Total Variation Spatial Regularization for Sparse Hyperspectral Unmixing

Abstract

Spectral unmixing aims at estimating the fractional abundances of pure spectral signatures (also called endmembers) in each mixed pixel collected by a remote sensing hyperspectral imaging instrument. In recent work, the linear spectral unmixing problem has been approached in semisupervised fashion as a
sparse regression one, under the assumption that the observed image signatures can be expressed as linear combinations of pure spectra, known a priori and available in a library. It happens, however, that sparse unmixing focuses on analyzing the hyperspectral data without incorporating spatial information. In this paper, we include the total variation (TV) regularization to the classical sparse regression formulation, thus exploiting the spatial–contextual information present in the hyperspectral images and developing a new algorithm called sparse unmixing via variable splitting augmented Lagrangian and TV. Our experimental results, conducted with both simulated and real hyperspectral data sets, indicate the potential of including spatial information (through the TV term) on sparse unmixing formulations for improved
characterization of mixed pixels in hyperspectral imagery.

Full text

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4484IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 11, NOVEMBER 2012
Total Variation Spatial Regularization for
Sparse Hyperspectral Unmixing
Marian-Daniel Iordache, José M. Bioucas-Dias, Member, IEEE, and Antonio Plaza, Senior Member, IEEE
Abstract—Spectral unmixing aims at estimating the fractional
abundances of pure spectral signatures (also called endmembers)
in each mixed pixel collected by a remote sensing hyperspectral
imaging instrument. In recent work, the linear spectral unmixing
problem has been approached in semisupervised fashion as a
sparse regression one, under the assumption that the observed
image signatures can be expressed as linear combinations of pure
spectra, known a priori and available in a library. It happens,
however, that sparse unmixing focuses on analyzing the hyper-
spectral data without incorporating spatial information. In this
paper, we include the total variation (TV) regularization to the
classicalsparseregressionformulation,thusexploitingthespatial–
contextual information present in the hyperspectral images and
developing a new algorithm called sparse unmixing via variable
splitting augmented Lagrangian and TV. Our experimental results,
conducted with both simulated and real hyperspectral data sets,
indicate the potential of including spatial information (through
the TV term) on sparse unmixing formulations for improved
characterization of mixed pixels in hyperspectral imagery.
Index Terms—Hyperspectral imaging, sparse regression, sparse
unmixing, spectral unmixing, total variation (TV) regularization.
I. INTRODUCTION
S
[2] is a standard technique for spectral mixture analysis that
infers a set of pure spectral signatures, called endmembers [3],
[4], and the fractions of these endmembers, called abundances
[5], in each pixel of the scene. This model assumes that the
spectra collected by the imaging spectrometer can be expressed
in the form of a linear combination of endmembers, weighted
by their corresponding abundances. Because each observed
spectral signal is the result of an actual mixing process, it is
expected that the driving abundances satisfy two constraints,
i.e., they should be non-negative [6], and the sum of abundances
PECTRAL unmixing is an important technique for hy-
perspectral data exploitation [1]. Linear spectral unmixing
Manuscript received July 15, 2011; revised December 26, 2011;
accepted February 22, 2012. Date of publication May 7, 2012; date of cur-
rent version October 24, 2012. This work was supported by the European
Community’s Marie Curie Research Training Networks Programme under
contract MRTN-CT-2006-035927, Hyperspectral Imaging Network (HYPER-
I-NET). Funding from the Spanish Ministry of Science and Innovation (CEOS-
SPAIN project, reference AYA2011-29334-C02-02) and from the Portuguese
Science and Technology Foundation, projects PEst-OE/EEI/LA0008/2011 and
PTDC/EEA-TEL/104515/2008, is also gratefully acknowledged.
M.-D. Iordache is with the Flemish Institute for Technological Research
(VITO), Centre for Remote Sensing and Earth Observation Processes (TAP),
2400 Mol, Belgium (e-mail: iordache@unex.es).
J. M. Bioucas-Dias is with the Instituto de Telecomunicações, Instituto
Superior Técnico, 1049-1 Lisbon, Portugal (e-mail: bioucas@gmail.com).
A. Plaza is with the Hyperspectral Computing Laboratory, Department of
Technology of Computers and Communications, Escuela Politécnica, Univer-
sity of Extremadura, 10071 Cáceres, Spain (e-mail: aplaza@unex.es).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TGRS.2012.2191590
for a given pixel should be unity [7]. Although the linear
model has practical advantages, such as ease of implementation
and flexibility in different applications, nonlinear unmixing
describes mixed spectra (in physical [8], [9], or statistical [10]
sense) by assuming that part of the source radiation is multiply
scattered before being collected at the sensor. The distinction
between linear and nonlinear unmixing has been widely studied
in recent years [11].
The exploitation of the more general linear mixture model,
in spite of its simplicity, has fostered a large amount of research
leading to a plethora of endmember identification algorithms
based on geometrical and statistical approaches [12]–[37]. A
recent trend is the design of algorithms that do not assume the
presence of pure signatures in the input data [38]–[42]. How-
ever, these algorithms provide virtual endmembers [43] (not
necessarily present in the set comprised by input data samples).
A recently developed approach to tackle the problems related
to the unavailability of pure spectral signatures is to model
mixed pixel observations as linearcombinations of spectrafrom
a library collected on the ground by a field spectro-radiometer.
Unmixing then amounts to finding the optimal subset of signa-
tures in a (potentially very large) spectral library that can best
model each mixed pixel in the scene [44]. In practice, this is a
combinatorial problem that calls for efficient sparse regression
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. To cope with these issues,
in previous work we have resorted to fast algorithms such
as the sparse unmixing via variable splitting and augmented
Lagrangian (SUnSAL) [45] and a constrained version of the
same algorithm (CSUnSAL) [45], which exploit the alternating
direction method of multipliers (ADMM) [46] in a way similar
to [47] and [48].
Sparse unmixing techniques have been increasingly used in
recentyears.In[49],anewmethodforsubpixelmodeling,map-
ping, and classification of hyperspectral images is presented. It
uses learned block-structured discriminative dictionaries [50],
where each block is adapted and optimized to represent a
specific material in a compact and sparse manner. In [51], a
sparsity constraint is included in non-negative matrix factoriza-
tion, a widely used technique to unmix hyperspectral images
and recover the material endmembers [52], [53]. Although
sparse unmixing techniques have been shown to exhibit good
potential for the characterization of mixed pixels using spectral
libraries, they focus mostly on exploiting the spectral infor-
mation available in the hyperspectral data. Such information
deals with pixel vectors in an independent manner from the
neighboring pixel values, whereas spatial information concerns
the relationship between each pixel vector and its neighbors.
0196-2892/$31.00 © 2012 IEEE

