endmember extraction algorithms from hyperspectral images

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ANNALS OF GEOPHYSICS, VOL. 49, N. 1, February 2006

Key words endmember extraction – hyperspectralmixtures – linear spectral unmixing

1. Introduction

Imaging spectroscopy is the measurementand analysis of spectra acquired as images. Theuse of hyperspectral imaging sensor data tostudy the Earth’s surface is based on the capabil-ity of such sensors to provide hundreds of spec-tral bands (high resolution spectra), providingone spectrum per pixel, along with the imagedata. This capability can be used to determineEarth’s surface constituent signatures from thespectral information provided by the sensor.

The imaging spectroscopy concept can beconsidered in the general problem of solvingunknowns from measurements. Unknowns in-

clude all the absorbing and scattering compo-nents of the atmosphere and the surface that af-fects the radiance (Green et al., 1998).

If a problem has more unknowns than meas-urements, then the problem is underdetermined.If there are more measurements than unknownsthen the problem is over determined and solv-able. The imaging spectroscopy approach leadsto over determined, therefore solvable prob-lems.

The problem must be solved taking into ac-count all the unknowns that may affect the ex-pression of interest (there is no magic band forany single material). As the number of equa-tions (bands) increases, the problem is moreeasily solved.

In most cases, the sub-scene represented byeach observed pixel is not homogeneous. As aresult, the hyperspectral signature collected bythe sensor at each pixel is formed by an integra-tion of signatures, associated with the purestportions of the sub-scene. These signatures,which can be considered macroscopically pure,are usually named «endmembers» in hyperspec-tral analysis terminology (Bateson et al., 2000).

Endmember extraction algorithms from hyperspectral images

Pablo J. Martínez, Rosa M. Pérez, Antonio Plaza, Pedro L. Aguilar, María C. Cantero and Javier Plaza Computer Science Department, University of Extremadura, Cáceres, Spain

AbstractDuring the last years, several high-resolution sensors have been developed for hyperspectral remote sensing ap-plications. Some of these sensors are already available on space-borne devices. Space-borne sensors are current-ly acquiring a continual stream of hyperspectral data, and new efficient unsupervised algorithms are required toanalyze the great amount of data produced by these instruments. The identification of image endmembers is acrucial task in hyperspectral data exploitation. Once the individual endmembers have been identified, severalmethods can be used to map their spatial distribution, associations and abundances. This paper reviews the Pix-el Purity Index (PPI), N-FINDR and Automatic Morphological Endmember Extraction (AMEE) algorithms de-veloped to accomplish the task of finding appropriate image endmembers by applying them to real hyperspec-tral data. In order to compare the performance of these methods a metric based on the Root Mean Square Error(RMSE) between the estimated and reference abundance maps is used.

Mailing address: Dr. Pablo J. Martínez, ComputerScience Department, University of Extremadura, Avda. de la Universidad s/n, 10071 Cáceres, Spain; e-mail: [email protected]

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Pablo J. Martínez, Rosa M. Pérez, Antonio Plaza, Pedro L. Aguilar, María C. Cantero and Javier Plaza

In this paper, we review some endmemberextraction techniques and use a novel approachto compare the accuracy of these methods.

The rest of the paper is structured as fol-lows: Section 2 provides a description of theconcepts of endmember and fully constrainedlinear spectral unmixing. Section 3 discussesthe Pixel Purity Index (PPI), N-FINDR and Au-tomatic Morphological Endmember Extraction(AMEE) techniques. Section 4 describes the re-sult achieved after applying the proposed algo-rithms to a real mineral mapping application,and establishes a comparison between them, us-ing real data collected by the NASA Jet Propul-sion Laboratory’s Airborne Visible InfraRedImaging Spectrometer «AVIRIS». Finally, Sec-tion 5 includes some concluding statements andcomments on plausible future research.

2. Endmembers in hyperspectral images

In order to understand the endmember con-cept, we will analyze the spectrum compositionmodel, considering that a spectrum is an N-di-mensional vector.

The radiation source F is characterized by aspectral radiance f(m); this radiation will cross amedium (atmosphere), until it reacts with a sam-ple i; this medium will be characterized by atransmittance, g(m) in such a way that the radia-tion that reaches the sample will be f(m)g(m).

