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Performance Analysis & Comparative Study of

Geometrical Approaches for Spectral Unmixing

Bijitha. S.R, Geetha. P, Soman.K.P

Centre for Excellence in Computational Engineering and Networking

Amrita Vishwa Vidyapeetham, Coimbatore, TamilNadu, India

Abstract— The hyperspectral cameras used for imaging is having low spatial resolution ,and thus the pixels in the captured image will b e mixtures of spectra of various materials present in the scene.Then further analysis of images becomes a tough task.Thus spectral unmixing comes as an unavoidable step in hyperspectral image processing.Spectral unmixing aims at finding out the no. of reference substances(endmembers),their spectral signatures and corresponding abundance maps of them in a hyperspectral image.This paper presents a comparative study and performance analysis of 5 geometrical algorithms for spectral unmixing ,namely AVMAX,SVMAX,ADVMM,SDVMM and N-FINDR.All the 5 algorithms are applied to the real hyperspectral data set (cuprite data,Nevada,U.S) and results are validated with reference to U.S.G.S spectral library.

Index Terms— ADVMM, AVMAX, Hyperspectral imaging, N-FINDR, SDVMM, Spectral signature, Spectral unmixing , SVMAX,

——————————  ——————————

1 INTRODUCTION

YPERSPECTRAL sensors collects the data in hundreds of very narrow contiguous bands,and this provides a good way for the identification of various materials over
the observed scene captured by the sensor.The various materi- als are discriminated on their unique spectral signa- tures.Hyperspectral imaging is having a wide range of appli- cations in various fields as in agriculture,planetary re- motesensing,military,environmental monitoring etc[1].The hyperspectral imaging sensors can capture many contiguous bands which is having very high spectral resolution and this will be covering not only visible regions but also the infra red regions of electromagnetic spectrum(0.3-
2.5µm)[2],[3].Advanced hyperspectral sensors like AVIRIS [4] of NASA is now able to cover the above mentioned wave- length region using about 200 spectral channels.
In the case of hyperspectral images ,depending upon the spatial resolution of sensor,the individual pixels in the cap- tured scene may comprise of more than one material.each pix- els will be the mixture of various materials of the surface patch and thus the spectra observed will contain multiple end- menbers (or spectral signatures) and thus the further analysis becomes difficult.This happens mainly because of the poor spatial resolution of the sensor used.Fig1 explains the concept of hyperspectral imaging[10].There comes the need of hyper- spectral unmixing.Hyperspectral unmixng aims at the decom- position of the observed spectra into a set of pure reference materials(endmembers)and their abundance fractions.Thus unmixing process gives both spectral signatures and corre- sponding abundance maps of materials present in the sce- ne[5]. This unmixing problem has been a subject to many in-

————————————————

• Bijitha S.R is currently pursuing masters degree program in Re- mote sensing and wireless sensor networks in AmritaVishwa vidyapeetham,India,

E-mail: bijitharajagopal@gmail.com

• Geetha.P is working as assistant professor at Dept of CEN at Am-

rita vishwavidyapeetham

vestigative studies for the past many years.
Fig1.Hyperspectral imagery.

1.1 Spectral unmixing

Hyper Spectral unmixing is basically a blind source separation problem [6],[35]. Hyperspectral sense contain sources which are statistically dependent and they may combine either in a linear or nonlinear fashion.This makes spectral unmixing problem to be placed in higher level compared to other source separation problems.
Unmixing can be classified to linear[7] and Non line- ar[8].Linear models assume that the mixing scale is macro- scopic,and the light which falls on the surface interacts with only one material.This type of mixing takes place due to the low spatial resolution of the sensor.Here multiple scatterings do not take place.
In the case of Nonlinear mixing models the interaction be- tween the light which is scattered by multiple materials oc- curs,and the mixed model becomes complicated.The interac- tions can be at Intimate or microscopic level.Thus non-linear unmixing becomes a difficult task.So here this paper concen- trates in linear unmixing due to its simplicity,and also it’s the basis of many algorithms for more than 30 years.

