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Vertex component analysis: a fast algorithm to unmix hyperspectral data

TL;DR: A new method for unsupervised endmember extraction from hyperspectral data, termed vertex component analysis (VCA), which competes with state-of-the-art methods, with a computational complexity between one and two orders of magnitude lower than the best available method.
Abstract: Given a set of mixed spectral (multispectral or hyperspectral) vectors, linear spectral mixture analysis, or linear unmixing, aims at estimating the number of reference substances, also called endmembers, their spectral signatures, and their abundance fractions. This paper presents a new method for unsupervised endmember extraction from hyperspectral data, termed vertex component analysis (VCA). The algorithm exploits two facts: (1) the endmembers are the vertices of a simplex and (2) the affine transformation of a simplex is also a simplex. In a series of experiments using simulated and real data, the VCA algorithm competes with state-of-the-art methods, with a computational complexity between one and two orders of magnitude lower than the best available method.
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898 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 43, NO. 4, APRIL 2005
Vertex Component Analysis: A Fast Algorithm to
Unmix Hyperspectral Data
José M. P. Nascimento, Student Member, IEEE, and José M. Bioucas Dias, Member, IEEE
Abstract—Given a set of mixed spectral (multispectral or hy-
perspectral) vectors, linear spectral mixture analysis, or linear
unmixing, aims at estimating the number of reference substances,
also called endmembers, their spectral signatures, and their
abundance fractions. This paper presents a new method for
unsupervised endmember extraction from hyperspectral data,
termed vertex component analysis (VCA). The algorithm exploits
two facts: 1) the endmembers are the vertices of a simplex and 2)
the affine transformation of a simplex is also a simplex. In a series
of experiments using simulated and real data, the VCA algorithm
competes with state-of-the-art methods, with a computational
complexity between one and two orders of magnitude lower than
the best available method.
Index Terms—Linear unmixing, simplex, spectral mixture
model, unmixing hypespectral data, unsupervised endmember
extraction, vertex component analysis (VCA).
I. INTRODUCTION
H
YPERSPECTRAL remote sensing exploits the electro-
magnetic (EM) scattering patterns of different materials at
specific wavelengths [1], [2]. Hyperspectral sensors have been
developed to sample the scattered portion of the EM spectrum
extending from the visible region through the near-infrared and
midinfrared, in hundreds of narrow contiguous bands [3], [4].
The number and variety of potential civilian and military appli-
cations of hyperspectral remote sensing is enormous [5], [6].
Very often, the resolution cell corresponding to a single pixel
in an image contains several substances (endmembers) [3]. In
this situation, the scattered energy is a mixing of the endmember
spectra. A challenging task underlying many hyperspectral im-
agery applications is then decomposing a mixed pixel into a col-
lection of reflectance spectra, called endmember signatures, and
the corresponding abundance fractions [7]–[9].
Depending on the mixing scales at each pixel, the observed
mixture is either linear or nonlinear [10], [11]. A linear mixing
model holds approximately when the mixing scale is macro-
scopic [12] and there is negligible interaction among distinct
endmembers [2], [13]. If, however, the mixing scale is micro-
scopic (or intimate mixtures) [14], [15] and the incident solar
Manuscript received January 6, 2004; revised December 21, 2004. This work
was supported in part by the Fundação para a Ciência e Tecnologia under the
Projects POSI/34071/CPS/2000 and PDCTE/CPS/49967/2003 and in part by
the Departamento de Engenharia de Electrónica e Telecomunicações e de Com-
putadores of the Instituto Superior de Engenharia de Lisboa.
J. M. P. Nascimento is with the Instituto Superior de Engenharia de Lisboa
and the Instituto de Telecomunicações, 1949-001 Lisbon, Portugal (e-mail:
zen@isel.pt).
J. M. Bioucas Dias is with the Instituto de Telecomunicações and the Instituto
Superiror Técnico, 1949-001 Lisbon, Portugal (e-mail: bioucas@lx.it.pt).
Digital Object Identifier 10.1109/TGRS.2005.844293
radiation is scattered by the scene through multiple bounces in-
volving several endmembers [16], the linear model is no longer
accurate.
Linear spectral unmixing has been intensively researched
in the last years [8], [9], [11], [17]–[20]. It considers that a
mixed pixel is a linear combination of endmember signatures
weighted by the correspondent abundance fractions. Under
this model, and assuming that the number of substances and
their reflectance spectra are known, hyperspectral unmixing is
a linear problem for which many solutions have been proposed
(e.g., maximum-likelihood estimation [7], spectral signature
matching [21], spectral angle mapper [22], subspace projection
methods [23], [24], and constrained least squares [25]).
In most cases, the number of substances and their reflectances
are not known and, then, hyperspectral unmixing falls into the
class of blind source separation problems [26]. independent
component analysis (ICA) has recently been proposed as a tool
to blindly unmix hyperspectral data [27]–[30]. ICA is based on
the assumption of mutually independent sources (abundance
fractions), which is not the case of hyperspectral data, since
the sum of abundance fractions is constant, implying statistical
dependence among them. This dependence compromises ICA
applicability to hyperspectral images as shown in [20] and
[31]. In fact, ICA finds the endmember signatures by mul-
tiplying the spectral vectors with an unmixing matrix which
minimizes the mutual information among channels. If sources
are independent, ICA provides the correct unmixing, since the
minimum of the mutual information corresponds to and only
to independent sources. This is no longer true for dependent
fractional abundances. Nevertheless, some endmembers may be
approximately unmixed. These aspects are addressed in [31].
Under the linear mixing model, the observations from a scene
are in a simplex whose vertices correspond to the endmembers.
Several approaches [32]–[34] have exploited this geometric fea-
ture of hyperspectral mixtures [33].
The minimum volume transform (MVT) algorithm [34] deter-
mines the simplex of minimum volume containing the data.
The method presented in [35] is also of MVT type, but by in-
troducing the notion of bundles, it takes into account the end-
member variability usually present in hyperspectral mixtures.
The MVT type approaches are complex from the compu-
tational point of view. Usually, these algorithms first find the
convex hull defined by the observed data and then fit a minimum
volume simplex to it. For example, the
gift wrapping algorithm
[36] computes the convex hull of
data points in a -dimen-
sional space with a computational complexity of
,
where
is the highest integer lower or equal than , and is
the number of samples. The complexity of the method presented
0196-2892/$20.00 © 2005 IEEE

