HYPERSPECTRAL UNMIXING WITH SIMULTANEOUS

DIMENSIONALITY ESTIMATION

Jos´e M. P. Nascimento

and Jos´e M. Bioucas-Dias

Instituto de Telecomunicac¸˜oes and Instituto Superior de Engenharia de Lisboa

R. Conselheiro Em´ıdio Navarro, N. 1, 1959-007 Lisboa, Portugal

Instituto de Telecomunicac¸˜oes and Instituto Superior T´ecnico, Technical University of Lisbon

Av. Rovisco Pais, Torre Norte, Piso 10, 1049-001 Lisboa, Portugal

Keywords:

Blind hyperspectral unmixing, Minimum volume simplex, Minimum Description Length (MDL), Variable

splitting augmented Lagrangian, Dimensionality reduction.

Abstract:

This paper is an elaboration of the simplex identiﬁcation via split augmented Lagrangian (SISAL) algorithm

(Bioucas-Dias, 2009) to blindly unmix hyperspectral data. SISAL is a linear hyperspectral unmixing method of

the minimum volume class. This method solve a non-convex problem by a sequence of augmented Lagrangian

optimizations, where the positivity constraints, forcing the spectral vectors to belong to the convex hull of the

endmember signatures, are replaced by soft constraints.

With respect to SISAL, we introduce a dimensionality estimation method based on the minimum description

length (MDL) principle. The effectiveness of the proposed algorithm is illustrated with simulated and real

data.

1 INTRODUCTION

Although, there have been signiﬁcant improvements

in the hyperspectral remote sensing sensors, there

are in an image pixels than contain more than one

material, i.e., the acquired spectral vectors are mix-

tures of the material spectral signatures present in

the scene (Bioucas-Dias and Plaza, 2010; Nascimento

and Bioucas-Dias, 2005a).

The linear mixing assumption has been widely

used to describe the observed hyperspectral vectors.

According to this assumption, a mixed pixel is a lin-

ear combination of endmembers signatures weighted

by the corresponding abundance fractions. Due to

physical considerations, the abundance fractions are

subject to the so-called non-negativity and a full-

additivity constraints (Bioucas-Dias and Plaza, 2010).

Hyperspectral unmixing, aims at estimating the

number of reference materials, also called endmem-

bers, their spectral signatures, and their abundance

fractions (Keshava and Mustard, 2002).

Approaches to hyperspectral linear unmixing can

be classiﬁed as either statistical or geometrical. Sta-

tistical methods very often formulate the problem

under the Bayesian framework (Nascimento and

Bioucas-Dias, 2011; Dobigeon et al., 2009; Arngren

et al., 2009; Moussaoui et al., 2008).

The geometric perspective just referred to has

been exploited by many algorithms. These algorithms

are based on the fact that, under the linear mixing

model, hyperspectral vectors belong to a simplex set

whose vertices correspond to the endmembers signa-

tures. Therefore, ﬁnding the endmembers is equiva-

lent to identifying the vertices of the referred to sim-

plex (Nascimento and Bioucas-Dias, 2005b).

Some algorithms assume the presence of, at least,

one pure pixel per endmember (i. e., containing just

one material). Some popular algorithms taking this

assumption are vertex component analysis (VCA),

(Nascimento and Bioucas-Dias, 2005b), the auto-

mated morphological endmember extraction (AMEE)

(Plaza et al., 2002), the pixel purity index (PPI),

(Boardman, 1993), and the N-FINDR (Winter, 1999)

(see (Chan et al., 2011) for recently introduced rein-

terpretations and improvements of N-FINDR). These

methods are followed by a fully constrained least

square estimation (Heinz and Chein-I-Chang, 2001)

or by a maximum likelihood estimation (Settle, 1996)

of the abundance fractions to complete the unmixing

procedure.

If the pure pixel assumption is not fulﬁlled, which

is a more realistic scenario, the unmixing process

is a rather challenging task, since some endmem-

bers are not in the dataset. Some recent methods,

438

M. P. Nascimento J. and M. Bioucas-Dias J..

HYPERSPECTRAL UNMIXING WITH SIMULTANEOUS DIMENSIONALITY ESTIMATION.