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1.2 Linear Spectral Unmixing

The linear mixing model assumes that the spectra of a pixel in the acquired scene is a linear combination of all pure mate- rials(endmembers) present in the scene.i it is assumed that the hyperspectral sensor used for capturing the scene has L spec- tral bands, linear mixing model can be mathematically repre- sented as follows.
Statistical approaches are rarely used in the case of spec- tral unmixing ,since their computational complexity level is very high when compared to other methods as sparse and geometrical methods.but still if the spectral mixtures are high- ly mixed then geometrical approaches provide poor results due to the lack of pure spectral vectors and all,these stastistical methods comes to play.under this,unmixing problem is for-

y = M α + n

(1)
mulated as a statistical inference problem and statistical de
.ICA(Independent component analysis)[11],DECA(dependent
Where y is an Lx1 column vector ,M is an Lxq matrix contain- ing q endmembers(pure reference materials) and α is a qx1 vector containing the fractional abundances of the endmem- bers in the pixel and n is another Lx1 vector indicating the errors which affect the measurements at each pixel[9].In this modelling both M and α have to be found by unmixing .Here

ANC(abundance non-negativity constraint) αi ≥ 0 ,where i=1,2….q and ASC(abundance sum to one constraint),1T α = 1

Are imposed to this model.This takes another fact into consid- eration as αi ,for i=1,2…q ,represent the fractions or propor- tions of the pure materials or endmembers present in the sce-

ne.In this Y ≡ { y ∈ R L , i = 1,...n} of n no. of observed spec-

tral vectors with dimension L.
Here in this paper all the 5 geometrical algorithms relay on
this linear model inorder to carry out the unmixing process.All
the geometrical approaches tries to solve the same linear un-
mixing problem shown in (1).Linear mixing concept is shown
in fig.2[9]

Fig2.Linear mixing model without any multiple scattering effects.

1.3 Endmember extraction algorithms-overview

Spectral unmixing algorithms can also be called as Endmember extraction algorithms,endmember identification algorithms etc.All the Spectral unmixing algorithms are main- ly classified in to three types.They are statistical approaches, sparse based methods and geometrical approaches.
component analysis)[12] etc are the main algorithms coming under this category.All these are formulated under a Bayesian framework.
Sparse based approaches is the another category of spectral unmixing algorithms.In this spectral unmixing is formulated in a semi-supervised fashion,and it is assumed that the spec- tral signatures observed can be expressed as the linear combi- nations of known pure spectral signatures from a spectral li- brary[13],[14].OMP(orthogonalmatchingpursuit)[15],ISMA(Ite rative spectral mixture analysis)[16] etc comes under this.The most popular algorithms coming under this category are SUNSAL(sparse unmixing via splitted and augmented la- grangian approach)[17],and SUNSAL-TV((sparse unmixing via splitted and augmented lagrangian-total variation)[18].
Geometrical approaches come as the third category of spectral unmixing algorithms.Basically it follows the fact that
,under the linear mixing model spectral vectors belong to the simplex set whose vertices correspond to the endmem- bers.Thus by finding out the vertices it is possible to find out the endmember in the hyperspectral image.There are two cat- egories in this approach.Algorithms which assume the pres- ence of pure pixels comes under the one category and algo- rithms which do not assume the presence of pure pixels comes under another category.MVSA(minimum volume sim- plex analysis)[19],MVES(minimum volume enclosing sim- plex)[20],SISAL(simplex identification via split and augment- ed lagrangian)[21],etc comes under the first category.In the second category to which this paper concen- trates,comesthefollowingalgorithmslikeSVMAX(successivevol umemaximization)[22],AVMAX(Alternating volume maximi- zation)[22],ADVMM(alternating decoupled volume max- min)[23],SDVMM(successive decoupled volume max- min)[23],N-FINDR[24],VCA(vertex component analysis)[25]
,IEA(iterative error analysis)[26],PPI(pixel purity in-
dex)[27],etc are some of the algorithms come under this sec-
tion.In this paper 5 popular algorithms of this category name-
ly AVMAX,SVMAX,ADVMM,SDVMM and N-FINDR is taken
in to account and their performance is evaluated and results
are compared to find out the good one giving the best result
among them.
The rest of the paper is organized as follows.Section 2 gives
the theoretical ideas behind the selected algorithms, Section3
explains the metrics used for performance evaluation ,Section
4presents the experiments with real hyperspectral data it’s
results and the performance analysis and finally conclusions are drawn in Section 5 .

