García, I.

Hendrix, E.M.T.

Plaza, A.

Plaza, J.

English

Wageningen Yield 

Proceedings of TOGO 2010, pp. 55 – 58.
Spectral unmixing based on aminimum volume simplicial
enclosure model∗
Inmaculada García,1 Eligius M.T. Hendrix,1 Javier Plaza,2 and Antonio Plaza2
1Universidad de Malaga, Dept. of Computer Architecture. 29080 Malaga, Spain, {eligius, igarciaf}@uma.es
2University of Extremadura, Dept. of Technology of Computers and Communications, Caceres, Spain, aplaza@unex.es
Abstract This work describes the minimum volume enclosing simplex problem, which is known to be a mul-
timodal Global Optimization problem. The problem has been used as a basis to estimate so-called
endmember and abundance data in unmixing spectral data of hyperspectral sensors. This estimation
problem is a big challenge. We explore the possibility of a new estimation algorithm using the min-
imum volume enclosing simplex problem. We investigate its behaviour numerically on designed
instances comparing its outcomes with a maximum volume enclosed simplex approach which is
used frequently in spectral unmixing.
Keywords: spectral unmixing, endmembers, principal components, optimization. minimum volume
1. Introduction
A challenging problem in having data from multispectral imaging sensors is to unfold them
into components. We study here the possibility to do so using a minimum volume enclosing
simplex approach. Hyperspectral sensors record scenes in which various disparate material
substances contribute to the spectrum measured from a single pixel.
Spectral unmixing ([5]) is a term to denote a procedure to decompose a measured spec-
trum of a mixed pixel into a collection of constituent spectra (endmembers) and a set of cor-
responding fractions (abundances) that indicate the proportion of each endmember present
in the pixel. Endmembers normally correspond to familiar macroscopic objects in the scene,
such as water, soil, metal, or any natural or man-made material.
Many methods have been developed and tested to perform endmember extraction and un-
mixing, see [3] for an overview. We will focus on what is called linear unmixing and ask the
question how one can recover the endmember and abundance data via unbiased estimators.
One typically sees least squares approaches with the additional complication that the abun-
dance estimate should lay on the unit simplex (nonnegativity). [4] takes an approach where
two conflicting objectives, that of least squares and minimizing the volume of an enclosing
simplex are combined in an objective function. Recently, [1] develop an approach where they
apply sequential Linear programming to solve the minimum volume enclosing simplex prob-
lem. In this paper we use standard available nonlinear optimization algorithms to solve the
problem.
The problem of enclosing a set of points with a minimum volume body leads usually to
a Global Optimization problem; we will illustrate that for the generic simplicial enclosure
∗This work has been supported by the SpanishMinistry of Science and Innovation through grants TIN2008-01117 and AYA2008-
05965-C04-02. Eligius Hendrix is a fellow of the Spanish "Ramon y Cajal" contract program, co-financed by the European Social
Fund.
56 Inmaculada García, Eligius M.T. Hendrix, Javier Plaza, and Antonio Plaza
this is the same. However, the use in spectral unmixing is far from worst case behaviour;
instances are characterised by low noise and pixel data is well spread. A local search from
a well designed starting body leads to the global optimum soon. We will take a hierarchical
vision: First to minimize least squares using principal component analysis, which is very
common in image data analysis and second, minimize the volume of an enclosing simplex in
the reduced space. The question is how to use such an approach such that for linear mixture
with white noise one obtains unbiased estimates of endmembers and abundance.
A benchmark method is to consider a maximum volume “inscribing” simplex looked for
by the so-called n-findr algorithm [6]. Given the reduced data, in principle one looks for
all combinations within the given pixels as candidate endmembers such that the resulting
volume of the spanning simplex is maximum. If indeed the endmembers are present in the
data and noise is low, the approach is very promising as analysed by [6]. We can use the
results of such an approach to compare methods numerically.
2. Unmixing and minimum enclosing simplex
Let us assume that a hyperspectral scene contains m spectral bands and r pixel vectors. Each
pixel vector in the scene (yk) can be modeled using the following expression:
yk = Xak + ǫ (1)
where yk ism×1 observation bands,X ism×n, bands of endmembers, ak is 1×n abundance
and ǫ is m × 1 white noise with a standard deviation of σ. Our goal is to design a method