DOI: 10.5220/0003877504380444

In Proceedings of the 1st International Conference on Pattern Recognition Applications and Methods (PRARSHIA-2012), pages 438-444

ISBN: 978-989-8425-98-0

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

in the vein of Craig’s work minimum Volume Trans-

form (MVT) (Craig, 1994) which ﬁnds the smallest

simplex that contain the dataset, are the iterated con-

strained endmember (ICE), (Berman et al., 2004), the

minimum volume simplex (MVSA) (Li and Bioucas-

Dias, 2008), the Minimum-Volume Enclosing Simplex

Algorithm (MVES) (Chan et al., 2009), and the alter-

nating projected subgradients (APS) (Zymnis et al.,

2007).

Fig. 1 illustrates three datasets raising different

degrees of difﬁculties in what unmixing is concerned:

the dataset shown in Fig.1(a) contains pure pixels, i.e.,

the spectra corresponding to the simplex vertices are

in the dataset. This is the easiest scenario with which

all the unmixing algorithms cope without problems;

the dataset shown in Fig.1(b) does not contain pure

pixels, at least for some endmembers. This is a much

more challenging, usually attacked with the minimum

volume based methods, note that pure-pixels based

methods are outperformedunder these circumstances;

Fig.1(c), contains a highly mixed dataset where only

statistical methods can give accurate unmixing re-

sults.

Most of these methods assume that the number of

endmembers are knowna-priori or estimated for some

method, such as, NWHFC (Chang and Du, 2004),

HySime (Bioucas-Dias and Nascimento, 2008), and

Second moment linear dimensionality (SML) (Ba-

jorski, 2011). The robust signal subspace estimation

(RSSE) (Diani and Corsini, 2010) have been proposed

in order to estimate the signal subspace in the pres-

ence of rare signal pixels, thus it can be used as a

preprocessing step for small target detection appli-

cations. Sparsity promoting ICE (SPICE) (Zare and

Gader, 2007) is an extension of ICE algorithm that

incorporates sparsity-promoting priors to ﬁnd the cor-

rect number of endmembers. The framework pre-

sented in (Broadwater et al., 2004) also estimates the

number of endmembers when it unmix the data. This

framework has the disadvantage of using the Unsu-

pervised Fully Constrained Least Squares (UFCLS)

algorithm proposed in (Heinz and Chein-I-Chang,

2001) which assumes the presence of at least one pure

pixel per endmember.

Simplex identiﬁcation via split augmented La-

grangian (SISAL) (Bioucas-Dias, 2009) belongs to

the minimum volume class methods. The non-convex

optimization problem is solved as a sequence of non-

smooth convex subproblems using variable splitting

to obtain a constraint formulation, and then applying

an augmented Lagrangian technique.

This paper proposes an improvement of SISAL

by introducing an algorithm based on the minimum

description length (MDL) principle. Thus, SISAL is

(a)

(b)

(c)

Figure 1: Illustration of tree scenarios: (a) with pure pixels

(solid line - estimated simplex by all methods); (b) with-

out pure pixels and with pixels in the facets (solid red line

- estimated simplex based on minimum volume; dashed

blue line - estimated simplex by pure-pixel based methods);

based on minimum volume; dashed blue line - estimated

simplex by pure-pixel based methods).

able to estimate the number of endmembers as it un-

mix the hyperspectral dataset.

This paper is organized as follows. Section 2

describes the fundamentals of the proposed method.

Section 3 presents the method to infer the number

of number of endmembers. Section 4 illustrates as-

HYPERSPECTRAL UNMIXING WITH SIMULTANEOUS DIMENSIONALITY ESTIMATION

439

pects of the performance of the proposed approach

with experimental data based on U.S.G.S. laboratory

spectra and with real hyperspectral data collected by

the AVIRIS sensor, respectively. Section 5 concludes

with some remarks.