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IORDACHE et al.: TOTAL VARIATION SPATIAL REGULARIZATION4485
This means that, despite the inherent spatial-spectral duality
that resides in hyperspectral scenes, the classic sparse unmixing
approach uses only spectral information and disregards spatial-
contextual information. Previous works showed that including
spatial information, both as a preprocessing step [54] and in
the unmixing procedure itself [55], [56], has a positive impact
on the accuracy of the estimated fractional abundances. On
the other hand, the success of unmixing when performed via
sparse regression techniques relies on the mutual coherence
[57] of the spectral library1and the number of materials in the
mixtures, i.e., the so-called degree of sparsity [44]. In favorable
scenarios, we have low degree of sparsity and low coherence.
In hyperspectral applications, the former is often true, but the
latter is not.
In order to sidestep the limitations imposed to sparse re-
gression by the high correlation of spectral libraries, in this
paper, we develop a new algorithm called sparse unmixing via
variable splitting augmented Lagrangian and total variation
(SUnSAL-TV). This method includes spatial information on
the sparse unmixing formulation by means of the TV reg-
ularizer [59]–[61]. This regularizer accounts for spatial ho-
mogeneity: it is very likely that two neighboring pixels have
similar fractional abundances for the same endmember. The
TV regularizer acts as a priori information, which improves
the conditioning of the underlying inverse problem [62]. At the
end, unmixing is obtained by solving a large nonsmooth convex
optimization problem. It should be noted that the work [63]
introduces a hyperspectral unmixing approach formally similar
to ours. There is, however, a major conceptual difference:
whereas we are solving a sparse regression problem based on
a library, in [63], the endmembers are jointly inferred with the
fractional abundances. Recall that the main rationale for our
approach is avoiding the endmember estimation step. A vector
TV was also used in [60] to unmix and increase the resolution
of hyperspectral images. Again, there is a major conceptual
difference with respect to our approach: whereas we are solving
a sparse regression problem based on a library, the endmembers
in [60] are inferred using an external algorithm, in this case the
N-FINDR [17].
Theremainderofthepaperisstructuredasfollows.SectionII
describes the linear versus the sparse unmixing formulations.
Section III describes the newly developed SUnSAL-TV algo-
rithm. Section IV describes our experimental results with sim-
ulated hyperspectral data sets. Section V describes experiments
with real hyperspectral data. Section VI concludes with some
remarks and hints at plausible future research lines.
II. LINEAR VERSUS SPARSE UNMIXING
A. Linear Spectral Unmixing
The linear mixture model assumes that the spectral response
of a pixel is a linear combination of all the pure spectral
signatures (endmembers) present in the pixel. For each pixel,
the linear model can be written as follows:
yi=
q
?
j=1
mijαj+ ni
(1)
1The mutual coherence of a library is closely related with the so-called
restricted isometric property [58],
where yiis the measured value of the reflectance at spectral
band i,mijis the reflectance of the jth endmember at spectral
band i,αjis the fractional abundance of the jth endmember,
and nirepresents the error term for the spectral band i (i.e., the
noise affecting the measurement process). If we assume that the
hyperspectral sensor used for data acquisition has L spectral
bands, (1) can be rewritten in compact form as:
y = Mα + n
(2)
where y is an L × 1 column vector (the measured spectrum
of the pixel), M is an L × q matrix containing q endmembers,
α is a q × 1 vector containing the fractional abundances of the
endmembersinthepixel,andnisanL × 1vectorcollectingthe
errorsaffectingthemeasurementsateachspectralband.Theso-
called abundance non-negativity constraint (ANC): αi≥ 0 for
i = 1,...,q, and the abundance sum-to-one constraint (ASC):
?q
patible with α) are often imposed into the model described in
(1), owing to the fact that αi, for i = 1,...,q, represent the
fractions of the endmembers present in the considered pixel [5].
In a typical hyperspectral unmixing scenario, we are given a
setY ≡ {yi∈ RL,i = 1,...,n} of nobserved L-dimensional
spectral vectors, and the objective is to estimate the endmember
matrix M and the fractional abundances α for every pixel
in the scene.
i=1αi= 1, respectively, represented in compact form by
α ≥ 0 and 1Tα = 1 (where 1Tis a line vector of 1’s com-
B. Sparse Unmixing
Sparse unmixing reformulates (2) assuming the availability
of a library of spectral signatures a priori as follows:
y = Ax + n
(3)
where x is the fractional abundance vector compatible with
library A ∈ RL×m. Due to the fact that only a few of the sig-
natures contained in A are likely contributing to the observed
spectrum, x contains many values of zero, which means that
it is sparse. An important indicator regarding the difficulty to
infer correct solutions for a linear system of equations is the so-
called mutual coherence, defined as the largest cosine between
any two columns of A. It has been shown that the quality of
the solution of a linear system of equations decreases when
the mutual coherence increases. As shown in [44], the mutual
coherence of hyperspectral libraries tend to be close to one.
In [44], we present detailed formulations of sparse unmixing
under the form of different convex optimization problems,
for both noiseless and noisy environments. In this paper, we
consider only noisy observations due to their most realistic na-
ture. With these considerations in mind, the unmixing problem
can be formulated as an l2− l0norm optimization problem,
in which the observations are affected by noise and the ANC
is enforced
min
x
1
2?Ax − y?2
2+ λ?x?0
subject to
x ≥ 0
(4)
where ?x?0represents the so-called l0norm of the vector x,
which simply counts the nonzero components of x, and λ is
a regularization parameter which weights the two terms of the
objectivefunction.Thisformulationhasasimpleinterpretation:

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4486 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 11, NOVEMBER 2012
welookforthesparsestnon-negativesolutionxwhichperfectly
explains the data y, given the spectral library A. It should be
noted that, in the SUnSAL-TV formulation developed in this
paper, we explicitly enforce the ANC constraint but not the
ASC constraint. This is because, following the reasoning in
[44], the ASC is prone to strong criticisms in the literature,
while the ANC alone automatically imposes a generalized ASC
(see [44] for more details). The optimization problem in (4) is
equivalent to
min
x
?x?0
subject to
Ax = yx ≥ 0.
(5)
Both problems (4) and (5) are nonconvex and difficult to
solve. However, under certain conditions [57], the l0norm can
be replaced by the l1norm. Moreover, since we are interested
in solving problems in which the observations are affected by
noise, we allow a small reconstruction error for the solution
in (5). With these assumptions in mind, the two objective
functions, respectively, become
min
x
1
2?Ax − y?2
2+ λ?x?1
subject to
x ≥ 0
(6)
min
x
?x?1
subject to
?Ax − y?2≤ δ,x ≥ 0
(7)
where δ is the tolerated reconstruction error. The optimization
problems in (6) and (7) are convex and equivalent. This means
that, for any given parameter λ, a corresponding parameter
δ can be found which leads to the same unmixing solution.
In the following section, we describe a new algorithm called
SUnSAL-TV which efficiently solves the sparse unmixing
problem taking into account the relationship between each pixel
vector and its neighbors, including the TV regularizer [59]–[61]
on top of the aforementioned sparse unmixing formulation.
III. TOTAL VARIATION FOR SPARSE UNMIXING—THE
SUNSAL-TV ALGORITHM
Let Y ∈ RL×nbe the observed data matrix (where each
column contains the observed spectrum of a given pixel); let
X ∈ Rm×nbe the fractional abundances matrix; let ?X?F≡
?
and λTV ≥ 0 be regularization parameters. With these defini-
tions in place, we can now carry out the sparse unmixing by
solving the following optimization problem
trace{XXT} be the Frobenius norm of X; let ?X?1,1≡
?n
i=1?xi?1(xidenotes the ith column of X); and let λ ≥ 0
min
X
1
2?AX − Y?2
F+ λ?X?1,1+ λTVTV (X),
subject to
X ≥ 0
(8)
where
TV (X) ≡
?
{i,j}∈ε
?xi− xj?1
(9)
is a vector extension of the nonisotropic TV [59], [61], which
promotes piecewise constant (or smooth) transitions in the frac-
tional abundance of the same endmember among neighboring
pixels, and ε denotes the set of horizontal and vertical neighbors
in the image. The minimization in (8) with λTV set to zero is
a constrained basis pursuit denoising (CBPDN) problem [64]
applied to each individual pixel. The application of CBPDN
and several other constrained sparse regression algorithms to
hyperspectral unmixing was extensively studied in [44].
It should be noted that the optimization problem in (8),
although convex, is very hard to solve owing to nonsmooth
terms and its huge dimensionality. For instance, if we
consider a 256 × 256-pixel image and a spectral library with
500 materials, then X ∈ R500×2562, which is about 40 million
variables! To solve the problem in (8), we have modified the
SUnSAL algorithm [45] following closely the methodology
introduced in [62]. The core idea is to introduce a set of new
variables per regularizer and then use the ADMM method [46]
to solve the resulting constrained optimization problem. By
a careful choice of the new variables, the initial problem is
converted into a sequence of much simpler problems. Next, we
formalize the SUnSAL-TV algorithm (more details are given
in an Appendix at the end of the paper).
Let Hh: Rm×n→ Rm×ndenote a linear operator com-
puting the horizontal differences between the components
of X corresponding to neighboring pixels; i.e., HhX =
[d1,d2,...,dn], where di= xi− xih, with i and ihdenoting
a pixel and its horizontal neighbor. We are assuming periodic
boundaries. Let Hv: Rm×n→ Rm×nbe defined in a similar
way for the vertical diferences; i.e., HvX = [v1,v2,...,vn],
where vi= xi− xiv, with i and ivdenoting a pixel and its ver-
tical neighbor. With these two difference operators, we define
?HhX
With these definitions in place, an equivalent way of writing
the optimization problem (8) is
HX ≡
HvX
?
.
min
X
where ιR+(X) =?n
belongs to the nonnegative orthant and +∞ otherwise).
Given the objective function (10), we write the following
(constrained) equivalent formulation:
1
2?AX−Y?2
F+λ?X?1,1+λTV?HX?1,1+ιR+(X) (10)
i=1ιR+(xi) is the indicator function (xi
represents the ith column of X and ιR+(xi) is zero if xi
min
U,V1,V2,V3,V4,V5
+ιR+(V5),
subject to
1
2?V1−Y?2
F+λ?V2?1,1+λTV?V4?1,1
V1=AU
V2=U
V3=U
V4=HV3
V5=U.
(11)
In (11), notice the asymmetry of the constraint V4= HV3,
compared with the remaining constraints. As seen below, this
asymmetry underlies large computational gains by decoupling
the optimization in the spatial domain from the optimization
in the spectral domain. Optimization (11) can be written in a
compact form as follows:
min
U,Vg(V)
subject to
GU + BV = 0
(12)