The interaction with the sample will be di-rected by its spectral reflectance t(m); the radi-

ation emerging the sample will be, therefore,f(m)g(m)t(m) and it will have to suffer a newinteraction with the propagation medium, insuch a way that the incident radiation over thesensor will be f(m)g(m)t(m)gl(m), agreeingwith spectrometer transfer function h(m), themeasurement result, will be

(2.1)

The endmember hyperspectral signature ti, is areflectance vector obtained from a measurementof a i pure canopy sample xi obtained in the sameconditions as the hyperspectral image (detectorh, source f and atmospheric conditions g and gl)

(2.2)

Usually the i components present in a non-uni-form sample do not participate in this process inthe same proportion; if we call ti(m) to the i com-ponent reflectance spectrum and ci to the abun-dance of i component in our sample, the measure-ment process will be like the one shown in fig. 1.

The composed reflectance spectrum t(m)can be expressed as a linear combination of theendmember reflectance signatures ti(m) wheneach isolated source i stimulates the detector sep-arately, i=1, ..., K, (where K is the number ofendmembers). According to the related notation,the superposition principle can be generalized by

(2.3)Rc ci i

i

K

1

= =t t=

/

( ) , , ..., ., , ,i i i i N1 2=t m t t t T6 @

( ) ( ) ( ) ( ) ( ) ( ) .x f g g h=m m m m m mt l

Fig. 1. Spectrum composition.

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where R=[t1, t2, ..., tk] and c=[c1, c2, ..., ck]T.For instance, a simple mixture model based onthree endmembers has the geometrical interpre-tation of a triangle whose vertices are the end-members. Cover fractions are determined bythe position of spectra within the triangle, andthey can be considered relative coordinates in anew reference system given by the endmem-bers, as shown in fig. 2.

Although the development of effective de-vices with new and improved technical capabili-ties stands out, the evolution in the design of al-gorithms for data processing is not as positive.Many of the techniques currently underway forhyperspectral data analysis and endmember ex-traction have their origins in classic spec-troscopy, and thus focus exclusively on the spec-tral nature of the data (Plaza et al., 2002). Avail-able analysis procedures do not usually take intoaccount information related to the spatial con-text, which can be useful to improve the obtainedclassification (Madhok and Langreb, 1999).

2.1. Full constraint linear spectral unmixing

The objective of Linear Spectral Unmixing(LSU) is to obtain the fractions of componentsin the mixture. Unmixing analysis generally be-

gins with a few basic endmembers or compo-nents. Endmembers such as green vegetation,soil, non-photosynthetic vegetation, and shadeare often used. The spectra of these endmem-bers may come from laboratory measurements,field measurements or from the imaging spec-trometer data set itself.

Each spectrum is then atmospherically cor-rected and the imaging spectrometer data set isthen decomposed into the fractions of the end-members that best replicate the measured spec-trum. The output of LSU analysis is a series ofendmember fraction images. These images re-port the derived distribution of the endmembersover the image.

Two constraints that are often consideredon this analysis are: that the fractions must sumto 1.0 and negative fractions are not allowed(FCLSU) (Green et al., 1998).

3. Background on endmember extractiontechniques

A number of algorithms based on the notionof spectral mixture modeling have been pro-posed to accomplish the complex task of find-ing appropriate endmembers for spectral un-mixing in multi/hyperspectral data.

Fig. 2. Mixture model based on three endmembers.

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3.1. PPI

One of the most successful approaches hasbeen the Pixel Purity Index or PPI (Boardmanet al., 1995), which is based on the geometry ofconvex sets (Ifarraguerri and Chang, 1999). PPIconsiders spectral pixels as vectors in a N-di-mensional space.

A dimensionality reduction is first applied tothe original data cube by using the MinimumNoise Fraction. The algorithm proceeds by gen-erating a large number of random N-dimensionalvectors, also called «skewers» (Theiler et al.,2000), through the dataset. Every data point isprojected onto each skewer, along which positionit is pointed out. The data points which corre-spond to extreme values in the direction of askewer are identified and placed on a list. Asmore skewers are generated, the list grows, andthe number of times a given pixel is placed on thislist is also tallied.

