快速分块反投影法实现合成孔径雷达图像处理
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"《利用快速分解背投影的合成孔径雷达处理》(Synthetic-Aperture Radar Processing Using Fast Factorized Back-Projection)是LARSM.H.ULANDER等作者于2003年发表的一篇经典论文，着重探讨了如何在雷达成像领域利用高效算法改进传统方法。论文的核心内容集中在解决合成孔径雷达(SAR)图像重建中的计算效率问题。
传统的合成孔径雷达成像通常通过反投影（back-projection）技术实现，但这种方法在处理具有复杂几何形状的孔径时，计算负担十分繁重，限制了其广泛应用。该论文提出了一种创新的方法，即通过快速变换技术（fast transform techniques），结合孔径分解（aperture factorization），将反投影积分进行递归分割，并据此设计出一种高效算法。这种方法的优势在于能够显著减少计算操作量，使其接近快速变换算法的性能。
作者特别关注的是应用在空中超宽带(CARABASSAR)雷达的数据处理上，结果显示，新算法能将处理时间降低一个到两个数量级，这在当时的雷达成像技术中是一项重大突破。整个研究过程经历了从最初的提交（2000年12月13日）到修订（2002年6月27日和11月19日），最终在2003年3月13日发布，展现了科研人员对提高雷达图像处理效率的深入理解和持续探索。
这篇论文不仅提供了理论上的优化策略，而且展示了在实际应用中的明显优势，对于推进合成孔径雷达技术的发展，尤其是在实时和高分辨率成像方面，具有重要的指导意义。"
(3) for the radar echo data as the starting point for
deriving the filtered back-projection inversion. The
fundamental principle is to generate a radar map of
two cylindrical coordinates, i.e. a map where each
object is located at its 2-parametric position (
x
,
½
). The
radar map is assumed to be a linear transformation
from the radar echo data so that superposition of
imaged point objects applies. We introduce the
back-projected signal
h
(
x
,
½
) according to
h
(
x
,
½
)=
=
+
g
(
x
,
R
)
Rdx
(4)
where
g
(
x
,
R
) is the radar echo data as a function
of along-track position
x
and range
R
,
R
=
S
(
x
x
)
2
+
½
2
,and
x
is the along-track integration
parameter. F or each image position (
x
,
½
), the
along-track integral sums the value from each radar
echo at the range corresponding to the distance
between the antenna and image position. In practise,
the radar echo data is a discrete-time signal which has
been sampled from the original continuous-time signal
according to the Nyquist criteria. The bandwidth
in the radial direction is determined by the pulse
bandwidth, whereas the bandwidth in the along-track
direction is determined by the maximum frequency
(see Fig. 1(a) and (1)). It follows that the radar echo
can be interpolated to arbitrarily accuracy using
an appropriate interpolator kernel, e.g. a weighted
sinc interpolator. Therefore, the integral (4) can
be evaluated for an arbitrary image position by
interpolating the discrete-time radar echo data.
We can also consider the integral (4) from the
back-projection perspective, i.e. each interpolated
radar echo is spread out over the imaged ground like
a sun fan. The back-projected data will thus have
constant values along concentric spheres, or circles in
the 2-dimensional case, centered on the antenna phase
center.
Evaluation of the Fourier transform of (4) is
straightforward and results in
H
(
k
x
,
k
½
)=2
¼
=
+
=
+
0
g
(
x
,
R
)
R
2
J
o
R
T
k
2
x
+
k
2
½
exp(
jk
x
x
)
dR dx:
(5)
Inserting the radar echo model (3) and assuming that
the pulse has infinite bandwidth, i.e. the pulse is a
Dirac delta-function
p
(
R
)=
±
(
R
), results in
H
(
k
x
,
k
½
)=4
¼
exp(
jk
x
x
o
)
=
+
0
J
o
T
k
2
x
+
k
2
½
S
x
2
+
½
2
o
cos(
k
x
x
)
dx:
(6)
Equation (6) is a standard cosine transform which
evaluates to
H
(
k
x
,
k
½
)=2
¼
exp[
j
(
k
x
x
o
+
k
½
½
o
)] + exp[
j
(
k
x
x
o
k
½
½
o
)]
M
k
½
M
:
(7)
After rearranging the ramp filter in the denominator to
the left-hand side and inverse Fourier transformation
we formally obtain the exact inversion result
F
1
IM
k
½
M
H
(
k
x
,
k
½
)
J
=2
¼
[
±
(2)
(
x
x
o
,
½
½
o
)+
±
(2)
(
x
x
o
,
½
+
½
o
)]
(8)
where
±
(2)
(
x
,
½
)=
±
(
x
)
±
(
½
) is the two-dimensional
Dirac-function. The back-projected and filtered
radar echo data thus produce two Dirac-functions,
one located at (
x
o
,
½
o
) and the other at (
x
o
,
½
o
).
The latter is discarded since
½>
0 by definition.
Since superposition applies we may add an arbitrary
number of point objects at different locations and with
different amplitudes, and (8) will provide us the exact
scene inversion.
In theory, the linear-aperture SAR problem can
therefore be solved exactly and unambiguously by
assuming that the Born approximation is valid. In
practise, however, the ground is three-dimensional
whereas the linear-aperture SAR is only capable
of reconstructing the two-dimensional cylindrical
projection of the scattering distribution, i.e. the
ground represented in cylindrical coordinates with
the two-dimensional distribution defined in terms of
x
and
½
. This means that the full three-dimensional
ground cannot be retrieved without ambiguities. Given
a topographic ground surface, ambiguities appear due
to shadowing, layover as well as left/right-ambiguity.
The latter are ambiguities arising from the ground on
the opposite side of the linear track. Such ambiguities
may be adequately reduced by introducing antenna
directivity to suppress the unwanted ambiguous
signals as mentioned earlier. The shadowing and
layover ambiguities, ho we ver, are not possible to
avoid but can be mitigated by using multiple flight
tracks which provide different illumination geometry.
B. Extension to Nonlinear Track
The back-projection integral is easily modified to
handle a general flight track geometry. A complication
for nonlinear tracks is that data inversion becomes a
3-parameter problem which strictly speaking would
require a two-dimensional aperture to solve exactly.
However, in most cases only a one-dimensional
aperture is available and the focusing becomes
dependent on ground topography. The back-projection
integral is therefore expressed as a function of the
three-dimensional image position vector
¯
r
o
according
to
h
(
¯
r
o
)=
=
g
(
x
,
R
)
Rdx
(9)
where
R
=
R
(
x
)=
M
¯
r
(
x
)
¯
r
o
M
is the range from
the antenna to the image position, and the other
parameters are defined as in (4). Equation (9) is able
to handle any track geometry including, for example,
ULANDER ET AL.: SYNTHETIC-APERTURE RADAR PROCESSING USING FAST FACTORIZED BACK-PROJECTION 763
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