IEEE Geoscience and Remote Sensing Magazine - June 2020 - 15

maintaining a modest computational cost and benefiting
from the intrinsic properties of CS.
In [83], the problem of scatterer detection has been addressed by discriminating between single and double scatterers on the basis of a Generalized Likelihood Ratio Test
(GLRT). This approach can be viewed as a tomographic
reconstruction technique, imposing the constraint that, at
most, two scatterers can be present in the same range/azimuth pixel. It provides a noncontinuous 3D reconstruction,
in the form of a point cloud, because only reliable reconstructed scatterers, according to a constant false alarm rate
(CFAR) criterion (for a given P FA level), are selected.
This approach was successively extended to a number of
scatterers K M larger than two and modified by introducing
a different sequential test [84], which improved detection
performance when the distance between the scatterers was
below the Rayleigh resolution criterion (superresolution
capability). In [84], the proposed GLRT is searching for the
support of the signal with the minimum , 0 pseudonorm
that best matches the data (Sup-GLRT). In other words, it
exploits the solution of (13) in the statistical test applied to
filter out false alarms. This method can exhibit a high numerical complexity for values of K M larger than two and for
high dimensions of the unknown vectors. A fast approximate version has been proposed in [85] and extended to
polarimetric data in [43].
Assuming that, at most, K M scatterers are present in the
same range/azimuth-resolution cell, the detection problem can
be formulated in terms of K M + 1 statistical hypotheses [84]:
H i: presence of i scatterers, with i = 0, f, K M .
The signal model in each hypothesis is
Z e
]
] Au + e
]]
...
v=[
] Au + e
] ...
] Au + e
\

under H 0
under H 1, with u 1-sparse
under H i, with u

i-sparse

,

(15)

under H K M, with u K M-sparse

where the dependence of A (h) on h has been dropped for
the sake of notation simplicity.
As far as the statistical model of (15) is concerned, the
noise vector e can be assumed to be a circularly symmetric
complex (or proper complex) Gaussian vector, with uncorrelated samples and mutually uncorrelated real and imaginary parts, having zero-mean and same variance. Then, in
each hypothesis H i, v is a circularly symmetric Gaussian
random vector with the same covariance matrix as e and a
mean Au, where u is an i-sparse vector.
The Fast-Support (Fast-Sup)-GLRT detector, depicted in
Figure 2, is a multistep detector derived for sequentially determining whether an additional scatterer is present or not
in the considered resolution cell and, if present, estimating
its intensity and elevation. At each step i, it applies the following binary hypothesis test, obtained by an approximation of the GLRT associated with (15) [85]:
JUNE 2020

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

K i (v) =

7v @ P X=t i - 1 vA Hi - 1
T Ti,
7v @ P X=t K vA HK $ i

(16)

M

t i - 1 is the estimated support of cardinality i - 1
where X
(i.e., the positions of the i - 1 elements different from zero
-1
=
in the vector u) and P Xt i - 1 = I N - A Xi - 1 ^ A @Xi - 1 A Xi - 1 h A @Xi - 1 is
a projection on the subspace orthogonal to A Xi - 1 (i.e., orthogonal to the subspace spanned by the steering vectors
associated with the i - 1 first scatterers). A Xi - 1 is a matrix
of size N # (i - 1) obtained by extracting from A only the
columns a l 1, f, a l i - 1, whose indices correspond to the sig=
nal support, and P Xt 0 = I N . The thresholds Ti at each step
i of the detection test can be derived by following a CFAR
approach, consisting of setting Ti in such a way that, at each
step i, an assigned PFA = R Kl =Mi P (H l /H 0) and false detection
PFDi = R Kl =Mi P (H l /H i - 1) with i = 2, ..., K M are obtained. All
of the thresholds can be numerically evaluated through a
Monte Carlo simulation.
If, for instance, K M = 2, the following procedure is
applied.
t 1 and X
t 2 are estimated,
◗ For each range/azimuth cell, X
@ =
t
minimizing v P Xi v with i = 1, 2 by means of iterative 1D
minimizations.
◗ For each range/azimuth cell, the two-step GLRT (16) is
applied, and it is decided which of the three hypotheses
is verified. Then, the elevation ht of one scatterer, in the
case of H 1, or the elevations of two scatterers, in case of
t 2.
t 1 and X
H 2, are determined based on X
t = A Xt i
◗ Finally, the reflectivity can be estimated by u
-1
_ A @Xt i A Xt i i A @Xt i v, with i = 1, 2 respectively, when either
H 1 or H 2 is chosen.
The SSF approach, presented in the "Weighted Subspace
Fitting" section, uses the same type of orthogonal projector
but involves the eigenvectors of the sample covariance matrix
of the data, corresponding to the D largest eigenvalues [see
(10)]. It requires estimation of the covariance matrix (hence,
some form of multilooking), which differs from the detection
approach. Moreover, SSF requires knowledge of the number
of sources D, whereas, in the Fast-Sup-GLRT approach, only
the maximum number of sources is fixed, and the effective
number of scatterers is locally identified by sequentially comparing the GRLT (16) to a threshold at each step.
The Fast-Sup-GLRT was applied to the Paris data set described in "Data and Results." The point clouds obtained
are shown in "Tomographic Reconstructions of Buildings
in Paris," Figure S9(a) and (b). The detected scatterers appear correctly located on the buildings and the ground,
thus confirming the reliability of the technique. The detector was able to identify the contributions of façades,
ground, and the roofs of the buildings, even when two of
them are interfering within the same resolution cell [see
Figure S9(b) in "Tomographic Reconstructions of Buildings in Paris," where the double scatterers are shown in
red and blue]. The representation of the surface interpolating the height position of the detected point cloud is
shown in Figure S9(c). Although the height reconstruction
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