IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 60

A Rationale for Backprojection in Spotlight Synthetic Aperture Radar Image Formation
g0 u þ p;x0ðÞ. Obviously, any contiguous span of distance
p would work mathematically, and for the radar geometry
presented earlier, a range ofp=2; p=2½Þmakes more sense.
Above in (39), the notation Q has been used to indicate any
support subset as necessarily defined by the operation of the
radar but which support is normally a contiguous subset
ranging from some umin to some other umax. And actually,
to be respectful of the radar scenario and recognizing that
only a discrete set ofpossibly sparse projections will be measured,
it is possible to collect the discrete projections from
½Þ0; 2p without any duplicates being measured; a discretized
version of(39) would allow for this possibility and establish
afinite Du for each projection, with care taken to account for
any situation where projections are measured from both u
and u þ p. Together, the combined actions of Pk
ðÞ and Q
act to cause spotlight SAR to be considered as a kind of
bandpass version ofCT ala Figure 16 with exemplar reconstruction
Figure 20 but with the additional difference that
SAR is coherent-it measures phase and thus collects complex-valued
data-while CT is incoherent, as noted in [1].
COMPUTATIONAL CONSIDERATIONS
We have so far developed an interpretation ofspotlight SAR
imaging as an additive combination ofweighted monochromatic
plane waves, clarifying the intimate connection
between wave theory and the signal processing concepts of
the Fourier and Radon transforms. This notion is the central
purpose of this article and should be kept firmly in mind.
The images of Figures 15, 17, 19, and 20 were constructed
literally in this way. However, this is a highly inefficient
method, requiring far more calculations than is necessary.
There have been many works written on the details of all
aspects of this topic, some ofwhich are listed in the section
" Introduction, " but here we shall touch only lightly on a few
conceptual matters of importance. In so doing, we shall
highlight the differences and the essential sameness of two
popular classes ofSAR image reconstruction methods, convolution-backprojection
and direct Fourier inversion also
known as the polar format algorithm (PFA).12
We have concentrated on the backprojection of monochromatic
waves as a tool to easily tie together simple
wave mechanics with signal processing, but by now it is
obvious that there is a more efficient way, and we have in
fact already seen it. Consider (36), rewritten here as
g xðÞ ¼
1
4p2
Zp Z1
1
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
12 " Polar format algorithm " seems to imply a single algorithm where
in fact there are many variations. " Direct Fourier inversion " seems
less presumptuous but as we shall see, it too is perhaps not so
unambiguously descriptive.
60
G kðÞejkx kjjdk du:
(40)
Discretizing the above in the manner of (35) helps to clarify
the processing. Allowing for possibly unequal Dks and
Dus-especially useful for the latter in cases where the
moving platform or pulse transmission program or
engagement scenario does not allow for equal angular
increments-allows the approximation ofg xðÞ
g^ xu;v ¼
1
4p2
X
n
X
m
G km;n ejkm;nxu;v
jjkm Dkm
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Dun
(41)
which approximation might also include band limiting in
wavenumber or view angle or both. The backprojection
operation is very expensive and should be done as rarely
as possible. Backprojecting each of the sinusoidal components
of (40) or (41) thus is not a good idea. Instead, consider
the inner integral in (40) or the inner summation
over m in (41) as indicated by the brace in either the continuous
or the discrete case. Backprojection is linear so a
much more efficient method is to sum these components
before backprojecting the sum only once. The backprojected
function is no longer a monochromatic wave but a
polychromatic wave as alluded to earlier. This summation
is a 1-D Fourier synthesis operation. Figure 16 is apropos
here as a mental picture, with the circled group ofk-plane
impulses representing a particular summation over m for
a fixed angular index n. This summation reduces the operation
count from ON4
to ON3ðÞ. We shall not explore
this concept deeply, but roughly, without summing, the
number of operations is number ofu pixels
number ofv
pixels
number of u angles
number of k frequency
points. The summation collapses the final component of
that count. Fast algorithms for backprojection have been
reported, e.g., [46], reducing the operation count to
ON2logNðÞ, but at least some of these algorithms require
approximations, potentially causing compromises in
image quality-Basu and Bresler [46] claimed minimal
degradation. However, the desire to reduce the operation
count is sometimes found in practice to be compelling.
The reconstructions in the image figures herein took
advantage of the fact that the backprojected functions
could be computed exactly at every point u; vðÞ, those
functions being sinusoids. In practice, this luxury is not
available because the filtered projections are known only
on a set of discrete points oriented at angle u and are
derived from the unknown ground patch and thus must be
treated to a 1-D interpolation to each image pixel during
backprojection. This step unavoidably induces errors and
has been itself a focus of several studies to find acceptable
interpolators. This 1-D interpolation adds to the computational
burden.
Switching attention now to direct Fourier inversion,
consider once again Figure 16 but also the rectangular format
rendering ofFigure 14. Again, each dot ofthese figures
represents a weighted impulse in the k-plane, a complexIEEE
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IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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