2 PROBLEM FORMULATION

Assuming the linear observation model, each pixel

y of an hyperspectral image can be represented as a

spectral vector in R

(l is the number of bands) and is

given by y = Ms + n, where M ≡ [m

,...,m

] is

an l × p mixing matrix (m

denotes the jth endmem-

ber spectral signature), p is the number of endmem-

bers present in the covered area, s = [s

,...,s

]

the abundance vector containing the fractions of each

endmember (notation (·)

stands for vector trans-

posed), and vector n holds the sensor noise and mod-

eling errors.

To ﬁx notation, let Y ≡ [y

, ...,y

] ∈ R

l×n

denote

a matrix holding the observed spectral vectors, S ≡

, ...,s

] ∈ R

p×n

a matrix holding the respective

abundance fractions, and N ≡ [n

, ...,n

] ∈ R

l×n

ac-

counts for additive noise. To be physically meaning-

ful, abundance fractions are subject to non-negativity

and constant sum constraints, i.e., {s ∈ R

: s 

0,1

s = 1

}

. Therefore

Y = MS+ N

s.t. : S  0, 1

S = 1

. (1)

Usually the number of endmembers is much lower

than the number of bands (p ≪ L). Thus, the observed

spectral vectors can be projected onto the signal sub-

space. The identiﬁcation of the signal subspace im-

proves the SNR, allows a correct dimension reduc-

tion, and thus yields gains in computational time and

complexity (Bioucas-Dias and Nascimento, 2008).

Let E

be a matrix, with orthonormal columns,

spanning the signal subspace. Thus

X ≡ E

Y+ E

= AS+ N

∗

, (2)

where X ≡ [x

, ...,x

] ∈ R

p×n

denote a matrix hold-

ing the projected spectral vectors, A = E

M is a p× p

square mixing matrix, and N

∗

accounts for the pro-

jected noise.

Linear unmixing amounts to infer matrices A and

S. This can be achieved by ﬁtting a minimum volume

simplex to the dataset (Craig, 1994). Finding a min-

imum volume matrix A subject to constraints in (1),

s  0 means s

≥ 0, for j = 1,... , p and 1

≡ [1,. . .,1]

leads to the non-convex optimization problem

Q = argmin

{−log|detQ|}

s.t. : QX  0, 1

QX = 1

, (3)

where Q ≡ A

−1

. The constraint 1

QX = 1

can be

simpliﬁed, by multiplying the equality on the right

hand side by X

(XX

)

−1

, resulting 1

QX = 1

⇔

Q = a

, where a

≡ 1

(XX

)

−1

SISAL aims to give a sub-optimal solution of (3)

solving the following problem by a sequence of aug-

mented Lagrangian optimizations:

∗

= argmin

{−log|detQ| + λkQXk

}

s.t. : 1

Q = a

, (4)

where kQXk

≡

∑

h(QX), h(x) ≡ max(−x,0) is

the so-called hinge function and λ is the regulariza-

tion parameter. Notice that kQXk

penalizes nega-

tive components of QX, thus playing the rule of a

soft constraint, yielding solutions that are robust to

outliers, noise, and poor initialization.(see (Bioucas-

Dias, 2009) for details).

3 NUMBER OF ENDMEMBERS

ESTIMATION

The MDL principle proposed by Rissanen (Rissanen,

1978) aims to select the model that offers the shortest

description length of the data. This approach can be

used to estimate the number of endmembers (Broad-

water et al., 2004). The well-known MDL criterion

for n i.i.d. observations, in general, is given by

MDL

= argmin



L (X|

) +

klogn



, (5)

where L (X|

) is a likelihood equation based on the

projected data X with parameters θ, and

klogn is

the penalizing term that increases with k (Figueiredo

and Jain, 2002).