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IORDACHE et al.: TOTAL VARIATION SPATIAL REGULARIZATION4487
Fig. 1.
endmember; (d) abundances of endmember; (e) abundances of endmember; (f) abundances of endmember.
True fractional abundances of endmembers in the simulated data cube 1 (DC1). (a) Simulated image; (b) abundances of endmember; (c) abundances of
where
V≡(V1,V2,V3,V4,V5),
g(V)≡1
⎡
⎣
The ADMM algorithm for the formulation (12) is shown in
Algorithm 1, where (see [62] and [65]):
L(U,V,D) ≡ g(U,V) +μ
2?V1−Y?2
A
I
I
0
I
F+λ?V2?1,1+λTV?V4?1,1+ιR+(V5)
⎡
⎣
G =
⎢⎢⎢
⎤
⎦
⎥⎥⎥
B =
⎢⎢⎢
−I
0
0
0
0
00
0
0
0
0
0
0
0
0
−I
0
0
0
−I
H
0
−I
0
−I
⎤
⎦.
⎥⎥⎥
(13)
2?GU + BV − D?2
F
(14)
is the augmented Lagrangian for problem (12), μ > 0 is a
positive constant, and D/μ denotes the Lagrange multipliers
associated to the constraint GU + BV = 0. In each iteration,
Algorithm 1 sequentially optimizes L with respect to U (step 3)
and V (step 4), and then updates the Lagrange multipliers
(step 5).
Algorithm 1 ADMM pseudocode for solving problem (12).
1. Initialization: set k = 0, choose μ > 0, U(0), V(0), D(0)
2. repeat:
3.
U(k+1)← argminUL(U,V(k),D(k))
4.
V(k+1)← argminVL(U(k+1),V,D(k))
5.
D(k+1)← D(k)− GU(k+1)− BV(k+1)
6. until some stopping criterion is satisfied.
Matrix G is full column rank and function g introduced
in (12) is closed, proper, and convex. Then, [46, Theorem 1]
ensures that, for any μ > 0, if (12) has a solution, say U∗, then
the sequence {U(k)}, converges to U∗. If (12) does not have a
solution, then at least one of the sequences {U(k)} or {D(k)}
diverges. As stopping criterion, we use ?GU(k)+ BV(k)?F≤
ε. The convergence speed of the ADMM algorithm depends
on a suitable choice of parameter μ. This is an active research
topic. For example, previous work in [66], [67] (see also [68])
uses an adaptive scheme (based on the primal and the dual
ADMM variables) that performs very well in our case. In this
scheme, μ is updated with the objective of keeping the ratio
between the ADMM primal and dual residual norms within a
given positive interval, as they both converge to zero.
We present in the Appendix the details of the optimizations
with respect to U and V of the ADMM Algorithm 1, which we
term SUnSAL-TV. Here, we make, however, a few pertinent
observations. The optimization U amounts at solving a linear
system of equations of size m × m. The matrix involved in
this system of equations is fixed and then can be precomputed
involving low complexity as the rank of A is min{L,m}. The
optimization with respect to V is decoupled with respect to V1,
V2, (V3,V4), and V5. The reason for the coupling between
V3and V4is the asymmetry already referred to in variable
splitting introduced in (11). If we had followed [62] exactly,
we would have V3= HU instead of the couple V3= U
and V4= HV3. However, in this case, the optimization with
respect to U would be unbearable, as we would have matrix A
acting over the spectral dimension and matrix H acting over the
spatial domain in the same system.
As shown in the Appendix, the optimizations with respect to
V3alone and to V4alone are very light (the same is true for
V1, V2, and V5). Therefore, a very simple way to achieve joint
optimization with respect to (V3,V4) is to cycle over these two
optimizations until convergence. In our SUnSAL-TV algorithm
we cycle only once. We have two reasons for this decision:
1) there is no need for an exact solution of optimizations with
respect to U (step 3) and V (step 4) [46, Theorem 1], as long as
the errors are summed; and 2) we have observed systematically
faster convergence with just one step than with more steps. A