The pixels with the highest tallies are consid-ered the purest ones, since a pixel count providesa «pixel purity index». It is important to empha-size that the PPI algorithm does not identify a fi-nal list of endmembers. PPI was conceived not asa solution, but as a guide; in fact, the author pro-posed comparing the pure pixels with target spec-tra from a library and successively projecting thedata to lower dimensional spaces while endmem-bers were identified. There are several interactivesoftware tools oriented to perform this task, butthe supervised nature of such tools leads to thefact that the analysis of a scene usually requiresthe intervention of a highly trained analyst. Then,interactive solutions may not be effective in situ-ations where large scenes must be analyzedquickly as a matter of routine. In addition, ran-domness in the selection of skewers has beenidentified by some authors as a shortcoming ofthe algorithm.

The original implementation of PPI proposesthe use of unitary vectors as skewers in randomdirections of the N-dimensional space. This im-plementation may be improved by a careful se-lection of existing vectors to skew the dataset. In-telligent selection of skewers may result in amore efficient behavior of the algorithm. Sometools based on variations of PPI concepts havebeen proposed (Theiler et al., 2000).

3.2. N-FINDR

The N-FINDR method (Winter, 1999) is anautomated approach that finds the set of pixelswhich define the simplex with the maximumvolume, potentially inscribed within the data-set. First, a dimensionality reduction of theoriginal image is accomplished by MNF. Next,randomly selected pixels qualify as endmem-bers, and a trial volume is calculated. In orderto refine the initial volume estimation, a trialvolume is calculated for every pixel in eachendmember position by replacing that end-member and recalculating the volume. If the re-placement results in a volume increase, the pix-el replaces the endmember. This procedure,which does not require any input parameters, isrepeated until there are no replacements of end-members left. It should be noted that both PPIand N-FINDR rely on spectral properties of thedata alone, neglecting the information related tothe spatial arrangement of pixels in the scene.

The N-FINDR method is sensitive to the ran-dom selection of an initial set of endmembers. Ifthe initial estimation is appropriate, the algorithmwill not require to do many iterations until itreaches the optimum solution. On the contrary, anerroneous initial estimation may lead to a highcomputational complexity of the algorithm.

On the other hand, the algorithm recalculatesthe volume of the simplex every time a new pix-el is incorporated to the endmember set. Thisproperty makes the algorithm quite sensitive tonoise.

3.3. AMEE

Mathematical morphology is the science ofshape and structure, based on set-theoretical,topological and geometrical concepts (Serra,1982, 1993). Morphology was originally definedfor binary images and has been extended to thegrayscale (Sternberg, 2000) and color (Lambertand Chanussot, 2000) image cases, but it has sel-dom been used to process multi/hyperspectralimagery.

We propose a new application of mathemat-ical morphology focused on the automated ex-traction of endmembers from multi-dimension-

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al data. This application is fully described in(Plaza et al., 2002). In addition, we list key ad-vantages in the use of morphology to performthis task:

– A first major point is that the extraction ofendmembers is basically a non-linear task.

– Furthermore, morphology allows the intro-duction of a local-to-global approach in thesearch of endmembers by using spatial kernels(structure elements in the morphology jargon).These items define local neighbourhoods aroundeach pixel. This type of processing can be ap-plied to the search of convexities or extremitieswithin a data cloud of spectral points and to theuse of the spatial correlation between the data.

– As a final main step, morphological oper-ations are implemented by replacing a pixelwith a neighbor that satisfies a certain condi-tion. In binary/grayscale morphology, the con-dition is usually related to the digital value ofthe pixel, and the two basic morphological op-erations (dilation and erosion) are respectivelybased on the replacement of a pixel by theneighbor with the maximum and minimum val-ue. Since an endmember is defined as a spec-trally pure pixel that describes several mixedpixels in the scene, extended morphological op-erations can obviously contribute to the loca-tion of suitable pixels that replace others in thescene, according to some desired particularityof the pixel, for example, its spectral purity.

