IEEE Geoscience and Remote Sensing Magazine - June 2019 - 18

Tyo et al. [66] evaluated several monovariate distributions
besides Gaussian for modeling both the single-band spectral
radiance and the total radiance (summed across all bands)
extracted from multiple instances of the same object in a
hyperspectral image. They observed that the extracted data
exhibited a skewed distribution and, therefore, used lognormal, gamma, and Weibull pdfs for fitting. Experiments
showed that none of the tested pdfs did a very good job in
accurately modeling the statistical variability of the spectral
signature (with a slight advantage for the log-normal distribution when the total radiance was modeled) and suggested
that other more complex distributions should be tested. In
[67], the univariate beta distribution was found effective at
modeling the spectral variability affecting target material
reflectances at each spectral dimension while incorporating
distributional skew and ensuring that the reflectance values
of the target materials are constrained to a physically realistic
range. To approximate any distribution that the target materials may exhibit, such as the multimodal distributions very
commonly found in real data, a mixture of Gaussian distributions was used in [68]. These models based on beta and a
mixture of Gaussian distributions were specifically used in
unmixing applications in the context of endmember spectral
variability [67], [68]. Simplex-based distributions have also
been suggested for anomaly detection [69].

where t ! R d is the material spectrum, T is the d # K target
subspace matrix, and a ! R K is a vector of coefficients. According to (13), the target material spectrum is expressed as
a linear combination of the subspace basis vectors " t k ,Kk =1
weighted by the elements of the coefficient vector a. If
rank {T} = K, then the basis vectors are linearly independent, and a given t will uniquely define the coefficient a k.
The affine model [72] is a useful extension to the subspace model that is obtained by adding a nonzero offset:

UNCONSTRAINED SUBSPACE MODELS
Subspaces have been widely utilized to model the spectral
variability of observed material spectra [36], [45], [46],
[70], [71]. With a subspace model, the target material vector is restricted to vary in a K-dimensional subspace of the
d-dimensional data space (with K # d). The amount of variability allowed increases as K increases from K = 1 (scalar
variability in one dimension) to K = d (the spectral vector is
allowed to vary over all of R d). This can be written as

(15)

K

t = / a k t k = Ta,

(13)

k =1

Band 3
Target
Subspace t
c
ti
t

Band 3
Target
pdf
tc
ti
t2

Band 2

t1

Band 1
(a)

CONSTRAINED SUBSPACE MODELS
In (13), the coefficients in vector a are unconstrained. If they
are to be interpreted as physical quantities (such as abundances), however, then constraints can be used to produce
more realistic models of target variability. In particular, nonnegativity and sum-to-one constraints for the " a k ,Kk =1 coefficients can be imposed, thus leading to a simplex model:
K
Z
] t = / a k e k = Ea
]
k =1
[
,
]] a <$ 0
1
a
=
1
\

where 1 is a K-dimensional column vector including all ones,
the superscript < denotes the transpose operator, and the
columns e k of the d # K matrix E can be interpreted as the K
vertices (or endmembers) of the K - 1 dimensional simplex,
inside which the target t is constrained to lie. This is essentially identical to the linear mixing model, first proposed by
Boardman [73] and now very widely used for background
modeling [61], [74]. If the target is being used in an additive
model, then the overall magnitude is not important, and

Band 3
Target
Manifold tc
t2
ti
Band 2

t1

Band 1
(b)

(14)

One scenario where this arises is when the coefficient a corresponds to the amount of direct solar illumination and t o
corresponds to the spectrum of the target in the shade.

Band 3
Target
Simplex tc
ti
t2

2

Band 2

t1

t = t o + Ta.

Band 2

t1

Band 1
(c)

Band 1
(d)

FIGURE 6. A sketch of the target variability models: (a) probabilistic, in which the target vectors are drawn from a probability distribution on

t ; (b) subspace, in which the target vectors are linear combinations of t 1 and t 2; (c) constrained subspace, in which the target vectors are
positive linear combinations of t 1 and t 2; and (d) manifold, in which the target vectors lie in a potentially intricate and convoluted structure
that locally appears to be Euclidean.
18

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

JUNE 2019
IEEE Geoscience and Remote Sensing Magazine - June 2019

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