IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 61

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valued sample of G kðÞ representing a magnitude and
phase. As described before, each dot can be converted to a
sinusoid and backprojected, with the rectangular grid suffering
a disadvantage because there is no opportunity to
group impulses along the radial variable k and thus no
opportunity to exploit the Projection Slice Theorem.
However, in either the rectangular case or the polar
case with kjj weighting, such backprojections of individual
sinusoids, summed in the image plane x,represent
a Fourier synthesis, an inverse (discrete) Fourier
transform. But in the rectangular format case there is
the opportunity to employ the inverse Fast Fourier
Transform since the samples lie on a grid, which can
immediately be accommodated into the FFT structure.
This has been an extremely important advantage historically
because of the efficiency of the FFT and reduces
the overall operation count to ON2logNðÞ.
The polar format therefore stands at a disadvantage
but polar-formatted data are the natural format for spotlight
SAR. To compensate, polar-formatted data can be
interpolated to a rectangular grid; once that is done, the
FFT can be used and its efficiency gain enjoyed. This does
not come for free since the 2-D interpolation is itself an
expensive operation. Still, practice has shown that the
combined operation of 2-D interpolation followed by 2-D
inverse FFT, direct Fourier inversion, is still faster on
most computing architectures than convolution-backprojection,
purpose-built, workstation clusters, or parallelprocessor
computers notwithstanding, but this is a complicated
subject and results vary [47], [48] and should be
considered in light of the aforementioned fast backprojection
algorithms.
There is a way to ease the burden of the 2-D interpolation
by placing a slight burden on the transmitter. Each
pulse can be modified from the preceding pulse by subtly
compressing or expanding it in time [44], [49] causing an
inverse amount of expansion or compression in its spectrum.
This can be programmed into the waveform synthesizer
of the transmitter. If this is done on a particular
schedule, a keystone format can be achieved as shown in
Figure 21. This then requires only a 1-D interpolation,
along lines parallel to the ky-axis, before applying the
inverse FFT. A similar effect can be achieved by slightly
altering the analog-to-digital sampling rate [49].
CONVOLUTION-BACKPROJECTION VERSUS DIRECT
FOURIER INVERSION
We see that convolution-backprojection and direct Fourier
inversion both operate on the same principle, that of summing
weighted plane waves-Fourier synthesis as represented
by (22). As such, they are fundamentally the same,
differing mainly in implementation details. Convolutionbackprojection
can work on any distribution ofplane-wave
MAY 2022
Figure 21.
Discretization points in the k-plane for a keystone format
achieved by adjusting the waveform slightly from pulse to pulse,
thus requiring only 1-D interpolation to adapt it for FFT
processing.
samples in the k-plane whatsoever as long as proper weight
is given to each sample's share of surrounding area, the
Jacobian or some similar measure if there is no appropriate
mapping to a uniform sampling grid. It works for example
on Figures 14, 16, 21, or even some haphazard or unorganized
distribution of impulses. It just happens that if the
data are polar-formatted, some computational savings
accrue. Indeed, we see that convolution-backprojection is
nothing but a " slow " inverse Fourier transform.
If the k-plane samples happen to lie on a rectangular
grid with not necessarily equal Dkx and Dky, then the
inverse FFT can be applied with typically, but not always,
great computational savings. If the data do not lie on such
a grid, it is necessary, if a well-focused image is desired,
to resample the data to such a rectangular grid. Some
authors state that this resampling is required in order to
obtain the image but it is required only if the efficiency of
the FFT is desired.
Convolution-backprojection works on polar data by
computing a 1-D inverse (discrete) Fourier transform,
DFT, after weighting by the Jacobian kjj, then doing a
1-D interpolation in the spatial domain x-plane as the
backprojection is performed. Direct Fourier inversion
does a kind of reversal of those steps, interpolating in
the k-plane before doing an inverse 2-D DFT probably
usinganFFT.
Some researchers have reported better images using
one method over the other as though they are fundamentally
different processes, backprojection usually being
considered better. Any differences between a convolutionbackprojection
reconstruction and a direct Fourier inversion
reconstruction are due to numerical choices such as
interpolation, windowing, assignment of Jacobian values
in nontrivial distributions, and use or nonuse of portions
of the polar data that do not get incorporated in the 2-D
interpolation of a direct Fourier inversion. Comparison
studies are reported in [47], [48], [50]-[52]. Some ruminations
on this topic are in [19].
The convolution-backprojection computation can proceed
apace as each pulse reflection is received. Direct
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