NASCIMENTO AND DIAS: VERTEX COMPONENT ANALYSIS 899
in [35] is even higher, since the temperature of the simulated an-
nealing algorithm therein used shall follow a
law [37] to
assure convergence (in probability) to the desired solution.
Aiming at a lower computational complexity, some al-
gorithms such as the
pixel purity index (PPI) [33] and the
N-FINDR [38] still nd the minimum volume simplex con-
taining the data cloud, but they assume the presence in the data
of at least one pure pixel of each endmember. This is a strong
requisite that may not hold in some datasets. In any case, these
algorithms nd the set of most pure pixels in the data.
The PPI algorithm uses the minimum-noise fraction (MNF)
[39] as a preprocessing step to reduce dimensionality and to
improve the signal-to-noise ratio (SNR). The algorithm then
projects every spectral vector onto skewers (large number of
random vectors) [33], [40], [41]. The points corresponding to
extremes, for each skewer direction, are stored. A cumulative
account records the number of times each pixel (i.e., a given
spectral vector) is found to be an extreme. The pixels with the
highest scores are the purest ones.
The N-FINDR algorithm [38] is based on the fact that in
spectral dimensions, the -volume dened by a simplex formed
by the purest pixels is larger than any other volume dened by
any other combination of pixels. This algorithm nds the set of
pixels dening the largest volume by inflating a simplex inside
the data.
ORASIS [42], [43] is a hyperspectral framework developed
by the Naval Research Laboratory consisting of several algo-
rithms organized in six modules: exemplar selector, adaptative
learner, demixer, knowledge base/spectral library, and spatial
postprocessor. The rst step consists in at elding the spectra.
Next, the exemplar selection module is used to select spectral
vectors that best represent the smaller convex cone containing
the data. The other pixels are rejected when the spectral angle
distance is less than a given threshold. The procedure nds the
basis for a subspace of a lower dimension using a modied
Gram-Schmidt orthogonalization. The selected vectors are then
projected onto this subspace, and a simplex is found by an MVT
process. ORASIS is oriented to real-time target detection from
uncrewed air vehicles using hyperspectral data [44].
In this paper we introduce the vertex component analysis
(VCA) algorithm to unmix linear mixtures of endmember
spectra. The algorithm is unsupervised and exploits two facts:
1) the endmembers are the vertices of a simplex and 2) the
afne transformation of a simplex is also a simplex. It works
with unprojected and with projected data. As PPI and N-FINDR
algorithms, VCA also assumes the presence of pure pixels in
the data. The algorithm iteratively projects data onto a direction
orthogonal to the subspace spanned by the endmembers already
determined. The new endmember signature corresponds to
the extreme of the projection. The algorithm iterates until all
endmembers are exhausted. VCA performs much better than
PPI and better than or comparable to N-FINDR; yet it has
a computational complexity between one and two orders of
magnitude lower than N-FINDR.
The paper is structured as follows. Section II describes the
geometric fundamentals of the proposed method. Sections III
and IV evaluate the proposed algorithm using simulated and real
data, respectively. Section V ends the paper by presenting some
concluding remarks.
II. V
ERTEX COMPONENT
ANALYSIS
ALGORITHM
Assuming the linear mixing scenario, each observed spectral
vector is given by
(1)
where
is an -vector ( is the number of bands),
is the mixing matrix ( denotes the
th endmember signature and is the number of endmembers
present in the covered area),
( is a scale factor
modeling illumination variability due to surface topography),
is the abundance vector containing
the fractions of each endmember (the notation
stands for
vector transposed) and
models system additive noise.
Owing to physical constraints [19], abundance fractions are
nonnegative
and satisfy the so-called positivity con-
straint
, where is a vector of ones. Each pixel
can be viewed as a vector in an
-dimensional Euclidean space,
where each channel is assigned to one axis of space. Since the
set
is a simplex, then the set
is also a simplex.
However, even assuming
, the observed vector set be-
longs to
that is a convex cone, owing to scale factor . Fig. 1(a) illus-
trates a simplex and a cone, projected on a two-dimensional sub-
space, dened by a mixture of three endmembers. The simplex
boundary is a triangle whose vertices correspond to the end-
members shown in Fig. 2. Small and medium dots are simulated
mixed spectra belonging to the simplex
and to the
cone
, respectively.
The projective projection of the convex cone
onto a prop-
erly chosen hyperplane is a simplex with vertices corresponding
to the vertices of the simplex
. This is illustrated in Fig. 1(b).
The simplex
is
the projective projection of the convex cone
onto the plane
, where the choice of assures that there is no ob-
served vectors orthogonal to it.
After identifying
, the VCA algorithm iteratively projects
data onto a direction orthogonal to the subspace spanned by the
endmembers already determined. The new endmember signa-
ture corresponds to the extreme of the projection. Fig. 1(b) il-
lustrates the two iterations of VCA algorithm applied to the sim-
plex
dened by the mixture of two endmembers. In the rst
iteration, data are projected onto the rst direction
. The ex-
treme of the projection corresponds to endmember
. In the
next iteration, endmember
is found by projecting data onto
direction
, which is orthogonal to . The algorithm iterates
until the number of endmembers is exhausted.
A. Dimensionality Reduction
Under the linear observation model, spectral vectors are in a
subspace of dimension
.If , it is worthy to project the
observed spectral vectors onto the subspace signal. This leads

