IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 59

Bauck
Figure 19.
Four impulses reconstructed from summing plane waves sampled
in the k-plane and numbering 124 in the radial direction and 1440
in angle.
derived graphical reconstructions using discretized versions
of the method. One purpose of this approach has
been to show that a backprojection is a plane wave
and to relate that idea to Fourier synthesis. We now
formalize this method a little.
Beginning with the 2-D Fourier transform of the
ground patch reflectivity function then converting to polar
coordinates in the wavenumber plane,
g xðÞ ¼
¼
4p2
1
¼
4p2
1
¼
4p2
1
4p2
1
Z
G kðÞejkxdk
Zp Z1
Zp Z1
Zp Z1
1
Gkcos u;k sin uðÞ
ejk x cos uþy sin uðÞkjjdk du
jjk dk du
Gk^x0ðÞejkx^x0
1
G0 kðÞejkx0
1
and
the last line following from the Projection Slice Theorem
(28) and that x ^x0 ¼ x0 from (4). Let F1 kjj
Then, the inner integral is
F1 G0 kðÞ kjj
Finally,
g xðÞ ¼
1
2p 0
Zp
g0 x0ðÞdu
(38)
which is a general form of the reconstruction ofprojections
ofg xðÞ.Notice that qx0ðÞ is not a function ofu, unlike g0 x0ðÞ,
and must be computed only once. Also, kjj is a generalized
function [36] and must be approximated in applications.
One method is to place a lowpass window over kjj,which
also tends to alleviate the emphasis ofhigh frequency noise.
Equation (38) is known as Radon's inversion formula. It
is interpreted as filtering-convolving-each projection
gx0ðÞ with a filter with frequency response kjj and then
MAY 2022
fg¼ g0 x0ðÞ qx0ðÞg0 x0ðÞ:
fg ¼ qx0ðÞ.
(37)
R0
x kðÞ ¼ G0 2kðÞPk
ðÞ kjj:
Here, the convolution with p 2x0ðÞ enters as being caused
by the physics of the propagating radar signal and convolution
with qx0ðÞ follows from the fact that the radar is
designed to sample the k-domain on a polar grid. This
modified receiver signal will not reconstruct g kðÞ exactly
because of the distorting influence of p 2x0ðÞ and Pk
ðÞ,so
define a new approximate reconstruction
g~ xðÞ ¼
1
Pk
Z
r0
2p Q
x x0ðÞdu:
(39)
ðÞ is normally made from a low-pass baseband signal
modulating a sinusoidal carrier and thus it effectively filters
G0 2kðÞ kjj into a bandpass signal centered on the carrier frequency.
The range of integration over u has been indicated
so far as 0; p½Þ because the radial integration limits are
1;1
½Þ and also because of the symmetry g0 u;x0ðÞ¼
IEEE A&E SYSTEMS MAGAZINE
59
jjk dk du
Figure 20.
Four impulses reconstructed from summing plane waves sampled
from a restricted region in the k plane and numbering 24 in the
radial direction and 25 in angle.
u-backprojecting each filtered projection and summing all
u-backprojections. This is the source of the commonly used
name, convolution-backprojection. The convolution indicated
by (37), in a discretized version, might be more efficiently
implemented using the Fast Fourier Transform in the
k-domain. Regardless of how the filtering is done, the
method is considered a spatial-domain algorithm, or sometimes,
rather oddly, it is called a time-domain algorithm.
The above discussion is again devoid of any radar
aspects but is appropriate for computerized tomography
(CT) applications. To cure this, notice that the only information
available to the radar is r0
(17). All we can do, apparently, is to backproject r0
(36)
adapted for polar coordinates-polar format:
r0
x x0ðÞg0 x0ðÞ p 2x0ðÞ
¼ g0 x0ðÞ p 2x0ðÞ qx0ðÞ
x x0ðÞ ¼ g0 x0ðÞ p 2x0ðÞ,
x x0ðÞ
IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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