Matrix Calculation of High-dimensional Cross Product and Its Application in Automatic Recognition of the Endmembers of Hyperspectral Imagary

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. RESEARCH PAPERS .

SCIENCE CHINAInformation Sciences

January 2011 Vol. 54 No. 1: 197–203doi: 10.1007/s11432-010-4074-x

c Sci ence China Pres s and Springer- Verlag Berli n Hei delberg 20 10 i nfo. sci china. co m ww w. springerlink. com

Matrix calculation of high-dimensional cross product

and its application in automatic recognition of

the endmembers of hyperspectral imagary

GENG XiuRui1∗, ZHAO YongChao1, LIU SuHong2 & WANG FuXiang3

1Key Laboratory of Technology in Geo-spatial Information Processing and Application System,

Institute of Electronics, Chinese Academy of Sciences, Beijing 100080, China;2State Key Laboratory of Remote Sensing Science, Jointly Sponsored by Beijing Normal University

and the Institute of Remote Sensing Applications of Chinese Academy of Sciences,

School of Geography, Beijing Normal University, Beijing 100875, China;3School of Electronics and Information Engineering, Beihang University, Beijing 10019, China

Received September 15, 2008; accepted July 26, 2009; published online September 14, 2010

Abstract This paper gives the definition of the high-dimensional cross product and its calculation by extending

the 3-D cross product definition into the high-dimensional vector space. Based on the properties of the cross

product, the volume variance index (VVI) is proposed to be used in extracting automatically the endmembersof the hypherspectral imagery which eliminates the shortcoming of the traditional method of using simplex

only where the extraction results were easily impacted by the abnormal pixels. A case study of endmembers

extraction experiment using the VVI method with the AVIRIS data for Cuprite has shown a very good result.

Keywords endmember, simplex, hyperspectral imagery, cross product

Citation Geng X R, Zhao Y C, Liu S H, et al. Matrix calculation of high-dimensional cross product and its

application in automatic recognition of the endmembers of hyperspectral imagary. Sci China Inf Sci, 2011, 54:

197–203, doi: 10.1007/s11432-010-4074-x

1 Introduction

The development of the high-spectral resolution remote sensing techniques has been one of the break-

throughs of the EOS in late 1990s. The study of the application of the imaging spectrometry is the

frontier of the remote sensing techniques. Due to the complexity and the diversity of the surface objects

as well as the space resolution limitation of the sensors, the mixed pixels exist ubiquitously in the re-

motely sensed images. Based on this, all the pixels in a hyperspectral image can be considered as the

linear mixing of the endmembers of the image. Therefore, the extraction of the endmembers of an image

is the prerequisite for understanding the hyperspectral data and performing further analysis on it.

How to extract the endmembers has been a hot issue in the hyperspectral imagery processing. There

are some methods and algorithms available. Earlier, Boardman [1] proposed the idea of extracting

the endmembers by using the method of the convex geometry analysis, pointing out that all the data

of a hyperspectral image in its feature space is covered by the simplex formed by the vertexes which

∗Corresponding author (email: [email protected])



198 Geng X R, et al. Sci China Inf Sci January 2011 Vol. 54 No. 1

corresponded to the pure pixels representing all the surface objects (endmembers). Together with Kruse

and Green [2], he developed the pure pixel index (PPI) algorithm to extract the endmembers. The minimal

volume transform (MVT) method proposed by Craig [3] is used to get the endmembers by calculating

the smallest volume of the simplex which can encompass the entire hyperspectral “data cloud”. Batesonand Curtiss [4] developed a man-machine interactive method (MEST) to extract the endmembers based

on the principal component analysis and the multi-dimension visualization software. The N-FINDR [5]

algorithm finds the set of endmembers with the largest possible volume by “inflating” a simplex within

the data. The iterative error analysis (IEA) [6] is also an algorithm of endmember extraction which does

not require the reduction of the dimensions or the removal of the redundancy of the original data. It first

sets the initial vector (generally this initial vector would be the mean value vector of all the spectra), then

conducts linear unmixing operation step by step with one endmember from the error image extracted in

each step and adds this endmember in the calculation during the next step, and so on and so forth, until

all the endmembers are calculated based on the given criteria.

