. RESEARCH PAPERS . SCIENCE CHINA Information Sciences Jan uary 2011 V ol. 54 No. 1: 197–203 doi: 10.100 7/s114 32-01 0-407 4-x c Sci en ce Ch ina Pr ess an d Spri ng er -V er lag Ber lin He id el berg 201 0 in fo . sci c hi n a. c om www .s pri ng erl ink . com Matrix calculation of high-dimensional cross product and its application in automatic recognition ofthe endmembers of hyperspectral imagary GENG XiuRui 1 ∗ , ZHAO YongChao 1 , LIU SuHong 2 & WANG FuXiang 3 1 Key Labor atory of Technolo gy in Geo-sp atial Information Process ing and Application System, Institute of Electronics, Chinese Academy of Sciences, Beijing100080, China; 2 State Key Laboratory of Remote Sensing Science, Jointly Sponsored by Beijing Normal Universityand the Institute of Remote Sensing Applications of Chinese Academy of Sciences, School of Geography, Beijing Normal University, Beijing100875, China; 3 School of Electronics and Information Engineering, Beihang University, Beijing10019, ChinaReceived 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 produc t definitio n into the high- dime nsional vector space. Based on the properties of the cross product, the volume variance index (VVI) is proposed to be used in extracting automatically the endmembers of the hypherspectral imagery whic h eliminates the shortc oming 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. Matri x calculatio n of high-dimension al cross produc t and its application in automati c recognition of the endmembers of hyperspec tral imaga ry. Sci China Inf Sci, 2011, 54: 197–203, doi: 10.1007/s11432-010-4074-x 1 In troduct ion The development of the high-spectral resolution remote sensing techniques has been one of the break- thr ough s of the EOS in late 1990s. The study of the application of the imaging spec tro metry 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 hyper spectral 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 met hods and alg ori thms av ail abl e. Earlier, Boa rdman [1] propos ed the idea of extr act ing 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])
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7/28/2019 Matrix Calculation of High-dimensional Cross Product and Its Application in Automatic Recognition of the Endmembers of Hyperspectral Imagary
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.
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 ]
7/28/2019 Matrix Calculation of High-dimensional Cross Product and Its Application in Automatic Recognition of the Endmembers of Hyperspectral Imagary
200 Geng X R, et al. Sci China Inf Sci January 2011 Vol. 54 No. 1
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
7/28/2019 Matrix Calculation of High-dimensional Cross Product and Its Application in Automatic Recognition of the Endmembers of Hyperspectral Imagary
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-
7/28/2019 Matrix Calculation of High-dimensional Cross Product and Its Application in Automatic Recognition of the Endmembers of Hyperspectral Imagary
202 Geng X R, et al. Sci China Inf Sci January 2011 Vol. 54 No. 1
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.
7/28/2019 Matrix Calculation of High-dimensional Cross Product and Its Application in Automatic Recognition of the Endmembers of Hyperspectral Imagary