900 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 43, NO. 4, APRIL 2005
(a)
(b)
Fig. 1. (a) Two-dimensional scatterplot of mixtures of the three endmembers
shown in Fig. 2. Circles denote pure materials. (b) Illustration of the VCA
algorithm.
Fig. 2. Reectances of carnallite, ammonioalunite, and biotite.
to signicant savings in computational complexity and to SNR
improvements.
Principal component analysis (PCA) [45], maximum-noise
fraction (MNF) [46], and singular value decomposition (SVD)
[47] are three well-known projection techniques widely used in
remote sensing. PCA, also known as KarhunenLoéve trans-
form, seeks the projection that best represents data in a least-
squares sense; MNF seeks the projection that optimizes SNR;
and SVD provides the projection that best represents data in the
maximum-power sense. PCA and MNF are equal in the case
Fig. 3. Scatterplot (bands
= 827
nm and
= 1780
nm) of the three
endmembers mixture. (a) Unprojected data. (b) Projected data using SVD. Solid
and dashed lines represent, respectively, simplexes computed from original and
estimated endmembers (using VCA).
of white noise. SVD and PCA are also equal in the case of
zero-mean data.
As discussed before, in the absence of noise, observed vec-
tors
lie in a convex cone contained in a subspace of
dimension
. The VCA algorithm starts by identifying by
SVD and then projects points in
onto a simplex by com-
puting
[see Fig. 1(b)]. This simplex is contained
in an afne set of dimension
. We note that the rational un-
derlying the VCA algorithm is still valid if the observed dataset
is projected onto any subspace
of dimension , for
, i.e., the projection of the cone onto followed
by a projective projection is also a simplex with the same ver-
tices. Of course, the SNR decreases as
increases.
For illustration purposes, a simulated scene was gener-
ated according to (1). Three spectral signatures (Abi-
otite, Bcarnallite, and Cammonioalunite) were selected
from the U.S. Geological Survey (USGS) digital spectral
library [48] (see Fig. 2); the abundance fractions follow a
Dirichlet distribution; parameter
is set to 1; and the noise
is zero-mean white Gaussian with covariance matrix
,
where
is the identity matrix and leading to
a SNR
dB. Fig. 3(a)
presents a scatterplot of the simulated spectral mixtures without
projection (bands
nm and nm). Two
triangles are also plotted whose vertices represent the true end-
members (solid line) and the estimated endmembers (dashed
line) by the VCA algorithm, respectively. Fig. 3(b) presents a
scatterplot (same bands) of projected data onto the estimated
afne set of dimension two inferred by SVD. Noise is clearly
reduced, leading to a visible improvement on the VCA results.
As referred before, we apply the rescaling
to get rid
of the topographic modulation factor. As the SNR decreases,
this rescaling amplies noise, being preferable to identify di-
rectly the afne space of dimension
by using only PCA.
This phenomenon is illustrated in Fig. 4, where data clouds
(noiseless and noisy) generated by two signatures are shown.
Afnes spaces
and identied, respectively, by PCA
of dimension
and SVD of dimension followed by projec-
tive projection are schematized by straight lines. In the absence
of noise, the direction of
is better identied by projective
projection onto
( better than ); in the presence of

