer positions as Figure 19 but restricts the bandwidth
of the k-plane support to something somewhat more
radar-like while keeping the same k-plane sampling density.
The angular variation in k is limited to6 centered on the
kx axis while the radial variation is limited to 4:8625; 6½,a
21% bandwidth. This bandwidth limitation is in the manner
ofFigure 16, but with different limits than shown there. For
convenience here, the radial bandwidth limitation has been
affected by a notional Pk
ðÞ from the section " Projections
and Slices " that perfectly limits the radial frequency range.
Both ofthese limited ranges are perhaps double that ofwhat
might normally be considered a wideband radar. Since the
k-plane support is roughly square, the reconstructed
impulses show a significant sincðÞsincðÞ character; the
radial and angular limitations were chosen to present a
roughly square support of the main lobes. A 2-D window
can be applied to the frequency domain data before the
image is calculated in order to reduce the sidelobe level but
with a widening of the main lobes. It should be noted that
the choice of image size to be constrained to50 in each
dimension is arbitrary and an actual ground patch image
would be larger so that the reconstructed impulses shown
here would occupy a smaller portion ofthe total image.
BACKPROJECTION GENERALLY
We have informally developed a method to reconstruct
a ground patch consisting of a single impulse, claimed
that ground patches of multiple impulses can be reconstructed
using the same method, and claimed further
that any ground patch, even a continuous one, can also
be so reconstructed. We have shown numerically
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IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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