Assuming that the additive noise is Gaussian dis-

tributed, i.e. n ∼ N (0,Λ)and given a set of n i.i.d.

observed samples, the likelihood equation is given by:

L (X|

) ≡

∑

i=1

−log p(x

)



plog(2π) + log| detΛ|



(X− AS)

−1

(X− AS)

(6)

ICPRAM 2012 - International Conference on Pattern Recognition Applications and Methods

440

where tr(·) denotes the trace of a matrix, matrices A

and S are replaced by their estimates using SISAL al-

gorithm, the noise covariance matrix, Λ is estimated

using the algorithm based on the multiple regression

theory proposed in (Bioucas-Dias and Nascimento,

2008) and the number of free parameters is k = p

The resulting optimization algorithm is an iterative

scheme that requires to compute the objective func-

tion and to estimate the matrices A, S, and Λ for each

value of p.

4 EXPERIMENTS

This section provides simulated and real data experi-

ments to illustrate the algorithm’s performance. The

proposed method is tested and compared with SPICE

(Zare and Gader, 2007) on different simulated sce-

narios concerning with different signal-to-noise ratio

(SNR), absence of pure pixels, and number of end-

members present in the scene. The proposed method

is also applied to real hyperspectral data collected by

the AVIRIS sensor over Cuprite, Nevada.

4.1 Evaluation with Simulated Data

In this section the proposed method is tested on sim-

ulated scenes. To evaluate the performance of the al-

gorithm the spectral angle distance (SAD) (Keshava

and Mustard, 2002) is used. This widely known met-

ric measures the shape similarity between the ith end-

member signature m

and its estimate

. Based on

this metric we deﬁne a spectral root mean square an-

gle error, given by:

≡

∑

i=1



arccos



1/2

. (7)

To measure the similarity between the observed data

and the unmix result it is also computed the residual

error between the observed pixels and their estimates:

≡ kY−

, (8)

where

M = E

A and

S are estimated by SISAL.

Concerning the simulated data creation a hyper-

spectral image composed of 10

pixels is generated

according to expression (1), where spectral signatures

where selected from the USGS digital spectral library.

The selection of endmember signatures is arbitrary

as long as they are linearly independent. The re-

ﬂectances contain 224 spectral bands covering wave-

lengths from 0.38 to 2.5µm with a spectral resolution

0 0.5 1

0.2

0.4

0.6

0.8

channel 50 (λ = 827nm)

channel 150 (λ = 1780nm)

data

True endmembers

Estimated Endmembers

Figure 2: Scatterplot of the three endmembers mixture:

Dataset (blue dots); true endmembers (black circles); Pro-

posed method estimates (red squares).

of 10nm. The abundance fractions are generated ac-

cording to a Dirichlet distribution given by

D(s

, ..., s

|µ

, ..., µ

) =

Γ(

∑

j=1

)

∏

j=1

Γ(µ

)

∏

j=1

−1

(9)

where {s

,..., s

} are subject to non-negativity

and constant sum constraints. This density, be-

sides enforcing positivity and full additivity con-

straints, displays a wide range of shapes, depend-

ing on the parameters of the distribution µ =

[µ

,...,µ

]. In this experiment the Dirichlet param-

eters are set to µ = [3,. . . ,3], concerning the addi-

tive noise, the SNR, which is deﬁned as SNR ≡

10log











, is set to 30dB.

Fig. 2 presents a scatterplot of the simulated scene

for the p = 3 case, where dots represent the pixels

and circles represent the true endmembers. This Fig-

ure also shows the endmembers estimates (squares).

Fig.3 shows the endmembers signatures (solid line)

and their estimates (dashed line). Note that, in this

experiment there is no pure pixels in the dataset, how-

ever, the endmembers estimates are very close to the

true ones.

Fig 4, presents the evolution of the cost function

[see expression (5)] as a function of the number of

endmembers. The minimum of the function occurs at

k = 3 which is the true number of endmembers in the

scene.

Table 1 presents the root mean square error dis-

tance ε

, the residual least squares error r

, and the

estimated number of endmembers for different exper-

iments where p is set to {3, 5,10} and the SNR is

set to {30,50} dB. Note that the estimated values are

exactly the number of endmembers in the scene and

HYPERSPECTRAL UNMIXING WITH SIMULTANEOUS DIMENSIONALITY ESTIMATION

441

0.5 1 1.5 2 2.5

0.2

0.4

0.6

0.8

Wavelength (nm)

Reflectances

Figure 3: Endmembers signatures (solid line) and their es-

timates (dashed line).