NASCIMENTO AND DIAS: VERTEX COMPONENT ANALYSIS 901
strong noise, the direction of is better identied by orthog-
onal projection onto
( better than ). As a conclu-
sion, when the SNR is higher than a given threshold SNR
,
data is projected onto
followed by the rescaling ;
otherwise data are projected onto
. Based on experimental
results, we propose the threshold SNR
dB.
Since for zero-mean white noise SNR
, then
we conclude that at SNR
, , i.e., the
SNR
corresponds to the xed value of the SNR mea-
sured with respect to the signal subspace.
B. VCA Algorithm
The pseudocode for the VCA method is shown in Algorithm
1. Symbols
and stand for the th column of
and for the th to th columns of , respectively. Symbol
stands for the estimated mixing matrix.
Algorithm 1: Vertex Component Analysis (VCA)
INPUT
p
,
R
[
r
1
;
r
2
;
...
;
r
N
]
1: SNR
th
=15+10log
10
(
p
)
dB
2: if SNR
>
SNR
th
then
3:
d
:=
p
;
4:
X
:=
U
T
d
R
;{
U
d
obtained by SVD}
5:
u
:=
mean
(
X
)
;{
u
is a
1
2
d
vector}
6:
[
Y
]
:
;j
:= [
X
]
:
;j
=
([
X
]
T
:
;j
u
)
; {projective projection}
7: else
8:
d
:=
p
0
1
;
9:
[
X
]
:
;j
:=
U
T
d
([
R
]
:
;j
0
r
)
;{
U
d
obtained by PCA}
10:
c
:= arg max
j
=1...
N
k
[
X
]
:
;j
k
;
11:
c
:= [
c
j
c
j
...
j
c
]
;{
c
is a
1
2
N
vector}
12:
Y
:=
X
c
13: end if
14:
A
:= [
e
u
j
0
j
...
j
0
]
;{
e
u
=[0
;
...
;
0
;
1]
T
and
A
is a
p
2
p
auxiliary matrix}
15: for
i
:= 1
to
p
do
16:
w
:=
randn
(0
;
I
p
)
;{
w
is a zero-mean random Gaussian
vector of covariance
I
p
}
17:
f
:= ((
I
0
AA
#
)
w
)
=
(
k
(
I
0
AA
#
)
w
k
)
;{
f
is a vector
orthonormal to the subspace spanned by
[
A
]
:
;
1:
i
.}
18:
v
:=
f
T
Y
;
19:
k
:= arg max
j
=1
;
...
;N
j
[
v
]
:
;j
j
;{nd the projection ex-
treme.}
20:
[
A
]
:
;i
:= [
Y
]
:
;k
;
21:
[
indice
]
i
:=
k
; {stores the pixel index.}
22: end for
23: if SNR
>
SNR
th
then
24:
M
:=
U
d
[
X
]
:
;indice
;{
M
is a
L
2
p
estimated mixing
matrix}
25: else
26:
M
:=
U
d
[
X
]
:
;indice
+
r
;{
M
is a
L
2
p
estimated mixing
matrix}
27: end if
Step 2 tests if the SNR is higher than SNR in order to decide
whether the data are to be projected onto a subspace of dimen-
sion
or . In the rst case the projection matrix is
Fig. 4. Illustration of the noise effect on the dimensionality reduction.
obtained by SVD from , where
and is the number of pixels. In the second case the projection
is obtained by PCA from
, where is the
sample mean of
, for .
Steps 4 and 9 assure that the inner product between any vector
and vector is nonnegative, a crucial condition for the
VCA algorithm to work correctly. The chosen value of
, assures that the colatitude angle be-
tween
and any vector is between 0 and 45 , then
avoiding numerical errors which otherwise would occur for an-
gles near 90
.
Step 14 initializes the auxiliary matrix
, which stores the
projection of the estimated endmembers signatures. Assume
that there exists at least one pure pixel of each endmember in
the input sample
[see Fig. 1(b)]. Each time the loop for is
executed, a vector
orthonormal to the space spanned by the
columns of the auxiliary matrix
is randomly generated and
is projected onto . Notation stands for the pseudoinverse
matrix. Since we assume that pure endmembers occupy the
vertices of a simplex, then
, for ,
where values
and correspond to and only to pure pixels. We
store the endmember signature corresponding to
.
The next time loop for is executed,
is orthogonal to the space
spanned by the signatures already determined. Since
is the
projection of a zero-mean Gaussian independent random vector
onto the orthogonal space spanned by the columns of
,
then the probability of
being null is zero. Notice that the
underling reason for generating a random vector is only to get
a non null projection onto the orthogonal space generated by
the columns of
. Fig. 1(b) shows the input samples and the
chosen pixels, after the projection
. Then a second
vector
orthonormal to the endmember is generated and the
second endmember is stored. Finally, steps 24 and 26 compute
the columns of matrix
, which contain the estimated end-
members signatures in the
-dimensional space.

