IEEE Geoscience and Remote Sensing Magazine - June 2020 - 11

should be performed along a circle. Such refinements are considered in [51] and [52] and could also be included in a straightforward way in the formulation presented in the "Tomographic
Inversion With Spatial Regularization" section.
Because of measurement noise (thermal noise) and
changes in volume seen in the resolution cell (spatial decorrelation due to the change of view angle and temporal decorrelation caused by the evolution of the scene between each
pass), some fluctuations are not described by the geometrical
model of (1). These fluctuations are modeled by describing
the N observations v 1- v n at a given pixel of the SAR images
as a complex-valued random vector v ! C N. If the decorrelation phenomena are negligible, the random vector is generally modeled as an additive perturbation to the geometrical
model given in (1). This perturbation is modeled by a circular complex Gaussian with a covariance proportional to the
identity matrix. (Thermal noise is uncorrelated from one image to another, and it has the same variance.) The discretization of (1) at a collection of N h heights, h = [h 1, f, h Nh]T, then
leads to the so-called deterministic or conditional scatterer
model [39], [49], [50], which can be written, in the absence
of motion (6h, T r (h) = 0), as
v = A(h) $ u + e with e ! C N + CN (v 2n I).

(2)

We discuss how this model is extended to scatterers under
slow motion in the "Extensions of TomoSAR for Urban
Monitoring" section.
The notation CN (/) stands for the circular complex
Gaussian distribution of covariance /, and matrix A (h) is
the sensing matrix, that is, the matrix whose columns are
the steering vectors a(h 1) to a(h Nh), with
a(h i) = [exp(- jp 1 h i) g exp(- jp N h i)]T,

(3)

and where p = [p 1, f, p N]T is the vector of angular frequencies defined by p n = 2rfn = 4rb n / (mR 0) (see "Multidimensional Imaging in Multipass Synthetic Aperture
Radar Tomography").
Because of decorrelation phenomena, the phase and amplitude of each elementary scattering volume may change
from one pass to another. This is captured using the stochastic or unconditional scatterer model [39], [49], [50], [53], [54]
by describing the complex reflectivities of a scatterer i as a
random vector u i distributed according to a complex circular
Gaussian distribution of covariance C i: u i ! C N + CN (C i),
where the diagonal of covariance matrix C i is constant for
well-calibrated SAR images. Each element of the random
vector u i defines the complex reflectivity of the elementary
scattering volume at height h i for a given image.
In the absence of decorrelation, the complex reflectivity remains constant for all images, u i = u i1, and the covariance matrix C i tends to s 2i 11T (where s 2i corresponds to
the power of the ith elementary scatterering volume along
the elevation axis in the radar frame). If speckle is totally
decorrelated from one image to the other, the complex
JUNE 2020

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

reflectivities behave like independent draws from a circular complex Gaussian distribution: C i = s 2i I. Phases are then
unrelated from one image to another, and no tomographic
or interferometric inversion is possible. Stochastic and determinist signal models are illustrated in "Signal Models in
Multipass SAR Tomography."
The noncentral covariance defined by / = E [vv @], where @
is the Hermitian transpose, captures information about the
spatial distribution of the scatterers. Under the deterministic
scatterer model, / = A(h) $ u $ u @ $ A(h)@ + v 2n I ; whereas, under
Nh
the stochastic model, / = / i = 1 s i2 C i 9 (a(h i) a @(h i)) + v 2n I,
where 9 is the element-by-element product (Hadamard
product; see [39]).
The tomographic inversion can be performed using several approaches. Some are based on an analysis of the structure of (an estimate of) the noncentral covariance matrix
/ ; these are described in the "Spectral Analysis" section.
Other approaches perform an inversion of the direct model
given in (1) for each pixel (x, r). If the baselines b n are uniformly distributed, the estimation of the reflectivity distribution along h can be obtained by computing the inverse
discrete Fourier transform of the vector v = [v 1 g v N] ! C N.
The reflectivity profile can then be retrieved with the spatial Rayleigh resolution d h = mR 0 /(2 max n ; b n ;) [23].
When superresolution is sought, that is, when N h is
larger than the number of images N, the inversion problem
becomes ill posed, and some form of regularization is necessary. The "Compressive Sensing" section describes compressed sensing (CS) approaches that regularize with an
, 1 norm to promote a sparse solution. The "Tomographic
Inversion With Spatial Regularization" section presents an
approach that also includes spatial regularization to couple
the inversion at different range and azimuth locations.
A different approach consists of detecting the presence
of scatterers. Application of the hypothesis tests of detection theory, the number of scatterers, and their location
can be retrieved for each resolution cell. This approach is
described in the "Scatterer Extraction" section and is extended to scatterers under slow motion in the "Extensions
of TomoSAR for Urban Monitoring" section.
On the one hand, high-resolution SAR images of urban areas are scarce, so only a few spatial frequencies are
measured; this strongly limits available information on the
height profile. On the other hand, there are generally only
a few dominant scatterers along the elevation direction in
urban areas (typically between one and three). This peculiarity can be exploited to improve the inversion by identifying and locating those few scatterers. Urban areas are
characterized by a high density of dihedral or trihedral objects, which backscatter a stable signal. Moreover, as their
intensity is several orders of magnitude higher than other
smooth or distributed elements, they tend to hide the contributions of the rest of the objects in the scene. The deterministic scatterer model is, then, adopted in these areas
at the price of speckle-affected scatterers. Generally, urban
tomography does not require a radiometric accuracy; this
11
IEEE Geoscience and Remote Sensing Magazine - June 2020

Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - June 2020