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4488 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 11, NOVEMBER 2012
Fig. 2.True fractional abundances of endmembers in the simulated data cube 2 (DC2).
proof of the convergence of SUnSAL-TV with just one step is,
however, beyond the scope of this paper.
Concerning computational complexity, we show in the Ap-
pendix that each iteration of SUnSAL-TV has complexity
O(L2n) + O(Lnlogn). Given that L is usually larger than
nlogn, the first term dominates the total complexity of the
algorithm.
Before concluding this section, it is important to emphasize
that, in this work, we test the SUnSAL-TV formulation
using an exhaustive range of values for parameters λ and
λTV, including zero. Hereinafter, we term the problem in
(8) with λ = 0 as nonnegative constrained least squares TV
(NCLS-TV) and only nonnegative constrained least squares
(NCLS) [5] when λ = λTV = 0.
Inthefollowingsections,weprovideexperimentalresultsus-
ing both simulated and real hyperspectral data sets to illustrate
the potential advantages of including TV regularization on top
of sparse unmixing formulations, including both SUnSAL and
NCLS.
IV. EXPERIMENTS WITH SIMULATED DATA
In this section, we illustrate the unmixing performance
achieved by including the TV regularizer on top of sparse
regression formulations for spectral unmixing using two sim-
ulated hyperspectral data sets.
The goal is to analyze the influence of the TV regularizer
in the unmixing results, both for solutions which are explicitly
constrained to be sparse, and also for the ones computed with-
out enforcing the sparsity explicitly. The section is organized
as follows. Section IV-A describes how the simulated data
sets have been generated. We consider only scenes affected by
noise, as the noiseless case is trivial. Section IV-B describes
the considered performance discriminators. Section IV-C con-
cludes with a summary of the most relevant aspects observed in
our simulated data experiments.
A. Simulated Data Sets
We have considered two spectral libraries in our simulated
image experiments: A1∈ R224×240and A2∈ R224×230. The
first library is generated from a random selection of 240 mate-
rials (different mineral types) from the USGS library, denoted
splib062and released in September 2007. It comprises spectral
signatures with reflectance values given in 224 spectral bands
and distributed uniformly in the interval 0.4–2.5 μm. The sec-
ondlibraryisobtainedfromarandomselectionof230materials
from the ASTER library,3a compilation of over 2400 spectra
2Available online: http://speclab.cr.usgs.gov/spectral.lib06
3Available online: http://speclib.jpl.nasa.gov

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IORDACHE et al.: TOTAL VARIATION SPATIAL REGULARIZATION4489
TABLE I
SRE(dB) VALUES ACHIEVED AFTER APPLYING DIFFERENT UNMIXING METHODS TO SIMULATED DATA (THE OPTIMAL
PARAMETERS FOR WHICH THE REPORTED VALUES WERE ACHIEVED ARE INDICATED IN THE PARENTHESES)
of natural and man-made materials. Specifically, each of the
members of this library has reflectance values measured for 224
spectral bands distributed uniformly in the interval 3–12 μm.
Themutualcoherences[57]ofthetwolibrariesareverycloseto
1. These libraries were used to generate two different simulated
hyperspectral data cubes.
1) Simulated Data Cube 1 (DC1): This simulated data cube
was generated using five randomly selected spectral signatures
from A1. DC1 has 75 × 75 pixels and 224 bands per pixel.
The data were generated using a linear mixture model, with five
randomly selected signatures as the endmembers and imposing
the ASC in each simulated pixel. In the resulting simulated
image, shown in Fig. 1(a), there are pure regions as well as
mixed regions constructed using mixtures ranging between
two and five endmembers, distributed spatially in the form of
distinct square regions. Fig. 1(b)–(f), respectively, show the
true fractional abundances for each of the five endmembers.
The background pixels are made up of mixtures of the same
five endmembers, but this time their respective fractional abun-
dances values were randomly fixed to values 0.1149, 0.0741,
0.2003, 0.2055, and 0.4051, respectively. After generating DC1
following the procedure described above, the scene was con-
taminated with white noise (on the one hand) and also with
spectrally correlated noise (on the other hand) resulting from

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4490IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 11, NOVEMBER 2012
TABLE II
psVALUES VALUES ACHIEVED AFTER APPLYING DIFFERENT UNMIXING METHODS TO SIMULATED DATA (THE OPTIMAL
PARAMETERS FOR WHICH THE REPORTED VALUES WERE ACHIEVED ARE INDICATED IN THE PARENTHESES)
low-pass filtering i.i.d. Gaussian noise, using a normalized
cutoff frequency of 5π/L, for three levels of the signal-to-
noise ratio (SNR ≡ E?Ax?2/E?n?2
and 40 dB.
2) Simulated Data Cube 2 (DC2): This simulated data cube
contains 100 × 100 pixels generated using nine randomly
selected signatures from A2. The fractional abundances satisfy
the ANC and the ASC and are piecewise smooth, i.e., they are
smooth with sharp transitions, as shown in Fig. 2. The resulting
observations exhibit spatial homogeneity as described in Fig. 2,
which shows the true abundances of the endmembers. After
2), i.e., 20 dB, 30 dB,
generating DC2 following the procedure described above, the
scene was again contaminated with both white and correlated
noise using the same SNR values adopted for DC1.
B. Performance Discriminators
The performance discriminator adopted in this work to
measure the quality of the reconstruction of spectral mixtures
is the signal to reconstruction error [44]: SRE ≡ E[?x?2
E[?x − ? x?2
2]/
2], measured in dB: SRE(dB) ≡ 10log10(SRE). We
use this measure instead of the classical root mean square error