In order to explain variability of the endmembers spectra, Roberts [7] developed the multiple end-

members spectral mixing analysis (MESMA). Its main point is that each endmember is represented by a

group of vectors rather than a single vector and when conducting the linear unmixing, the best suitable

vector is chosen from its representative vector group to make the mean square root error minimal where

the endmember can be chosen from the image data or the spectral library of the region. Noticing the

variability of the endmembers, Bateson proposed the concept of the endmember bundles and generated

the endmember bundles by using the simulated annealing algorithm. Plaza [8] proposed an algorithm

to extract endmembers automatically based on the morphology which made good use of the space cor-

relation of the pixels while utilizing the spectral information. The vertex components analysis (VCA)

method, proposed by Nascimento [9] to extract the endmembers of an image, has the advantage in the

speed of the extraction. Based on quick convergence property of the non-negative matrix decomposition,

Miao [10] designated a novel endmembers extraction method which does not require the hypothesis that

there are pure pixels in an image. The aforementioned endmember extraction methods are all based on

linear mixture of the pixels in a hyperspectral image, which was equivalent to the use of the simplexcharacteristics of the scatter points of the hyperspectral image data in its feature space. However, there

are abnormal pixels of an image due to the impacts caused by various elements during the acquirement

of the hyperspectral image data. The algorithms based on the linear mixture models or simplex volume

would extract automatically those abnormal pixels as part of the endmembers, which is obviously not

beneficial to the further processing and analyzing of an image. This paper extends the 3-D cross product

definition into the high-dimensional vector space and proposes a volume variance index to be used in the

automatic extraction of the endmembers which takes into consideration both the geometrical properties

and statistics information of the data distribution based on the properties of the high-dimensional cross

product.

2 The problem background

In general, each pixel in a hyperspectral image can be considered as the linear mixture of the pixels of

the image, which can be expressed as

p =N

i=1

ciei + n = Ec + n, (1)

N

i=1

ci = 1, (2)

0 ci 1, (3)

where N is the number of endmembers in the image, and ci is a scalar value representing the fractional

abundance of endmember vector ei in the pixel corresponding to spectral vector p. E = [e1e2, . . . , eN ]



Geng X R, et al. Sci China Inf Sci January 2011 Vol. 54 No. 1 199

Figure 1 The triangular structure of three endmembers in a 2-D scatterplot.

is an L × N (L is the number of bands for the original data) matrix composed of all endmember vectors.

c = (c1, c2, . . . , cN )T and n are respectively abundance vector and noise vector.

There are three cases in the linear mixture model. Case 1 is expressed in eq. (1) in which there is no

restriction. If adding the restriction condition expressed by eq. (2), it becomes case 2 which is called

partial restricted linear mixture model. Case 3 is called the full restricted mixture model by adding

the restriction condition expressed by eq. (3) on top of case 2. The linear unmixing is to calculate

their corresponding proportions of the given endmembers in representing a pixel of an image so that the

proportion coefficients diagram of each endmember in that image can be obtained. Using the method of

least squares, the nonrestraint solution to eq. (1) is given as follows:

c = (E TE )−1E T p. (4)

When the error item, n, is very small, the set of all the points which meet eqs. (1)–(3) can just form a

convex set of high-dimensional space with the endmembers sitting on those vertexes of the simplex. Figure

1 illustrates the geometrical relationship between the endmembers by taking 2 bands and 3 endmembers

as an example. As shown in this figure, the endmembers, a, b and c, are located respectively at the three

vertexes of the triangle and the points inside the triangle would correspond to the mix pixels of an image.

As a result, the problem of extracting the endmembers of a hyperspectral image is turned into a problem

of obtaining the vertexes of the simplex.

As is noticeable, the area of the triangle formed by the endmembers a, b and c in Figure 1, is the

maximal among all the triangles formed by any three pixels of this image, expressed as

S (a , b, c) = max{S (i , j, k)}, (5)

where S (i , j, k) represents the area of the triangle formed by the pixels i, j and k. Therefore, the problem

of obtaining the vertexes of the simplex becomes a problem of seeking for the maximal volume of the

simplex. There are many algorithms based on this as mentioned in the previous section. Apparently those

algorithms utilize only the geometrical information of the image distribution in its feature space without

taking into account the statistic information of the scattering distribution of points. By extending the 3-D

cross product definition into the high-dimensional vector space, a volume variance index is proposed in

the next section for the automatic extraction of the endmembers of the hyperspectral imagery which takes

into consideration both the geometrical properties and statistics information of the data distribution.