902 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 43, NO. 4, APRIL 2005
III. EVALUATION OF THE
VCA A
LGORITHM
In this section, we compare VCA, PPI, and N-FINDR al-
gorithms. N-FINDR and PPI were coded accordingly to [38]
and [33], respectively. Regarding PPI, the number of
skewers
must be large [39], [40], [49][51]. Based on Monte Carlo
runs, we concluded that the minimum number of skewers
beyond which there is no unmixing improvements is about
1000. All experiments are based on simulated scenes from
which we know the signature endmembers and their frac-
tional abundances. Estimated endmembers are the columns of
. We also compare estimated abun-
dance fractions given by
,( stands
for pseudoinverse of
) with the true abundance fractions.
To evaluate the performance of the three algorithms,
we compute vectors of angles
and
with
1
(2)
(3)
where is the angle between vectors and ( th end-
member signature estimate) and
is the angle between vectors
and (vectors of formed by the th lines of ma-
trices
and , respectively). The symmetric
Kullback distance [52], a relative entropy-based distance, is an-
other error measure used to compare similarity between signa-
tures, namely under the name spectral information divergence
(SID) [53]. SID is dened by
SID
(4)
where
is the relative entropy of with respect to
given by
(5)
and
and .
Based on
, , and SID SID
SID , we estimate the following rms error distances:
(6)
(7)
(8)
where
denotes the expectation operator. The rst two quan-
tities measure distances between
and , for ;
the third is similar to the rst, but for the estimated abundance
1
Notation
h
x
;
y
i
stands for the inner product
x y
.
Fig. 5. First scenario (
N
= 1000
,
p
=3
,
L
=224
,
=
=
=1
=
3
,
=20
,
=1
). (a) rmsSID as function of SNR. (b) rmsSAE as function of
SNR. (c) rmsFAAE as function of SNRs.
fractions. Herein we name , , and as rmsSAE, rmsSID,
and rmsFAAE, respectively (SAE stands for signature angle
error and FAAE stands for fractional abundance angle error).
Mean values in (6)(8) are approximated by sample means
based on 100 Monte Carlo runs.
In all experiments, the spectral signatures are selected
from the USGS digital spectral library [48]. Fig. 2 shows
three of these endmember signatures. Abundance fractions
are generated according to a Dirichlet distribution given by
(9)
where
, , is the
expected value of the
th endmember fraction, and denotes
the Gamma function. Parameter
is Beta distributed,

Citations
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Journal ArticleDOI
TL;DR: This paper presents an overview of un Mixing methods from the time of Keshava and Mustard's unmixing tutorial to the present, including Signal-subspace, geometrical, statistical, sparsity-based, and spatial-contextual unmixed algorithms.
Abstract: Imaging spectrometers measure electromagnetic energy scattered in their instantaneous field view in hundreds or thousands of spectral channels with higher spectral resolution than multispectral cameras. Imaging spectrometers are therefore often referred to as hyperspectral cameras (HSCs). Higher spectral resolution enables material identification via spectroscopic analysis, which facilitates countless applications that require identifying materials in scenarios unsuitable for classical spectroscopic analysis. Due to low spatial resolution of HSCs, microscopic material mixing, and multiple scattering, spectra measured by HSCs are mixtures of spectra of materials in a scene. Thus, accurate estimation requires unmixing. Pixels are assumed to be mixtures of a few materials, called endmembers. Unmixing involves estimating all or some of: the number of endmembers, their spectral signatures, and their abundances at each pixel. Unmixing is a challenging, ill-posed inverse problem because of model inaccuracies, observation noise, environmental conditions, endmember variability, and data set size. Researchers have devised and investigated many models searching for robust, stable, tractable, and accurate unmixing algorithms. This paper presents an overview of unmixing methods from the time of Keshava and Mustard's unmixing tutorial to the present. Mixing models are first discussed. Signal-subspace, geometrical, statistical, sparsity-based, and spatial-contextual unmixing algorithms are described. Mathematical problems and potential solutions are described. Algorithm characteristics are illustrated experimentally.

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Cites methods from "Vertex component analysis: a fast a..."

  • ...• The vertex component analysis (VCA) algorithm [123] iteratively projects data onto a direction orthogonal to the subspace spanned by the endmembers already determined....