for recovering “real" matrix X and abundance ak of observed pixels yk. To do so, usually
two objectives are minimized: noise in a least squares way and the volume of the simplex
spanned by the columns of matrix X. Moreover, the abundance should be positive for each
pixel. The question is how to deal with least squares and minimum volume in such a way that
the estimation is unbiased, i.e. the expected value of the estimator is the real value.
One should keep in mind that instances of the problem consisting of real images are char-
acterized by pixels being mixtures of less than 4 constituents, i.e. vectors ak have only a few
positive values. The idea of least squares in the estimation procedure is, that often it is not
know exactly how many endmembers, constituents, are involved in the data. Therefore ap-
plication of principal component analysis is popular. Having n endmembers gives that one
should discover an n− 1 dimensional subspace that is responsible for the main variation and
the rest of them dimensional space is considered noise.
First of all the data are centralized by the mean y, such that the columns of Y consist of cen-
tralized observations yk−y. The observed variation in the spectral data Y
TY is approximated
by (CZ)TCZ where C is an m × (n − 1) matrix of principal components and Z is (n − 1) × r
a so-called score matrix. In direction c1 we have the biggest variation, in direction c2 the sec-
ond biggest etc. Essentially we have reduced model (1) to z = V a + ξ, where we expect the
endmembers X to lay in the space < C > +y spanned by the columns of C . C represents an
estimate of the space in which the endmember spectraX are located,X = CV +y. To say it in
another way, with absence of noise the estimate ofC represents the space spanned byX−y1T ,
where 1 is the all-ones vector of appropriate dimension. With noise, ξ is now the projection of
ǫ on < C > and therefore its components also form white noise. To be consistent, we should
theoretically notice that y = Cz + y + ζ where ζ is the part of ǫ projected on the orthoplement
of < C >; ǫ = ξ + ζ . We will use the idea that the noise of z is componentwise independent.
We follow a two step approach often found in literature. First we estimate the space in
which the n endmembers are lying. Secondly, in that space, we minimize the volume of the
resulting simplex such that it encloses the projections of the observed bands of the pixels. The
n-findr algorithm follows an approach where on the projected plane the volume of a simplex
is maximized.
Spectral unmixing 57
3. Minimum volume versus maximum volume simplices
The estimate of the matrix of endmembersX = CV + y appears from an estimate of V based
on the projected bands (scores) Z . The problem of finding the minimum enclosing simplex of
a set of points zk, k = 1, . . . , r in (n− 1)-dimensional space is
min
V
{f(V ) := det
(
V
1
T
)
} subject to ak =
(
V
1
T
)
−1(
zk
1
)
≥ 0, k = 1, . . . , r (2)
Enclosing with shapes may lead to GO problems. [2] give several examples for enclosing
with spheres (the Chebychev problem) and with hyper-rectangles. The use of the minimum
volume problem for endmember identification is illustrated next.
In general, we will call V the real values of endmembers defining simplex S = conv(V )
and use for the outer enclosing estimate Vˆ o and corresponding simplex Sˆo. In case all pixels
would be convex combinations of (few) endmemberswithout any noise, the enclosing simplex
Sˆo obtains the endmembers V as vertices despite they do not appear in the pixels. Literature
on spectral unmixing also uses a maximum volume simplex perspective. The idea is that pure
pixels representing the endmembers are present in the data set Z . Consider the pixel data
as a set Z. One wants to find a subset V with |V| = n such that the corresponding simplex
has maximum volume; i.e. maxV⊂Z f(V ), where V is a matrix with the columns of V. This
defines a combinatorial optimization problem. The n-findr algorithm is a so-called local
search heuristic in combinatorial optimisation context. We used a matlab implementation of
n-findr as reference method to compare to minvest described in Section 4.
4. Minimum volume estimation procedure: minvest
The minimum volume simplex Sˆo gives an accurate estimate of the endmembers if noise is
absent. That is, sufficiently many pixels should lay on the boundary of S. Mathematically,
this means that abundance values aj,k = 0; i.e. pixel k does not contain any constituent j. In
the hyperspectral image area, it is known that a pixel spectrum consists of a mix of at most
4 constituents. As soon as noise is added, one can approximate with probability theory the
chance that a pixel lays outside S. Let ρ be an estimation of the fraction of pixels we expect to
be interior with respect to S. An initial matrix V that does not include all pixels is generated.
