IEEE Geoscience and Remote Sensing Magazine - June 2020 - 13

Nonlocal (NL) approaches, such as the NL-SAR algorithm [58], weight the samples v , based on patch similarity,
depending on their statistics. Covariance-based tomographic techniques have proven to be very effective for the tomographic reconstruction of forested areas [59]. In dense urban
scenarios, however, identifying enough samples that could
be safely combined to estimate the covariance matrix without mixing pixels with different reflectivity profiles becomes
difficult when the number of images increases. (The vertical
resolution improves so that differences grow due to the curse
of dimensionality [60].) A possible workaround is to independently estimate the elements of the covariance matrix R
based on NL filtering of the interferometric pairs [61].
CONVENTIONAL BEAMFORMING
Single-look (conventional) beamforming [34], [62] is the
simplest estimator that can be used to retrieve the reflectivity for a given elevation h. It is given by the matching filter, which maximizes the SNR under additive white noise.
The backscattering profile can then be reconstructed by
performing the following estimation for all of the components in h:
ut (h) = a @(h) v.

(5)

If the baselines are uniformly sampled, this corresponds to
computing the inverse Fourier transform of the vector of
backscattered values v depicted in Figure 1. Although fast to
compute, this approach generally produces high sidelobes
and spurious outliers, depending on the data quality and spatial configuration of the baselines. The power spectral density
of the signal obtained through beamforming is more robust
to noise and generally preferred for analyzing reflectivity:
L

1
t a(h).
pt (h) = L / | a @(h) v ,|2 = a @(h) R
,=1

(6)

With this estimator, L samples are combined to comt [corresponding to the case w , = 1 in
pute the covariance R
(4)]. The reduction of speckle fluctuations obtained by the
spatial multilooking comes at the expense of a loss of resolution along the azimuth and slant-range directions. Singlelook and multilook beamforming resolutions dh are limited by the largest available baseline.
CAPON BEAMFORMING
Capon beamforming [51], [63], [64] is not derived from a
priori knowledge of the noise distribution but, rather, by
attenuating the power from all heights other than h under
a unit-gain constraint. The Capon beamforming estimate
of the signal is
t -1 a(h))-1.
pt (h) = (a @(h) R

(7)

Decorrelation phenomena with large temporal baselines
often induce variations of reflectivity over the acquisitions. This is particularly observed with volume or surface
JUNE 2020

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

scattering. Compared to other tomographic methods, Capon
beamforming generally performs favorably when decorrelated scatterers are encountered.
Capon beamforming has empirically exhibited better
performance than conventional beamforming for target
unmixing and sidelobe suppression [29], [63]. It is regarded as a high-resolution estimator, but its practical efficiency is variable.
MULTIPLE SIGNAL CLASSIFICATION
The multiple signal classification (MUSIC) estimator [29],
[51], [63], [65] is designed to retrieve the localization of a
finite number of sources D from a mixture corrupted by
additive white noise. The scatterer localization is obtained
through analysis of subspaces of the covariance matrix.
Because only a few scatterers dominate the backscattered
signal in a given resolution cell (D is much less than the
number of images N in the stack), the corresponding eigenvectors span a subspace of small dimension, and its
orthogonal complement, E n, of dimension N -D, contains
only noise. (E n is generally referred to as the noise subspace.) The height h of each scatterer can be identified by
measuring how orthogonal the steering vector a(h) is to
the subspace E n associated with the N -D smallest eigent
values of R:
pt (h) = (a @(h) E n E @n a(h))-1.

(8)

The scatterer position h in the cell is then obtained by identifying the maxima in pt (h), the MUSIC pseudospectra.
The MUSIC algorithm requires that the number of sources D be set. Because MUSIC is based on an analysis of the
t , some form of spatial averaging is percovariance matrix R
formed in a preprocessing step. In [66], a knowledge-based
baseline interpolation MUSIC method that does not require
multilooking is proposed. Capon and MUSIC methods are
extended to polarimetric data for 3D imaging in [41].
WEIGHTED SUBSPACE FITTING
Weighted subspace fitting methods [67]-[70] are very close
to MUSIC. Rather than giving all eigenvectors in E n the
same influence, they introduce weights based on the eigenvalues. In the same spirit as MUSIC, the noise subspace
fitting aims at minimizing the weighted projection of the
steering matrix on the noise subspace:
ht = argmin < E n A(h)


IEEE Geoscience and Remote Sensing Magazine - June 2020
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