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IORDACHE et al.: TOTAL VARIATION SPATIAL REGULARIZATION4491
Fig. 3.
(white noise); (b) SNR = 30 dB (white noise); (c) SNR = 40 dB (white noise); (d) SNR = 20 dB (correlated noise); (e) SNR = 30 dB (correlated noise);
(f) SNR = 40 dB (correlated noise).
SRE(dB) as a function of parameters λ and λTV for DC1 with different SNR levels (top: white noise; bottom: correlated noise). (a) SNR = 20 dB
Fig. 4.
(white noise); (b) SNR = 30 dB (white noise); (c) SNR = 40 dB (white noise); (d) SNR = 20 dB (correlated noise); (e) SNR = 30 dB (correlated noise);
(f) SNR = 40 dB (correlated noise).
SRE(dB) as a function of parameters λ and λTV for DC2 with different SNR levels (top: white noise; bottom: correlated noise). (a) SNR = 20 dB
[54] as it gives more information regarding the power of the
error in relation with the power of the signal. The higher the
SRE(dB), the better the unmixing performance.
Wealsocomputeaso-calledprobabilityofsuccess,ps,which
is an estimate of the probability that the relative error power be
smaller than a certain threshold. This metric is a widespread

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4492 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 11, NOVEMBER 2012
Fig. 5.Abundance maps obtained by different unmixing methods for endmember #5 in DC1 (white noise).
one in the sparse regression literature, and is formally defined
as ps≡ P(?? x − x?2/?x?2≤ threshold). For example, if we
ability one, less than 1/10. This gives an indication about the
stability of the estimation that cannot be directly derived from
the SRE (which is an average). In the following, we assume,
based on experiments reported in our previous work [44],
that the estimation result can be considered successful when
?? x − x?2/?x?2≤ 3.16 (5 dB). In all the tests using the TV
hood system.
set threshold = 10 and get ps= 1, this means that the total
relative error power of the fractional abundances is, with prob-
regularizer on top of sparse unmixing approaches (SUnSAL-
TV and NCLS-TV), we considered a first-order pixel neighbor-
C. Results and Discussion
In this section, we test the performance of the proposed
TV regularizer combined with sparse unmixing formulations
using the two simulated data cubes DC1 and DC2. We also
include the original SUnSAL and NCLS formulations. The
algorithms were tested using different values of the parameters
λ and λTV : 0, 5 · 10−4, 10−3, 5 · 10−3, 0.01, 0.05, 0.1, 0.3,
0.5, and 1. All possible combinations of these parameters were
considered. Table I shows the SRE(dB) results achieved by
the different tested methods with the two considered simulated
data sets, using all considered SNR levels. On the other hand,
Table II shows the psresults achieved by the different methods
for the same data cubes. In both tables, we only report the
best scores obtained across the considered parameter range (the
optimal parameters for which the reported values were obtained
are indicated in the parentheses).
From Tables I and II, we can conclude that the inclusion
of the TV regularizer offers the potential to improve unmixing
performance in two different analysis scenarios, i.e., when the
sparsity is imposed explicitly (SUnSAL), and also when it is
not enforced (NCLS). For high SNR values, the improvements

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IORDACHE et al.: TOTAL VARIATION SPATIAL REGULARIZATION4493
Fig. 6.Abundance maps obtained by different unmixing methods for endmember #9 in DC2 (white noise).
obtained with regard to the standard sparse unmixing for-
mulations (SUnSAL and NCLS) are not significant. This is
due to the fact that, with low noise conditions, NCLS and
the l2− l1norm optimization solution are able to recover the
fractional abundances with good accuracy. However, as the
noise increases, the spatial term becomes more important and
improves significantly the quality of unmixing results as it
can be observed in the results obtained by SUnSAL-TV and
NCLS-TV for SNR = 20 dB in Tables I and II. Note also
that the NCLS-TV and SUnSAL-TV provide results which
are not very different, which means that the TV regularizer
imposes intrinsically a kind of sparsity in the solutions. Al-
though this is expected to happen, the sparse regularizer is
still important as it can be seen in the scenarios dominated by
high noise, in which SUnSAL-TV performs clearly better than
NCLS-TV.
For illustrative purposes, Figs. 3 (DC1) and 4 (DC2) show
the computed values of SRE(dB) (as a function of both pa-
rameters λ and λTV) for observations affected by both white
and correlated noise. Here, we do not consider values larger
than 0.3 for the two aforementioned parameters as we have
experimentally observed that such values lead to poor perfor-
mances in our experiments. Since the plots obtained for the
ps metric are very similar, we do not report them here for
space considerations. From Figs. 3 and 4, it can be observed
that, for the two considered scenes with different SNR levels,
the best unmixing performances were achieved for relatively
small values of the parameters. The improvements in unmixing
performance resulting from the inclusion of the TV regularizer
are more apparent when the SNR is low, while the performance
of all methods becomes more similar as the SNR is increased.
These observations are in line with those already reported in
Tables I and II.
Figs. 5 and 6, respectively, show the abundance maps esti-
mated for one randomly selected endmember in DC1 and DC2
(considering different noise levels). Since the abundance maps