3 The algorithm

As is well known, being one of the basic calculations in the 3-D space, the cross product of two 3-D

vectors is a new vector with its direction vertical to the plane formed by the two vectors, and the value




equal to the area of the parallelogram formed by the two vectors. Supposing a = (a1a2a3), b = (b1b2b3),

their cross product, c, can be expressed as

c = a × b =

i j k

a1 a2 a3

b1 b2 b3

, (6)

where | · | is the determinant calculator, and i, j, k are the unit vectors of the three coordinate axes

respectively. In order for eq. (6) to make sense, a, b must be 3-D vectors. To extend this concept

into the high dimension space, we assume that there are N − 1 N -dimension vectors, represented by

ei = (ei1, ei2, . . . , eiN ), i = 1, 2, . . . , N − 1. In the N -dimension vector space, we define their cross

product as

d = e1 × e2 × · · · × eN −1 =

i1 · · · · · · iN

e11 · · · · · · e1N

.

.. · · · · · ·...

eN −1,1 · · · · · · eN −1,N

, (7)

where i1, i2, . . . , iN are the unit vectors of the N -dimensional coordinates axes. It is obvious that d is a

new vector with its direction vertical to the super plane (represented as span(e1, e2, . . . , eN −1)) formed by

e1, e2, . . . , eN −1, and its value equal to the volume of the (N − 1)-dimension parallel polyhedron formed

by the vectors, e1, e2, . . . , eN −1. The volume of the simplex with e1, e2, . . . , eN −1 being its vertexes can

be calculated as

V (e1, e2, . . . , eN −1) =1

(N − 2)!d. (8)

Apparently there must be N − 1 N -dimension vectors, ei = (ei1, ei2, . . . , eiN ), participating in the cal-

culation in order for eq. (7) to make sense. That is to say, the cross product in the N -dimension space

does not happen with two vectors. It instead requires N − 1 vectors to participate in the calculation.As a special case, the cross product in the 2-D space would be the unit calculation. For example, given

x = (x1, x2) as a non-zero vector in the 2-D space, its cross product could be described as y = | ix1

jx2

|.

And obviously y⊥x and they have the same length.

Assuming that e1, e2, . . . , eN −1 are all the endmembers retrieved from an image, we definitely hope that

the majority of the information of this image would be distributed in the super plane, span(e1, e2, . . . ,

eN −1), and the less the information distributed in the orthogonal complementary set of span(e1, e2, . . . ,

eN −1) the better. The variance of the image in the d direction can be expressed as

var(d) =dTΣd

d2, (9)

where Σ is the co-variance matrix of the N -dimension hyperspectral data.

Considering the value and direction of the cross product, d, of the vectors e1, e2, . . . and eN −1, the

volume variance index to be used to extract the endmembers is defined as

p(e1, e2, . . . , eN −1) =d

var(d)=

d3

dTΣd. (10)

The N − 1 N -dimension vectors, e1, e2, . . . , and eN −1, will be the endmembers if they can meet the

requirements to maximize the value of p(e1, e2, . . . , eN −1). This index is meaningful because of the larger

volume as 1(N −2)! d of the simplex formed by the endmembers, e1, e2, . . . , and eN −1, and the more

information projected on the super plane. Because the abnormal pixels of an image are normally isolated

far away from the “data cloud”, they would be included as part of the endmembers if extracted by the

algorithms only based on the volume of the simplex. This can be avoided by the introduction of the

variance in the calculation proposed in this paper.In many cases, the band numbers of a hyperspectral image to be processed is much bigger than its

latent dimension of the image. However, the latent dimension of an image depends on the total number




of the meaningful surface objects based on the linear mixing theory. If there are N − 1 endmembers of an

image, it is required to reduce the dimension of the original data to N dimension. This can be achieved

via the method of principal component analysis (PCA) which can eliminate the redundant information

while keeping the information content unchanged and thus reduce the number of dimension of the originalimage to the level required.

