IEEE Geoscience and Remote Sensing Magazine - June 2019 - 22

have invoked a variety of models, as described in the "Models for Spectral Variability" section.
Through the exploitation of probabilistic models
(see the "Probabilistic Models" section), GLRT invariance has been widely investigated [100], [113]-[116].
The GLRT has been found to be invariant (in terms of
the condition stated earlier) with respect to a group of
transformations if, upon transformation, the pdfs conditioned to the two hypotheses remain in the same family and the parameter spaces are preserved [100], [114].
Formally, we can write [114]
f!F

* p (x; H) = p ( fx; fH)

& D ( fx) = D (x) ,

(34)

f: X " X

where H is the parameter vector that belongs to the parameter spaces X 0 and X 1 under the null and alternative hypotheses, respectively, and f ! F is the set of transformations with respect to which the GLRT is invariant. A proof is
given in [114]. In [115], the relationships between the GLRT
and the UMP invariant (UMPI) tests are described. Other
approaches to find UMPI tests are illustrated in [113].
Aside from the GLRT and UMPI tests, the most widely
used invariant approaches are invariant in a broader sense,
in that they are robust to variability (i.e., their performances are not as degraded as those of conventional detectors)
while not strictly ensuring the same detector output upon
variation of the target signature. Nonetheless, in the literature they are called invariant methods, and we will adhere to
this nomenclature.
One straightforward way to impose invariance is by
invoking (unconstrained) subspace models (see the "Unconstrained Subspace Models" section). If T is a subspace,
then it can be parameterized by a matrix M, so that t ! T
is equivalent to the existence of a vector of coefficients p
with t = Mp .
One general approach for finding this target subspace is
to obtain (by measurement or simulation or both) a discrete
C
(and ideally large) set T = " t (i) ,i =1 of radiance spectra for a
given target material. This set should span the range of variation over which robustness is desired, and it can include
intrinsic, extrinsic, and/or environmental variability [13],
[35], [36], [45]-[47], [117]. For example, Figure 4 illustrates
spectral variability under a range of atmospheric conditions
by producing such an ensemble of individual spectra.
The idea of using a low-dimensional subspace is that, for
some Q % C , we can find a model characterized by a Q × d
matrix M which spans a subspace that approximates each of
the target spectra in the set, that is,
t (i) .

Q

/ p k(i) m k = Mp (i).

(35)

k =1

From the subspace defined by M, we can invoke, for instance, subspace versions of the AMF or ACE [97], [116]. This
approach has been extensively pursued in the literature [13],
22

[35], [36], [45]-[47], [71], [117], [118]. Here, the target radiance spectra have generally been synthetically generated by
radiative transfer modeling, and variability has been incorporated by varying the environmental, acquisition geometry, and illumination parameters, together with variations
of the BRDF, by exploiting suitable physics-based models
[13] and, in some cases, also exploiting in-scene information [117] or large spectral library databases [3].
The need to identify a low-dimensional linear subspace
poses the perennial challenge of estimating the "optimal"
subspace dimensionality. If the dimension is too small,
then the approximation in (35) becomes poor, but as the
dimension increases, the subspaces can harbor ever larger
numbers of false alarms [71]. Another general drawback of
this approach is that, regardless of its dimensionality, the
size of the subspace spanned by the basis vectors is not con(i)
strained because the coefficients p k can take any (even unphysical) values [119].
One way to deal with subspace high-dimensionality issues is to constrain the subspace by imposing conditions on
(i)
the coefficients p k , as, for instance, is done with constrained
subspace or simplex modeling (see the "Constrained Subspace Models" section). Various approaches have been explored for simplex-based variable target detection [71], [119],
[120], e.g., by enforcing only the additivity constraint [71],
enforcing both constraints [119], or performing simultaneous linear unmixing on the target and background spectra
together to obtain physical abundances [120]. This has led
to methods such as simplex ACE or simplex AMF [71], which
have exhibited detection performances that, in contrast to
their unconstrained subspace-based counterparts, are stable
with respect to increasing amounts of spectral variability
and increasing sizes of the target subspace.
More recent approaches to building invariant algorithms resort to nonlinear models, such as manifolds (see
the "Topological Manifolds" section). As previously noted, not all variability effects on spectra can be adequately
captured using low-dimensional linear models, but the
variable target spectra often "live" in a lower-dimensional nonlinear manifold. In [89], the target radiance specC
tra " t (i) ,i =1 are assumed to lie within a manifold, and a
graph-based manifold learning approach is used to perform invariant target detection. Resorting to nonlinear
manifolds is often necessary in underwater hyperspectral
remote sensing as well, where water spectra change nonlinearly upon variations of depth and water-inherent optical properties [81]. Nonlinear dimensionality reduction
by manifold learning has been successfully exploited in
several hyperspectral underwater remote sensing applications, such as underwater object detection [81] and hyperspectral bathymetry [90].
One challenge with using radiative transfer modeling
to create a very large set of variable target spectra is that
the computation can be prohibitive. It is worth mentioning
that some "emulation" approaches have been proposed, in
which a reduced number of available variable target spectra
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

JUNE 2019
IEEE Geoscience and Remote Sensing Magazine - June 2019

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