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4494 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 11, NOVEMBER 2012
Fig. 7.
with SNR of 30 dB (leftmost column), and the same results for 100 selected pixels marked in a green rectangle in the upper-leftmost plot (rightmost column).
Ground-truth and estimated abundances obtained for each endmember material in the A1library in the the scene DC1 affected by white noise, simulated
estimated for other endmembers exhibited similar behavior, we
only report the results observed for endmember #5 in DC1
(see Fig. 1) and for endmember #9 in DC2 (see Fig. 2). The
abundance maps displayed in Figs. 5 and 6 were obtained
using optimal values for parameters λ and λTV (see Tables I
and II). From Figs. 5 and 6, it can be seen that the spatial
term based on the TV regularizer improves both the NCLS
and the SUnSAL solutions. In qualitative fashion, we can

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IORDACHE et al.: TOTAL VARIATION SPATIAL REGULARIZATION4495
observe that the regions with high fractional abundance of
the considered endmember are better delineated, while mixed
regions with low concentration of the considered endmember
are more homogeneous in nature. No matter the level of noise,
the solutions get spatial consistency and the spatial distribution
of the materials is determined with good accuracy in both cases.
Sparse solutions are particularly needed when the SNR is low,
as reported in Tables I and II.
Fig. 7 shows the fractional abundance estimations obtained
for each endmember material in the A1 library (as a func-
tion of the pixel index in the scene DC1 simulated with
SNR of 30 dB and white noise) by the methods, along with
the ground-truth abundances. To facilitate the visualization,
background pixels are removed from the plots in the leftmost
column of Fig. 7. In order to better visualize the impact of
applying the TV regularizer, the rightmost column of Fig. 7
shows the same abundances for only 100 selected pixels (in-
dicated by the green rectangle in the upper-leftmost plot in
Fig. 7). The parameters used correspond to the ones reported
in Table I.
FromFig.7,itcanbeseengraphicallythattheTVregularizer
imposes spatial consistency in the unmixing results. The lines
(denoting the abundance of a certain endmember in all pixels
of the image) estimated by SUnSAL-TV are more similar to
those in the ground-truth than the ones estimated by SUnSAL.
After applying SUnSAL, there are many low abundance values
estimated for endmembers which are not actually present in
the image, but the TV regularizer vanishes those values and
provides an overall estimate which is closer to ground-truth val-
ues. Note that, although NCLS provides fractional abundances
larger than one, the accuracy of the results is clearly improved
when the TV regularizer is used. It is also remarkable that,
at first sight, it would appear that all the methods encounter
problems in accurately locating one endmember (i.e., the one
situated around line 150 of the upper-leftmost plot in Fig. 7).
However, we must note that the spectral signature of this
endmember corresponds to the muscovite mineral, and there are
11 spectra in the library corresponding to different variations
of this mineral (all very similar in spectral terms—within
2.52◦—and located around line 150 of the upper-leftmost plot
in Fig. 7). As a result, we conclude that the abundance estimates
providedbyNCLS-TVandbySUnSAL-TVarehighlyaccurate
and indeed correlated with the ground-truth ones. With the
aforementioned observations in mind, we believe that the com-
bination of the TV regularizer with sparse unmixing methods
offers promising results. Although the results obtained with
simulated data sets are quite encouraging and revealing of the
potential of including spatial information in sparse unmixing
formulations, further experiments should be conducted with
real hyperspectral scenes in order to fully substantiate our
findings in real analysis scenarios.
To conclude this section, Table III illustrates the per-pixel
execution times for the algorithms compared in this section.
The values reported correspond to the average times, measured
after processing 600 pixels randomly chosen from DC1 and
DC2 (300 pixels from each datacube, 100 for each noise
level), while the spectral libraries used were the ones shown in
Section IV-A. The algorithms were implemented using Matlab7
on a desktop PC equipped with an Intel Core 2 Quad CPU (at
2.33 GHz) and 4 GB of RAM memory.
TABLE III
AVERAGE PER-PIXEL PROCESSING TIMES (IN SECONDS) MEASURED
AFTER APPLYING THE CONSIDERED UNMIXING ALGORITHMS TO
600 RANDOMLY CHOSEN PIXELS FROM SIMULATED IMAGES
DC1 AND DC2, ON A DESKTOP PC EQUIPPED WITH
AN INTEL CORE 2 QUAD CPU (AT 2.33 GHz)
AND 4 GB OF RAM MEMORY
V. EXPERIMENTS WITH REAL DATA
The scene used in our real data experiments is the well-
known AVIRIS Cuprite data set, available online in reflectance
units.4This scene has been widely used to validate the perfor-
mance of endmember extraction algorithms. The portion used
in experiments corresponds to a 350 × 350-pixel subset of the
sector labeled as f970619t01p02_r02_{s}c03.a.rfl in the online
data. The scene comprises 224 spectral bands between 0.4 and
2.5 μm, with nominal spectral resolution of 10 nm. Prior to
the analysis, bands 1–2, 105–115, 150–170, and 223–224 were
removed due to water absorption and low SNR in those bands,
leaving a total of 188 spectral bands. The Cuprite site is well
understood mineralogically and has several exposed minerals
of interest, all included in the USGS library considered in
experiments, denoted splib065and released in September 2007.
In our experiments, we use spectra obtained from this library
as input to the unmixing methods described in Section III. For
illustrative purposes, Fig. 8 shows a mineral map produced in
1995 by USGS, in which the Tricorder 3.3 software product
[69] was used to map different minerals present in the Cuprite
mining district6. It should be noted that the Tricorder map is
only available for hyperspectral data collected in 1995, while
the publicly available AVIRIS Cuprite data was collected in
1997. Therefore, a direct comparison between the 1995 USGS
map and the 1997 AVIRIS data is not possible. However, the
USGS map serves as a good indicator for qualitative assessment
of the fractional abundance maps produced by the different
unmixing algorithms discussed in Section III.
Before unmixing the AVIRIS Cuprite hyperspectral data,
we addressed possible calibration mismatches between the real
image spectra and the spectra available in the library. This is
because, even though we are working with atmospherically
corrected data in reflectance units, these calibration interferers
are still present due to the rather different acquisition conditions
of the two data types. In order to minimize these mismatches,
we applied a band-dependent correction strategy to the original
data set, which amounts at replacing the data set Y with CY,
where C is a diagonal matrix that minimizes the modeling
error, i.e.,
?C = argmin
C,X≥0,1T
mX=1T
n
?A1X − CY?2
(15)
where X ≥ 0 is the fractional abundance matrix. The problem
in (15) is nonconvex and, thus, very hard to solve exactly.
In this paper,we have computed a suboptimal solution to this
4http://aviris.jpl.nasa.gov/html/aviris.freedata.html
5http://speclab.cr.usgs.gov/spectral.lib06
6http://speclab.cr.usgs.gov/cuprite95.tgif.2.2um_map.gif