Iteratively the endmembers Vˆ are estimated from the minimum volume problem by solving
(2) and the active pixels at its boundary are removed up to a ρ fraction is left over.
To recover the abundance values from the estimated endmembers V the term linear spec-
tral unmixing (LSU) is used when nonnegativeness of estimated abundance is not taken into
account. For the fraction of pixels located within simplex Sˆ we have automatically positive
abundance values. For pixels zk outside Sˆ, we have at least one corresponding ajk < 0. The
term fully constrained linear spectral unmixing (FCLSU) is used if we want to force abundance
values to be nonnegative. To do so we consider that the noise of zk is componentwise inde-
pendent we choose to project zk on the facet of Sˆ closest to zk and determine the abundance
for the endmembers in the plane of that facet.
5. Computer simulated data experiments and conclusions
Computer simulations have been carried out in order to evaluate the accuracy of minvest in
comparison with n-findr in highly controlled analysis scenarios. The quality of estimation Vˆ
(Aˆ) of V (A) is measured as the standard deviation estimate assuming Vˆ (Aˆ) is unbiased, also
called root mean squared error (RMSE). To distinguish, we will use σA if Aˆ is generated by
LSU and σAp if Aˆ is generated by FCLSUWe show the results obtained from a case with n = 5
58 Inmaculada García, Eligius M.T. Hendrix, Javier Plaza, and Antonio Plaza
endmembers and r = 500 pixels. To mimic the idea of combinations of a few constituents, a
ground truth abundance matrix A is generated consisting for 50% of mixtures of 2 endmem-
bers and for 50% of mixtures of 3 endmembers. They are generated uniformly over the unit
simplex. The score matrix Z as input for the estimation is taken as Z = V A + σ · ξ, where
ξ is standard white noise. The choice of the parameter value for ρ is determined by the data
ρ = 18.25%. Given that performance indicators depend on pseudo-randomly drawn white
noise, for each ground-truth matrix A we replicated white noise 100 times.
Table 1. (RMSE) of endmembers V and abundance A obtained by n-findr and minvest given noise σ.
n-findr minvest
σ 0.01 0.1 0.2 0.5 0.7 0.01 0.1 0.2 0.5 0.7
σV .030 .118 .233 .857 1.359 .013 .111 .194 .486 .922
σA .011 .063 .114 .259 .323 .007 .058 .105 .204 .266
σAp .008 .048 .092 .224 .281 .005 .048 .086 .174 .234
The measured performance for n-findr and minvest is given in Table 1. It shows standard
deviation estimates σV of endmembers and σA of fractional abundances calculated via LSU
and via fully constrained spectral unmixing (FCLSU). One can observe that the standard de-
viation of the estimates is in the same order of magnitude as that of noise. This means that
the procedures give results as accurate as the input data. Deviation of endmembers and abun-
dances estimations provided by n-findr are higher than those obtained with minvest. Other
scenarios with and without pure pixels have been generated and evaluated.
The following can be concluded: (1) The problem of unmixing hyperspectral data may be
a hard to solve problem. (2) The minimum volume simplicial enclosure problem is a Global
Optimization problem where the number of optima depends in worst case on the number of
points in the convex hull of the instance. (3) The resulting simplex of the (combinatorial) max-
imum volume simplex problem is enclosed in the result of the minimum volume enclosing
simplex problem. (4) Local search from a good starting simplex leads in general to the global
optimum for the case of spectral unmixing due to well spread data in the originating simplex
and low noise in practice. (5) The new minvest algorithm does not require pure pixels to be
present in the scene of the instance unlike the n-findr algorithm. (6) In the case of having
no noise and well spread data over the boundary of the spectral simplex, minvest recovers
the original endmembers and ground truth abundance. (7) The RMSE performance indica-
tor is sensitive to scaling in its use for measuring abundance discrepancies. (8) The results of
minvest seems more correlated to ground truth abundance data than the ones of n-findr.
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A quantitative and comparative analysis of endmember extraction algorithms from hyperspectral data.

A survey of spectral unmixing algorithms.

Convex analysis based minimum-volume enclosing simplex algorithm for hyperspectral unmixing.

Endmember extraction from highly mixed data using minimum volume constrained nonnegative matrix factorization.

Introduction to Nonlinear and Global Optimization.

N-FINDR: an algorithm for fast autonomous spectral end-member determination in hyperspectral data.

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