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2 PURE PIXEL BASED GEOMETRICAL ALGORITHMS

As discussed in the previous sections geometrical algo- rithms with pure pixel assumption assumes the presence of
Thus (4) is modified as follows
Where
atleast one pure pixel per endmember.These pure pixel algo-

max

w ,.....,w ∈R

f1 (w1 ) f2 (w1 , w2 ).... f N (w1 ,...., wN ) (6)

rithms still belong to minimum volume class.This assumption
of pure pixels make these algoriths very efficient but still cre-
ates difficulty in some datasets.In this section a brief theoreti-

1 N

s.t wi ∈ F ,

i = 1,........, N .

cal side of each of the 5 algorithms namely

f1 (w1 ) =

w1 2

AVMAX,SVMAX,ADVMM,SDVMM and N-FINDR is given .
.

f j (w1 ,.....w j ) =

P ⊥

1:( j −1) j 2 ,

j = 2,...., N

2.1 AVMAX (Alternating volume maximization)

Altenating volume maximization algorithm[22] is based on winter problem described in [28].in winter’s work he pro-
Thus the following procedure is followed. For j=1:N solve the problem
posed that the ground-truth endmembers can be located by finding a collection of pixel vectors whose simplex volume is

W j = arg max w ∈F

f j (w� 1,....... w� j −1 , w j ) (7)

the largest.The optimization formulation of winters problem is as follows.

At last we will get (w� 1 ,.........., w� N ) as the approximate solu-

tion of(6).It’s similar to VCA[25] in some aspects.But unlike
VCA algorithm SVMAX considers thewhole subspace when

max

ν1 ,,,,ν N

∈R N −1

vol (ν1 , .....,ν N )

the data is projected orthogonally whereasVCA takes random

s.t

ν i ∈ conv{x[1], ......, x[ L]},

i = 1, ...., N

(2)

direction in subspace.

2.3 ADVMM (Alternating decoupled volume max-min)

Where according to winters work each endmember esti- mateν i is restricted to be any vector in{x�[1],......, x�[L]} .when

alternating volume maximization is applied to this it maxim- izes in a cycic fashion,the volume of the simplex defined by the pure members(endmembers) but with respect to only one endmember at a time.This is explained as follows in[22].
In this winter’s problem shown in(2)is formulated as a max- min problem and alternating optimization [23]is used to solve it.This winter’s worst case problem is given as a max-min problem as follows in[30].
Where y[1],…y[L] is the data cloud inside which maximum
The starting point is taken as

(ν1 ,.....,ν N ) .The following

volume simplex is situated .From the vertices of this simplex
alternating cycle is repeated as for j=1…N solve the problem

max N −1 {min

det(∆(v1 − u1,....., vN − uN )) (8)

i

i =1,..., N

i

i =1,..., N

maxν ∈F

det(∆(ν1 , ...ν j −1 ,ν j ,ν j +1..,ν N ))

(3)

s.t vi ∈ conv{y[1], ...., y[ L]}, i = 1, ....., N

endmembers are to be found out.
By taking

vi = Y�θi ,

Y� = [ y[1],...., y[L]] ∈ R

( N −1)×L

and
And update ν j
as the solution of (3).we have to continue

for any permutation matrix p, det(P∆) = ± det(∆) we can

this until the stopping criterion is satisfied. The algorithm is
explained in detail in [22].Avmax is somewhat similar to SC-
N-FINDR which is a modified version of N-FINDR described
in[29].
write the problem in (8) as
Then by doing the cofactor expansion and simplification

2.2 SVMAX (Successive volume maximization)

maxθ ∈s

{min u ≤r ,

det(� (Y�θ1 − u1 , ....., Y�θ N − uN ))} (9)

Successive volume maximization [22] is another strategy of optimization for the winter’s problem shown in(2).This re- quires the winter’s problem to be written in a modified fash- ion as follows in[22].

i i

i =1,..., N i =1,..., N

of (9) as in[30] it is reduced to

max {min

k T (� (Y�θ

− u ))} (10)

max

w1 ,....., wN ∈R N

det(w)

θ j ∈s

u j ≤ r ,

j j j

s.t wi ∈ F ,

Where

i = 1, ...., N

(4)

The above problem can be solved by solving the 2 decou-
pled problems shown below.

F ={w ∈ R N | w = [ν T 1]T ,ν ∈ F}

Thus ADVMM solves the max-min problem of spectral unmix-
Then according to rules |det (w)| can be written as fol-

u j = arg max

k T u

= rk / k

(11)

u

lows.

j ≤r

j j j j

θ� = arg max

k T Y�θ

= e , l = arg max

k T y[n] (12)

det(w) =

det(wT w) (5)

j

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ing.
number of endmembers to be identified.
2)Take some randomly selected endmembers from the dataset

2.4 SDVMM(Successive decoupled volume max-min) as

( 0) ( 0) ( 0) ( 0)