0 5 10 15

200

400

600

800

number of endmembers (p)

Cost function

Figure 4: Cost function evolution as a function of the num-

ber of endmembers.

Table 1: Results for different scenarios as a function of the

SNR and of the number of endmembers (p).

Proposed Method SPICE

SNR p

k ε

3 3 0.048 4.76 3 0.293 4.82

30dB 5 5 0.053 6.41 5 0.198 6.47

10 10 0.929 6.99 6 0.258 7.18

3 3 0.042 0.47 3 0.141 1.06

50dB 5 5 0.059 0.64 5 0.432 1.30

10 10 0.196 0.70 6 0.268 1.70

the unmix error increases with increasing values of p

and with noise level. The results achieved by SPICE

in terms of residual error are similar to the proposed

method results, although the errors between endmem-

bers signatures and their estimates are worst.

Figure 5: Band 30 (wavelength λ = 655.8nm) of the subim-

age of AVIRIS Cuprite Nevada dataset (rectangle denotes

the image fraction used in the experiment).

Table 2: Results for Cuprite dataset.

Proposed method SPICE

k 6 7

3.13 3.27

4.2 Experiments with Real

Hyperspectral Data

In this section, the proposed method is applied to a

subset (50× 90 pixels and 224 bands) of the Cuprite

dataset acquired by the AVIRIS sensor on June 19,

1997, Fig. 5 shows band 30 (wavelength λ =

667.3nm) of the subimage of AVIRIS cuprite Nevada

dataset. The AVIRIS instrument covers the spectral

region from 0.41µm to 2.45µm in 224 bands with a

10nm band width. Flying at an altitude of 20km, it

has an IFOV of 20m and views a swath over 10km

wide. This site has been extensively used for remote

sensing experiments over the past years and its geol-

ogy was previously mapped in detail (Swayze et al.,

1992).

Table 2 present the residual error and the esti-

mated number of endmembers for SPICE and for the

proposed method. The results of both methods are

comparable.

Fig.6 shows the estimated signatures, which are

compared with the nearest laboratory spectra, to vi-

sually distinguish the different endmembers an off-

set has been added to each signature. Note that, this

endmembers are known to dominate the considered

subimage (Swayze et al., 1992).

ICPRAM 2012 - International Conference on Pattern Recognition Applications and Methods

442

0.5 1 1.5 2 2.5

λ (µm)

Figure 6: Comparison of the estimated signatures (dashed

line) with the nearest USGS spectra (solid line): a) Alunite;

b) Desert vanish; c) Dumortierite; d) Sphene; e) Kaolinite;

f) Montmorillonite.

Figure 7: Abundance maps estimates: a) Alunite; b) Desert

vanish; c) Dumortierite; d) Sphene; e) Kaolinite; f) Mont-

morillonite.

Fig.7 presents the estimated abundance maps for

the extracted endmembers. A visual comparisonshow

that these maps are in accordance with the known

ground truth. Note that for this region Desert vanish

(Fig.7b)) and Sphene (Fig. 7d)) abundance maps are

very similar. These results show the potential of the

proposed method to simultaneously select the number

of endmembers, estimate the spectral signatures, and

their abundance fractions.

5 CONCLUSIONS

In this paper, a new method is proposed to blindly

unmix hyperspectral data and simultaneously infer

the number of endmembers based on the minimum

description length (MDL) principle. The endmem-

bers spectra estimates is based on SISAL algorithm

(Bioucas-Dias, 2009) which solves a non-convex

problem by a sequence of augmented Lagrangian op-

timizations. The experimental results achieved shows

the potential of the proposed method.

ACKNOWLEDGEMENTS

This work was supported by the Instituto de

Telecomunicac¸˜oes and by the Fundac¸˜ao para a

Ciˆencia e Tecnologia under project HoHus.

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