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Posted Content
TL;DR: An overview of unmixing methods from the time of Keshava and Mustard's tutorial as mentioned in this paper to the present can be found in Section 2.2.1].
Abstract: Imaging spectrometers measure electromagnetic energy scattered in their instantaneous field view in hundreds or thousands of spectral channels with higher spectral resolution than multispectral cameras. Imaging spectrometers are therefore often referred to as hyperspectral cameras (HSCs). Higher spectral resolution enables material identification via spectroscopic analysis, which facilitates countless applications that require identifying materials in scenarios unsuitable for classical spectroscopic analysis. Due to low spatial resolution of HSCs, microscopic material mixing, and multiple scattering, spectra measured by HSCs are mixtures of spectra of materials in a scene. Thus, accurate estimation requires unmixing. Pixels are assumed to be mixtures of a few materials, called endmembers. Unmixing involves estimating all or some of: the number of endmembers, their spectral signatures, and their abundances at each pixel. Unmixing is a challenging, ill-posed inverse problem because of model inaccuracies, observation noise, environmental conditions, endmember variability, and data set size. Researchers have devised and investigated many models searching for robust, stable, tractable, and accurate unmixing algorithms. This paper presents an overview of unmixing methods from the time of Keshava and Mustard's unmixing tutorial [1] to the present. Mixing models are first discussed. Signal-subspace, geometrical, statistical, sparsity-based, and spatial-contextual unmixing algorithms are described. Mathematical problems and potential solutions are described. Algorithm characteristics are illustrated experimentally.

1,808 citations

Journal ArticleDOI
TL;DR: A tutorial/overview cross section of some relevant hyperspectral data analysis methods and algorithms, organized in six main topics: data fusion, unmixing, classification, target detection, physical parameter retrieval, and fast computing.
Abstract: Hyperspectral remote sensing technology has advanced significantly in the past two decades. Current sensors onboard airborne and spaceborne platforms cover large areas of the Earth surface with unprecedented spectral, spatial, and temporal resolutions. These characteristics enable a myriad of applications requiring fine identification of materials or estimation of physical parameters. Very often, these applications rely on sophisticated and complex data analysis methods. The sources of difficulties are, namely, the high dimensionality and size of the hyperspectral data, the spectral mixing (linear and nonlinear), and the degradation mechanisms associated to the measurement process such as noise and atmospheric effects. This paper presents a tutorial/overview cross section of some relevant hyperspectral data analysis methods and algorithms, organized in six main topics: data fusion, unmixing, classification, target detection, physical parameter retrieval, and fast computing. In all topics, we describe the state-of-the-art, provide illustrative examples, and point to future challenges and research directions.

1,604 citations


Cites background or methods from "Vertex component analysis: a fast a..."

  • ...The endmembers identified by VCA and N-FINDR area also represented....

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  • ...Representative algorithms of class a) are pixel purity index (PPI) [43], vertex component analysis (VCA) [44], simplex growing algorithm (SGA) [45] successive volume maximization (SVMAX) [46], and the recursive algorithm for separable NMF (RSSNMF) [47]; Representative algorithms of class b) are N-FINDR [48], iterative error analysis (IEA), [49], sequential maximum angle convex cone (SMACC), and alternating volume maximization (AVMAX) [46]. c. non-puRe pixel BAseD AlgoRithMs Fig....

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  • ...9 show the identified endmember signatures with the VCA algorithm [44]....

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  • ...Soil 1 Soil 2 Soil 3 Data N-FINDR VCA Grass Grass Shade Trees Trees (a) (b) (c) (d) (e) (f) Soil 1 Soil 2 Soil 3 Grass Shade Trees Soil 1 june 2013 ieee Geoscience and remote sensinG maGazine 17 The bilinear model is valid when the scene can be partitioned in successive layers with similar scattering properties....

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  • ...Representative algorithms of class a) are pixel purity index (PPI) [43], vertex component analysis (VCA) [44], simplex growing algorithm (SGA) [45] successive volume maximization (SVMAX) [46], and the recursive algorithm for separable NMF (RSSNMF) [47]; Representative algorithms of class b) are N-FINDR [48], iterative error analysis (IEA), [49], sequential maximum angle convex cone (SMACC), and alternating volume maximization (AVMAX) [46]....

