IEEE Geoscience and Remote Sensing Magazine - June 2019 - 19

there is no reason to enforce the sum-to-one constraint. In
this case, the constraining geometry becomes an uncapped
simplex (i.e., a polyhedral cone with K edges) [71].
With constraints imposed, it is not strictly necessary that the target subspace be of lower dimension than
the ambient dimension of the data. The simple geometric approach adopted by Yang et al. [75] defines a ball
B (t o, f) = # t t - t o # f - centered on a nominal target
spectrum t O and having a radius f corresponding to the
variability of the target. The target spectrum is assumed to
lie within the ball, and, when a detection algorithm is derived, the optimization is constrained by t ! B.
A different way to constrain a subspace model is by imposing a sparsity constraint on the coefficients. The model
becomes a union of lower-dimensional subspaces and has
been used to produce target-detection algorithms (e.g., [76]).
TOPOLOGICAL MANIFOLDS
Although the variability of material spectra can sometimes
be modeled well by low-dimensional linear subspaces, the
inherent nonlinearities of many physical processes that lead
to this variability (e.g., those discussed in the "Sources of
Target Spectral Variability" section) suggest that a more suitable model might be a "curved subspace" or nonlinear manifold [77], [78]. Manifolds can be linear or nonlinear, and
manifold learning refers to approaches that attempt to recover
(i.e., "learn") a lower-dimensional manifold assumed to be
embedded in a higher-dimensional space [79]. Although hyperspectral data are more commonly modeled with linear
manifolds (i.e., constrained and unconstrained subspaces),
some hyperspectral data sets have been shown to be more
effectively fit with nonlinear manifolds [80], [81].
Within this framework, nonlinear manifold learning attempts to derive a coordinate system that parameterizes the
manifold by, in the words of Bachmann et al. [77], "following its intricate and convoluted structure with the hope of
achieving a better data representation." In nonlinear manifold learning, the concept of linear distance is replaced by
that of geodesic distance [82], which corresponds to the
length of the shortest path on the manifold between two data
points. In practice, this means that the manifold coordinate
system resides on the manifold itself, so that the distances
are measured by following the curves of the manifold trajectory; thus, any geodesic distance along the manifold turns
out to be a simple linear distance in the manifold coordinate
system [77]. Formally, an m-dimensional manifold M is defined by stating that, for each point y ! M, there is an open
subset (often called a neighborhood) S 1 M, with y ! S, that
is homeomorphic to an open subset (or neighborhood) Sl
g
in an m-dimensional Euclidean space R m; that is, S $ Sl,
where g is the homeomorphic mapping function. Thus, even
though a manifold might have a complex nonlinear global
structure, locally it looks like Euclidean space [77].
Well-known manifold-learning algorithms include kernel principal component analysis [83], isometric mapping
[84], locally linear embedding [85], [86], and Laplacian
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IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

eigenmaps [87]. They have been applied to a variety of hyperspectral image exploitation tasks [77], [80], [81], [88]-
[95], but manifold learning in hyperspectral imaging is still
a growing research topic [78].

MANAGING SPECTRAL VARIABILITY
IN HYPERSPECTRAL ANALYSIS
Stay on target.
-Gold Five
Spectral variability takes many forms, and there have
been many approaches for dealing with this variability
in hyperspectral data analysis. We divide those strategies
into two categories. The first, which we explore in the
"Developing Algorithms That Are Robust to Spectral Variability" section, is to design algorithms that are robust to
this variability. In this section, we emphasize algorithms
for target detection, but we note that classification, segmentation, and unmixing tasks also require attention to
spectral variability of materials. The second kind of strategy is to preprocess the data to reduce (in some cases, to
"project out") the variability due to factors that are not
part of our analysis. In the "Mitigating Spectral Variability in Hyperspectral Data" section, we describe both
physics-based approaches, such as atmospheric compensation (AC), and data-driven approaches, such as in-scene
target characterization.
Modeling the spectral variability of materials is particularly important for target detection. Much of the target-detection literature has been focused on how to incorporate variability of the background into the detection
algorithms [61], but the variability of the target material
must be accounted for as well. Even though variability effects for the target material may be milder than those of
the background, if not properly accounted for, they will
unavoidably lead to performance degradation. One reason
target variability is more difficult to model is the scarceness of labeled training samples, which often consist of a
single library spectrum for each material. When, instead, a
greater number of labeled training samples is available (as
in classification), these may be representative, albeit partially, of the spectral variability to be experienced in the
scene. Another difficulty with target detection is the need
to transform between the reflectance and radiance domains (spectral libraries are in reflectance; remote sensing
measurements are in radiance), and this leads to a host of
environmental sources (e.g., atmospheric absorption and
scattering, angle of illumination, clouds, shadows, and so
on) of variability.
DEVELOPING ALGORITHMS THAT ARE
ROBUST TO SPECTRAL VARIABILITY
The most mathematically straightforward way to express
variability is as an explicit probability density function, and
the basic organizing principle for obtaining target detectors is the likelihood ratio test (LRT) [96], which optimally
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IEEE Geoscience and Remote Sensing Magazine - June 2019

Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - June 2019