IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 62

A Rationale for Backprojection in Spotlight Synthetic Aperture Radar Image Formation
Fourier inversion must wait until the last reflection to be
processed is received and then process all of the data at
once, as a batch, regardless of whether a FFT is used. This
means that there is the potential for convolution-backprojection
to finish sooner and the potential that maximum
computational throughput can be less than direct Fourier
inversion. Details vary by application, of course.
Related to the above, the convolution-backprojection
image can be viewed as it develops, at first in low resolution
and then with increasing resolution as more data are
received and backprojected.
The convolution-backprojection does not have to
have pixels on a rectangular grid (for example, (34)
indicates how a polar image can be developed) or on
any particular pixel arrangement at all; the points
selected in the x-plane can be selected at will, although
the normal method is indeed a regular pixel grid. This
does lead to an advantage, however. If only a portion
of the image contains objects of interest, the remaining
part does not need to be developed, or it can be developed
with lower resolution. Direct Fourier inversion is
limited to the strictures of the FFT-regular rectangular
sampling in both domains.
While we have concentrated on the plane wave
approximation, convolution backprojection can be
directly, easily, and naturally modified to account for any
amount of wavefront curvature in both monostatic and
bistatic radars [27]-[32], [52]. This is accomplished without
the separate focusing steps required of typical direct
Fourier inversion methods, which present a substantial
additional processing burden and often obtain only a partial
correction. In bistatic geometries, wavefront curvature
manifests as elliptical contours of equal time-offlight
and thus elliptical line integrals, for a flat Earth,
requiring elliptical backprojections. A comment in [53]
reads like this: " Among its classical advantages, [backprojection]
focusing accuracy does not depend on the
carrier wavelength, the desired resolution, the scene size
or the imaging configuration. Time-domain image formation
[sic] offers a further advantage particularly useful in
the case of bistatic systems: precise accommodation of
irregular sampling schemes. ...another strong advantage
of [backprojection] with respect to Fourier-domain [sic]
techniques is the possibility of precise range- and azimuth-variant
antenna filtering and weighting. " There is
also this comment in [32]: " In order to evaluate frequency-domain
processors that all have to use some kind
of approximations, a flexible bistatic time-domain processor
was implemented. This processor can be used as a
reference processor because it is based on the matchedfilter
principle and models the SAR geometry exactly
[29, local reference]. Provided that the platform tracks
are known to the order of fractions of the wavelength,
such a processor can be applied universally to any
62
bistatic configuration, even if the platforms move on
curved orbits. " Backprojection methods have been
adapted for arbitrary monostatic and bistatic flight paths
including full-circle, plus propagation attenuation correction,
nonflat topography, and antenna shading [27], [29],
[30], [52], [54], [55]. There is some reason to believe
that convolution-backprojection could be adapted for
near field (nonspherical, nonplanar) applications as well,
as long as the equal time-of-flight paths are known. Other
advantages of backprojection methods include the possibility
to account for quasi-fan beam geometries, a potentially
valuable feature where high-speed platform motion
or long pulses can cause intrapulse distortion [27]. Convolution-backprojection
methods can be implemented on
parallel computing architectures [47], [56], the method
being somewhat of a natural fit.
As the bandwidth of the transmitted signal increases
and as the range of look angles u increases, the interpolation
required for direct Fourier inversion becomes more
complicated as the samples deviate more from nearby
rectangular grid points and the increasing included angle
of the arc fits more awkwardly into a nice FFT rectangle,
whereas reconstruction by backprojection proceeds without
further complications; indeed, Bauck [27] showed
many simulation examples of full-circle spotlight SAR
using an impulsive signal.
Direct Fourier inversion tends to have a reduced
computational advantage as the image size is reduced. It
has been said [46] that the advantage tends to be lost for
image sizes less than 1000
1000, but this is no doubt a
rough guide depending on many factors.
The above claims for advantages of convolution-backprojection
are not meant to imply that direct Fourier inversion
can not be adapted to advantage. For example, CT
scanners can sort discrete line integrals from separate fan
projections into groups of equal u for processing as parallel-beam
CT. Direct Fourier inversion has also been modified
to other spotlight SAR scenarios including correction
for monostatic wavefront curvature using range migration
[14], [57], [58].
As a minisummary of the two methods, consider
that convolution-backprojection can readily handle
arbitrary distributions of wavenumber domain data and
is especially good with polar format data, requires a
Jacobian weighting or at least an appropriate DkDu for
each data point, performs the required summation in
the spatial domain, and can be slow unless special fast
algorithms or hardware are employed. Direct Fourier
inversionrequiresdatatobeonarectangulargrid and
therefore requires 2-D interpolation from arbitrary or
polar data thus starting the inversion process from a
different set of points, performs the summation in the
wavenumber domain, requires no Jacobian correction,
and enjoys the efficiency of the FFT.
IEEE A&E SYSTEMS MAGAZINE
MAY 2022
IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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