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Journal ArticleDOI
TL;DR: The experimental results, conducted using both simulated and real hyperspectral data sets collected by the NASA Jet Propulsion Laboratory's Airborne Visible Infrared Imaging Spectrometer and spectral libraries publicly available from the U.S. Geological Survey, indicate the potential of SR techniques in the task of accurately characterizing the mixed pixels using the library spectra.
Abstract: Linear spectral unmixing is a popular tool in remotely sensed hyperspectral data interpretation. It aims at estimating the fractional abundances of pure spectral signatures (also called as endmembers) in each mixed pixel collected by an imaging spectrometer. In many situations, the identification of the end-member signatures in the original data set may be challenging due to insufficient spatial resolution, mixtures happening at different scales, and unavailability of completely pure spectral signatures in the scene. However, the unmixing problem can also be approached in semisupervised fashion, i.e., by assuming that the observed image signatures can be expressed in the form of linear combinations of a number of pure spectral signatures known in advance (e.g., spectra collected on the ground by a field spectroradiometer). Unmixing then amounts to finding the optimal subset of signatures in a (potentially very large) spectral library that can best model each mixed pixel in the scene. In practice, this is a combinatorial problem which calls for efficient linear sparse regression (SR) techniques based on sparsity-inducing regularizers, since the number of endmembers participating in a mixed pixel is usually very small compared with the (ever-growing) dimensionality (and availability) of spectral libraries. Linear SR is an area of very active research, with strong links to compressed sensing, basis pursuit (BP), BP denoising, and matching pursuit. In this paper, we study the linear spectral unmixing problem under the light of recent theoretical results published in those referred to areas. Furthermore, we provide a comparison of several available and new linear SR algorithms, with the ultimate goal of analyzing their potential in solving the spectral unmixing problem by resorting to available spectral libraries. Our experimental results, conducted using both simulated and real hyperspectral data sets collected by the NASA Jet Propulsion Laboratory's Airborne Visible Infrared Imaging Spectrometer and spectral libraries publicly available from the U.S. Geological Survey, indicate the potential of SR techniques in the task of accurately characterizing the mixed pixels using the library spectra. This opens new perspectives for spectral unmixing, since the abundance estimation process no longer depends on the availability of pure spectral signatures in the input data nor on the capacity of a certain endmember extraction algorithm to identify such pure signatures.

956 citations

Journal ArticleDOI
TL;DR: A novel method without the pure-pixel assumption is presented, referred to as the minimum volume constrained nonnegative matrix factorization (MVC-NMF), for unsupervised endmember extraction from highly mixed image data, which outperforms several other advanced endmember detection approaches.
Abstract: Endmember extraction is a process to identify the hidden pure source signals from the mixture. In the past decade, numerous algorithms have been proposed to perform this estimation. One commonly used assumption is the presence of pure pixels in the given image scene, which are detected to serve as endmembers. When such pixels are absent, the image is referred to as the highly mixed data, for which these algorithms at best can only return certain data points that are close to the real endmembers. To overcome this problem, we present a novel method without the pure-pixel assumption, referred to as the minimum volume constrained nonnegative matrix factorization (MVC-NMF), for unsupervised endmember extraction from highly mixed image data. Two important facts are exploited: First, the spectral data are nonnegative; second, the simplex volume determined by the endmembers is the minimum among all possible simplexes that circumscribe the data scatter space. The proposed method takes advantage of the fast convergence of NMF schemes, and at the same time eliminates the pure-pixel assumption. The experimental results based on a set of synthetic mixtures and a real image scene demonstrate that the proposed method outperforms several other advanced endmember detection approaches

870 citations


Cites background from "Vertex component analysis: a fast a..."

  • ...First, this site has been extensively used for remote-sensing experiments since the 1980s, and many research works have been published with high-accuracy ground truth available [11], [35], [36]; second, the Cuprite area is a relatively undisturbed hydrothermal system with many well-exposed minerals....

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  • ...It is apparent that under the pure-pixel assumption [9]–[11], the best simplex is uniquely determined by the pure pixels, which are the vertices of the simplex....

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  • ...To speed up the process, some algorithms [9]–[11] assume the presence of pure pixels, i....

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  • ...Vertex component analysis (VCA) [11] is one of the most advanced convex-geometry-based endmember detection methods with the pure-pixel assumption....

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References
More filters
Journal ArticleDOI
TL;DR: The analogy between images and statistical mechanics systems is made and the analogous operation under the posterior distribution yields the maximum a posteriori (MAP) estimate of the image given the degraded observations, creating a highly parallel ``relaxation'' algorithm for MAP estimation.
Abstract: We make an analogy between images and statistical mechanics systems. Pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice-like physical system. The assignment of an energy function in the physical system determines its Gibbs distribution. Because of the Gibbs distribution, Markov random field (MRF) equivalence, this assignment also determines an MRF image model. The energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the MRF. For a range of degradation mechanisms, including blurring, nonlinear deformations, and multiplicative or additive noise, the posterior distribution is an MRF with a structure akin to the image model. By the analogy, the posterior distribution defines another (imaginary) physical system. Gradual temperature reduction in the physical system isolates low energy states (``annealing''), or what is the same thing, the most probable states under the Gibbs distribution. The analogous operation under the posterior distribution yields the maximum a posteriori (MAP) estimate of the image given the degraded observations. The result is a highly parallel ``relaxation'' algorithm for MAP estimation. We establish convergence properties of the algorithm and we experiment with some simple pictures, for which good restorations are obtained at low signal-to-noise ratios.

18,761 citations


"Vertex component analysis: a fast a..." refers methods in this paper

  • ...in [35] is even higher, since the temperature of the simulated annealing algorithm therein used shall follow a law [ 37 ] to assure convergence (in probability) to the desired solution....