This is also another algorithm which follows winter’s

{E1

, E2

, E3 ( 0.)...E(p0) }

( 0) ( 0)

problem shown in (8).This solves the decoupled max-min

{E1

, E2

, E3

....E p }

problem in a successive optimization method. By assuming
3)At each iteration k≥0,calculate the volume by this set of endmembers as follows.

w = [vT 1]T , z

= [uT

0]T & y[n] = [ y[n]T

1]T

i i i i

( k ) ( k ) ( k )

1 1 ...........1 

Problem(8) can be written as follows.

v(E1

, E2

,...E p

) = | det  E ( k )

E ( k ) ......E ( k ) 

| (19)

max w ∈F , {min z ≤r

i =1,... N eT z =0,∀i

| det([w1 − z1 ,....wN − zN ]) | (13)

 1 2 p 

( p −1)!

Where F=conv{y[1],…y[L]}.The problem (13) can be modified and written as

N

4) Replacement- For each and every pixel the volume correspond ing to it is checked by this way,if this pixel replaces one of the given endmember positions in matrix shown above.. If the repla cement of pixel results in an increase in volume,the pixel replaces

max ∈

min

∏ f ((w , z ), ...., (w , z

)) (14)

the endmember.This process continues until there are no

wi F ,

zi ≤r

1 1 j j

endmember replacements in the given data.

i =1,... N

where

eT z =0,∀i

j =1

3 METRIC USED FOR PERFORMANCE EVALUATION

f ((w1 , z1 ), ...., (w j , z j )) =

P ⊥

H 1:( j −1)

(w j − z j ) , (15)

As the result of spectral unmixing we will get spectral signa- tures of endmembers and also their corresponding abun-
The detailed explanation of terms and simplification steps is given in [30].The problem can be approximated by succes- sive optimization as follows.
daance maps. Here in this paper The metric used for the vali- dation of results is Spectral Angle Mapper (SAM)[34]. It’s measured between the original library spectra which we will get from U.S.G.S library [36], and the spectra obtained by the unmixing process.

(w� j , z� ) = arg max

, min

f ((w� 1 , z1 ),...., (w� j , z j )) (16)

The basic equation for the spectral angle is given as follows.

j wi ∈F zi ≤r

i =1,...N eT z =0,∀i

The solution to (16) is given in [30] as follows.

θ ( x, y) = arccos < x, y >

x y

(20)

z j = arg min

z j 2 ≤r

P H� 1 : ( j − 1) (wj − z j ) (17)

Where, x is the library spectra and y is the spectra obtained from unmixing.As the spectral angle (SA) decreases, the result becomes more good and when we get a high value for SA we can infer that the performance of algorithm is poor.In that way

w� j

= y[l ], l = arg max

P H� 1 : ( j − 1) wj

(18)

we calculate the SA for each mineral spectra and finally aver- age the results to obtain average SA for each algorithms.This
Thus it solves the max-min problem by successive optization.

2.5 N-FINDR

This is another popular algorithm used for spectral unmix- ing.This also works according to winter’s belief [24].This is a pure pixel based algorithm and this search for the set of pixels with largest possible volume by inflating a simplex inside the given dataset.
The original n-findr algorithm [31] is having 4 steps as fol- lows.
1) Feature reduction-In this the dimension of data is reduced from n toP-1 by some PCA [32] or MNF[33],where P is the
can be seen in detail in the following sections

4 EXPERIMENTS ON REAL DATA-RESULTS AND PERFORMANCE ANALYSIS

In this part,all the 5 mentioned algorithms AVMAX,SVMAX,ADVMM,SDVMM and N-FINDR are ap- plied to the real hyperspectral data set taken over Cuprite mining site,Nevada,in 1997[37].We consider only a subimage of the hyperspectral data as a region of interest ,which is of size 250x191 pixels(L=47750).This contains 224 bands over the wavelength region of 0.4µm to 2.5 µm.In this set, the bands 1-