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4496IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 11, NOVEMBER 2012
Fig. 8.
http://speclab.cr.usgs.gov/cuprite95.tgif.2.2um_map.gif.
USGS map showing the location of different minerals in the Cuprite mining district in Nevada. The map is available online at
Fig. 9.
spectral libraries: (a) A1, and (b) A.
Plot of diagonal values of the correction matrix C for the following
problem by alternating the minimization with respect to C
and to X [44]. In our experiments, we used the matrix A1
described in Section IV-A and also a library (denoted by A)
which contains 498 minerals randomly selected from the USGS
library. The correction matrices corresponding to A1and A
are, respectively displayed in Fig. 9(a) and (b).
Fig. 10 shows a visual (qualitative) comparison between
the fractional abundance maps estimated for the minerals:
alunite, buddingtonite and chalcedony by applying the NCLS,
NCLS-TV, SUnSAL, and SUnSAL-TV algorithms to the
AVIRIS Cuprite scene using the library A1. These minerals
are known to be present (in prominent fashion) in the Cuprite
mining district. For comparative purposes, the spatial distri-
bution maps of these materials extracted from the Tricorder
software product in Fig. 8 are also displayed in Fig. 10. On
the other hand, Fig. 11 displays the abundance maps estimated
for the same minerals using the different unmixing algorithms
considered in this work and the spectral library A. The pa-
rameters used in all cases (after empirical optimization) were
λ = λTV = 10−3for NCLS-TV, SUnSAL, and SUnSAL-TV.
For the NCLS, both λ, and λTV were set to zero.
From Figs. 10 and 11, it can be observed that the abundance
maps obtained by the methods with the TV spatial regularizer
exhibit good spatial consistency of minerals of interest and less
outliers than the maps without such regularizer. This can be
appreciated, for instance, in the buddingtonite maps in both
figures. In this particular mineral, the TV term helps both
NCLS-TVandSUnSAL-TVtoreducetowardzerosomeoutlier
values of high abundance of the mineral in isolated regions of
the image that can be observed for the NCLS and SUnSAL
results. In this case, the TV term allows obtaining a better
characterization of the buddingtonite mineral (which appears
as an anomaly in the scene) in accordance with the Tricorder
3.3 software product.
Another important observation from Figs. 10 and 11 is that
the results obtained by sparse unmixing methods using A1and
A are very similar, indicating that sparse unmixing is quite
insensitive to the number of spectra in the reference library, and
regardless of the inclusion of the TV term in the solution of
the sparse unmixing problem. Although the effect of the TV

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IORDACHE et al.: TOTAL VARIATION SPATIAL REGULARIZATION4497
Fig. 10.
algorithms to the AVIRIS Cuprite scene using the library A1.
Abundance maps estimated for the minerals: alunite, buddingtonite, and chalcedony by applying the NCLS, NCLS-TV, SUnSAL and SUnSAL-TV
regularizer is not as apparent in the real image experiments as
in the simulated image experiments (which is probably due to
the fact that the AVIRIS Cuprite image used in experiments is
characterized by high SNR), the behavior observed is in line
with the results obtained for the experiments carried out on
simulated data, indicating that the inclusion of spatial

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4498IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 11, NOVEMBER 2012
Fig. 11.
algorithms to the AVIRIS Cuprite scene using the library A.
Abundance maps estimated for the minerals: alunite, buddingtonite, and chalcedony by applying the NCLS, NCLS-TV, SUnSAL, and SUnSAL-TV
information in sparse unmixing by means of the proposed TV
regularizer can assist in obtaining piecewise smooth abundance
maps with less outliers and high spatial consistency. Further
experiments with additional hyperspectral scenes and libraries
should be conducted in future work in order to fully objectify
our findings.