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Book
01 May 1986
TL;DR: In this article, the authors present a graphical representation of data using Principal Component Analysis (PCA) for time series and other non-independent data, as well as a generalization and adaptation of principal component analysis.
Abstract: Introduction * Properties of Population Principal Components * Properties of Sample Principal Components * Interpreting Principal Components: Examples * Graphical Representation of Data Using Principal Components * Choosing a Subset of Principal Components or Variables * Principal Component Analysis and Factor Analysis * Principal Components in Regression Analysis * Principal Components Used with Other Multivariate Techniques * Outlier Detection, Influential Observations and Robust Estimation * Rotation and Interpretation of Principal Components * Principal Component Analysis for Time Series and Other Non-Independent Data * Principal Component Analysis for Special Types of Data * Generalizations and Adaptations of Principal Component Analysis

17,446 citations

Book
01 Jan 1959

7,235 citations

Book
01 Jan 1979
TL;DR: In this article, the authors present a textbook for introductory courses in remote sensing, which includes concepts and foundations of remote sensing; elements of photographic systems; introduction to airphoto interpretation; air photo interpretation for terrain evaluation; photogrammetry; radiometric characteristics of aerial photographs; aerial thermography; multispectral scanning and spectral pattern recognition; microwave sensing; and remote sensing from space.
Abstract: A textbook prepared primarily for use in introductory courses in remote sensing is presented. Topics covered include concepts and foundations of remote sensing; elements of photographic systems; introduction to airphoto interpretation; airphoto interpretation for terrain evaluation; photogrammetry; radiometric characteristics of aerial photographs; aerial thermography; multispectral scanning and spectral pattern recognition; microwave sensing; and remote sensing from space.

6,790 citations

Journal ArticleDOI
TL;DR: In this article, the concept of remote sensing elements of photogrammetry was introduced. Butterfly, thermal, and hyperspectral sensors were used to interpret multispectral, thermal and hypererspectral images.
Abstract: Concepts and Foundations of Remote Sensing Elements of Photographic Systems Basic Principles of Photogrammetry Introduction to Visual Image Interpretation Multispectral, Thermal, and Hyperspectral Sensing Earth Resource Satellites Operating in the Optical Spectrum Digital Image Processing Microwave and Lidar Sensing Appendix A: Radiometric Concepts, Terminology, and Units Appendix B: Remote Sensing Data and Information Resources Appendix C: Sample Coordinate Transformation and Resampling Procedures

6,547 citations

Frequently Asked Questions (11)
Q1. What are the contributions in "Vertex component analysis: a fast algorithm to unmix hyperspectral data" ?

This paper presents a new method for unsupervised endmember extraction from hyperspectral data, termed vertex component analysis ( VCA ). 

Principal component analysis (PCA) [45], maximum-noise fraction (MNF) [46], and singular value decomposition (SVD) [47] are three well-known projection techniques widely used in remote sensing. 

VCA is more robust to topographic modulation, since it seeks for the extreme projections of the simplex, whereas N-FINDR seeks for the maximum volume, which is more sensitive to fluctuations on . 

Since is the projection of a zero-mean Gaussian independent random vector onto the orthogonal space spanned by the columns of , then the probability of being null is zero. 

In order to estimate the number of endmembers present in the processed area, the authors resort to the virtual dimensionality (VD), recently proposed in [61]. 

The authors can see that the signal energy contained in the first eight eigenvalues is higher than 99.93% of the total signal energy, meaning that the other six endmembers only occurs in a small percentage of the subimage. 

Notice that the underling reason for generating a random vector is only to get a non null projection onto the orthogonal space generated by the columns of . 

The Dirichlet density, besidesenforcing positivity and full additivity constraints, displays a wide range of shapes, depending on the parameters . 

Three spectral signatures (A—biotite, B—carnallite, and C—ammonioalunite) were selected from the U.S. Geological Survey (USGS) digital spectral library [48] (see Fig. 2); the abundance fractions follow a Dirichlet distribution; parameter is set to 1; and the noise is zero-mean white Gaussian with covariance matrix , where is the identity matrix and leading to a SNR dB. Fig. 3(a) presents a scatterplot of the simulated spectral mixtures without projection (bands nm and nm). 

The chosen value of , assures that the colatitude angle between and any vector is between 0 and 45 , then avoiding numerical errors which otherwise would occur for angles near 90 . 

In the third experiment, the number of pixels of the scene varies, in order to illustrate the algorithm performance with the size of the covered area: as the number of pixels increases, the likelihood of having pure pixels also increases, improving the performance of the unmixing algorithms; in the fourth experiment, the algorithms are evaluated as function of the number of endmembers present in the scene; finally, in the fifth experiment, the number of floating-point operations (flops) is measured, in order to compare the computational complexity of VCA, N-FINDR, and PPI algorithms.