The following comes with the detailed steps of the algorithm:

(1) Use PCA to reduce the number of dimension of the original data to N dimension if needed

(2) Take any N − 1 vectors of the image as the initial vectors, to calculate the volume variance index,

p0 by using eq. (10)

(3) Sequentially traverse all the pixels of the image and replace the elements in the initial vectors with

the newly picked up pixels, to calculate the volume variance indexes in turn. Assume the maximal value

among the calculated variance indexes is p1, and if p1 > p0, then the corresponding pixel in the initial

vector set is replaced by this new pixel and p0 is replaced by p1. Otherwise, continue traversing to the

next pixel of the image

(4) Continue step (3) until the value of the volume variance index no longer changes. Then the resultingN − 1 vectors would be the endmembers.

4 The experiment validation

The experiment is conducted to extract the surface objects by using AVIRIS data obtained for Cuprite

from ENVI (Figure 2). AVIRIS is a hyperspectral instrument of high quality and low noise with 224

bands (the wave length ranged from 400 to 2500 nm). Only 50 consecutive bands of infrared shortwave

(from 1978 to 2478 nm) are selected for the algorithm validation. First, the image is transformed by

using the PCA method. Because the sum of the eigenvalues of the first 7 principal components takes

99.43% of the total summary of all the eigenvalues, the first 7 bands resulting from the PCA are selected

to conduct the endmembers extraction experiment.Based on the algorithm proposed in this paper, 6 endmembers should be extracted after the decrease

of the dimension. The comparison has been given between the method of using the volume only (as

expressed by eq. (8)) and the method of using the volume variance index (as expressed by eq. (10)). As

illustrated in Figure 3, there are two endmemebers in common out of the 6 endmembers. Out of the rest 4

endmembers, three of them are quite similar (Figure 4(a)–(c)) and leaving one with significant difference in

the spectral curve (Figure 4(d)). From the obtained endmembers, the endmember 2 represents kaolinite,

the endmember 3 represents alunite and the endmember 4, calcite. These endmembers spectra pretty

much match the measured spectra of those surface objects in the spectra database. It is worthy of notice

that there is an abnormal peak at band 44 for the endmember 6 (Figure 4(d)). If stretching the image

at the band 44, it is obvious that this endmember is an abnormal pixel (Figure 5). The introduction

of the volume variance index can enable us to detect this abnormality and thus exclude such abnormal

pixels in the endmembers (Figure 4(d)). Based on the real distribution of the surface objects in this area,

the endmember 6 obtained by using the volume variance index is corresponding to the surface objects,

such as silicon substances and rhyolitic tuff which are abroadly distributed in the region (as shown in the

red ellipse in Figure 2). This indicates that the abnormal pixels are excluded from the endmembers by

the introduction of the volume variance index, and therefore, what are extracted by this method is the

endmembers representing meaningful surface objects.

5 Conclusions

This paper extends the 3-D cross product concept into the high-dimensional vector space, and gives the

definition of the high-dimensional cross product and its calculation. Based on the properties of the high-dimensional cross product, a volume variance index is proposed to be used in the automatic extraction

of the endmembers for the hyperspectral images. Because it takes into account both the geometrical as-




Figure 2 False color composite image of the Cuprite Figure 3 Results of endmembers extraction from volume

sample data. (eq. (8)) and VVI (eq. (10)).

Figure 4 Endmembers extracted from the image by volume (eq. (8)) and VVI (eq. (10)). (a) The 2nd endmember; (b)

the 4th endmember; (c) the 5th endmember; (d) the 6th endmember.

pect and the statistical aspect (e.g., the volume and the variance), it is free of the shortcomings of

the traditional geometry focused methods in extraction of the endmembers. Both the analysis and

the experiment result show that our method can better avoid the abnormal pixels, indicating that the

introduction of the volume variance index is quite valuable. As any method has its application boundary,the method proposed herein may not be beneficial if the interest of an application is in the abnormal

pixels or the small objects of the imagery data.




Figure 5 The abnormal pixel in band 44 (the left image is the 44 band gray image, the right shows the locality of the

image, where the bright point is the abnormal pixel).

Acknowledgements

This work was supported by the National Basic Research Program of China (Grant Nos. 2007CB714406,

2007CB714401, 2007CB403507), the National Natural Science Foundation of China (Grant No. 40501041), and

the National Science Foundation of USA (Grant No. 0421530).

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Matrix Calculation of High-dimensional Cross Product and Its Application in Automatic Recognition of the Endmembers of Hyperspectral Imagary