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2,104-113,148-167 and 221-224 were removed due to low SNR effect which occurs due to the effect of water vapour.Thus a total of 188 bands were used for the implementation of the algorithms.As the next step,we want to know how many endmembers are located in this region of interest.For this we applied Hyperspectral subspace identification by minimum error (HySime) [38] and thus estimated the number of endmembers in this region as N=18.
The abundance maps corresponding to each mineral was obtained by using fully constrained least square (FCLS) [39] method.The minerals obtained were then identified by the visual comparison of the abundance maps obtained with the abundance maps shown in [20],[25],[40] , and [41].As said ear- lier spectral angle is used for as measure for the perfor- mance.The value of SA for the estimated endmembers ob- tained by all the 5 algorithms are shown in Table1,2,3,4 and
5..The numbers in parantheses denote the value of SA for the estimated endmember which is repeated.Due to the space lim- it here we have shown the estimated endmember signatures and abundance maps of N-FINDR algorithm only.The abun- dance maps and spectral signatures are shown below.
Dumortierite Kaolinite3
Buddingtonite
Andradite1
Pyrope#1 Dumortierite
Alunite
Dumortierite
Chalcedony Kao
Alunite
Nontronite2
Andradite



muscovite
Nontronite3

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Nontronite2

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Fig.3 eighteen abundance maps obtained by N-findr algorithm
The following figures show the spectral signatures obtained by N-FINDR algo- rithm.
Muscovite
AluAnliutenite


Nontronite 3
Dumortierite

Dumortierite
Chalcedony
Chalcedony
Kaolinite 3
Kaolinite 2
org

Int me 4, Issue 7, July-2013 441

ISS

Buddingtonite Andradite Smectite
Nontronite 2


Sphene
Dumortierite

Alunite

Montmorillonite 1

Alunite


Nontronite 2
Fig.4 Endmember spectral signatures estimated by
N-FINDR algorithm.

Minerals SA(In degrees)

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Average SA 9.2418
TABLE2
SA values for SVMAX algorithm


Minerals SA (In degrees)


Average SA 8.0094
TABLE 3
SA values for ADVMM algorithm

Minerals SAD (In degrees)

Average SA 8.3777
TABLE 4
SA values for SDVMM algorithm


Minerals SA (In degrees)

Average SA 8.2898

TABLE 5
SA values for N-FINDR algorithm

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Minerals SA (In degrees)
1)Pyrope 1 3.6341
2)Dumortierite 5.0531(11.9325)(9.9103)
3)Chalcedony 5.222
4)Kaolinite 2 10.8546
5)Muscovite 7.6819
6)Nontronite 3 7.9865
7)Kaolinite 3 10.4473
8)Buddingtonite 4.3752
9)Andradite 8.9945
10)Smectite 4.7384
11)Nontronite2 6.5617(3.9298)
12)Sphene 6.8655
13)Alunite 5.3319(15.3751)
14)Montmorillonite 1 7.5419


Average SA 7.5745
The tables shown above gives the Spectral angle between the estimated endmember signatures and the Library spectra of U.S.G.S library.When analyzing the tables we can see that the Average Spectral angle is different for all the five algorithms and AVMAX(Alternating Volume maximization)algorithm gives the high value for the average SA which is 9.24.This gives the indication of poor performance when compared to other algorithm.In between the algorithms N-FINDR gives the appreciable result with Average SA of 7.4131 .This points to- wards a good result of spectral unmixing of the given data set.All the other algorithms could identify only 13 minerals out of 18 whereas N-FINDR could identify 14 minerals out of
18.(In the table it can be seen that Alunite and Nontronite 2 were repeated ,and Dumortierite was repeated 3 times). Moreover N-FINDR was able to detect the rare mineral “sphene” where other 4 algorithms was not able to detect the presence of it.In between N-findr and Avmax, comes the rest of algorithms and in this Svmax gives the good performance after N-findr,Sdvmm comes as the third and Advmm comes as the second last when looking in to the performance level.
As the N-findr algorithm works on the replacement of pix- els and endmembers it does not miss any endmembers in the given data set and can give the better result when compared to all other algorithms.It solves the winter’s problem in an effi- cient way compared to all other algorithms and comes out with the best result among the other methods.

5 CONCLUSION

In this paper, the implementation and comparative study of five geometrical approaches which are popularly used for spectral unmixing is done.The spectral unmixing experiment was done with Cuprite dataset(Nevada,U.S).The performance comparison of algorithms is done based on Spectral Angle mapper and it used U.S.G.S library as the reference.By the comparative study it’s found that N-FINDR algorithm gives
the better performance compared to the other four algorithms. AVMAX (Alternating volume maximization) is the one which gives much lower performance compared to others.As a future work the geometrical based spectral unmixing techniques can be compared with the statistical methods and sparse methods and it can make out new results. This can open new ways to researches also.

ACKNOWLEDGMENT

The authors wish to sincerely thank Dr.J.M Biouscas for his valuable comments and suggestions in this work and imple- mentation of codes, and Mr.Tsung –Han Chan for the sugges- tions in this work and Dr.K.A Narayanankutty for his help in completing this work.

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