Therefore, we propose a mathematical frame-work to extend mathematical morphology to themulti-dimensional domain, which results in thedefinition of a set of spatial/spectral operationsthat can be used to extract reference spectral sig-natures.

The dilation and erosion operations have beenextended to the grayscale image case (Sternberg,2000). In grayscale morphology, images are han-dled as continuous valued sets. Let f(x, y) be agray level function representing an image, and Kanother set which comprises a finite number ofpixels. This set is usually referred to as a kernel orstructuring element in morphology terminology.Erosion and dilation of image f(x, y) by structur-ing element K are respectively written as

=( , ) ( , ) ( , )Minf k x y f x s y t k s t( , )s t k

7 + + -!

^ h " ,

(3.1)

where k(s, t) denotes the weight associated tothe different elements of the kernel. The maincomputational task in dealing with grayscalemorphological operations is the query for localmaxima or minima in a local search area aroundeach image pixel. This area is determined bythe size and shape of structuring element K.

If a dilation operation using K is applied overa grayscale image, then the local effect of the op-eration is the selection of the brightest pixel in a3×3-pixel search area around the target pixel.The constraints imposed on the kernel definitioncause all pixels lying within the kernel to be han-dled equally, i.e. no weight is associated with thepixels according to their position along the ker-nel, and the pixel with maximum digital value isselected. The previous operation is repeated withall the pixels of the image, leading to a new im-age (with the same dimensions as the original)where lighter zones are developed concerning thesize and shape of the structuring element. In con-trast, grayscale erosion has the global effect ofshrinking the lighter zones of the image.

3.3.1. Extending mathematical morphology to multi-spectral images

In grayscale morphology, the calculation ofmaximum and minimum gray values relies onthe definition of a simple ordering relation giv-en by the digital value of pixels. In the case ofmulti-spectral datasets, this natural assumptionis not straightforward, since there is no naturalmeans for the total ordering of multivariate pix-els. Hence, the greatest challenge in the task ofextending morphology operations to multi-spectral images is the definition of a vector or-dering relation related to pixel purity that al-lows us to determine the maximum and mini-mum elements in any family of N-dimensionalvectors (Plaza et al., 2002).

We have checked one possibility that relieson the calculation of the distance between everyspectral point in the kernel and the center of thekernel data cloud. Then, a local eccentricity

=( , ) ( , ) ( , )Maxf k x y f x s y t k s t( , )s t k

7 - - +!

^ h " ,

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measure for a particular N-dimensional point ofthe kernel f(x, y) can be obtained by calculatingthe distance dist(f, C) in relation to the center C(dist is a point-wise linear distance function –spectral angle). Distance Dl=dist(f, C) defines apartial ordering relation in the data cloud that al-lows the identification of the maximum as thatelement which is most «eccentric» or distantfrom the center (fig. 3a), while the minimum isthe closest element to the center (fig. 3b).

The result of applying an erosion/dilationoperation to a multi-spectral dataset is a newdata cube, with exactly the same dimensions asthe original, where each pixel has been replacedby the maximum/minimum of the kernel neigh-borhood considered.

In order to account for the «eccentricity» ofthe maximum pixel, we propose the combina-tion of the output provided by dilation and ero-sion operations. While dilation selects the purestpixel (the maximum is the more extremelyplaced pixel in the data cloud), erosion selectsthe most highly mixed pixel (the minimum isthe pixel that is more exactly placed in the cen-ter of the data cloud) in a certain spatial neigh-borhood. The distance between the maximumand the minimum (a widely used operation inclassic mathematical morp÷hology) provides aneccentricity value for the selected pixel that al-lows an evaluation of the quality of the pixel interms of its spectral purity. In order to incorpo-rate this idea, a new quality measure must be in-troduced. We define the Morphological Eccen-tricity Index (MEI) as the distance between themaximum element (obtained by a dilation) andminimum element (obtained by an erosion) inthe kernel.

3.3.2. Algorithm description

The Automated Morphological EndmemberExtraction (AMEE) method (Plaza et al., 2002)is described as follows: the input to the methodis the full data cube, with no previous dimen-sionality reduction or pre-processing.

Parameter L refers to the number of itera-tions that are performed by the algorithm. Para-meters Smin and Smax denote the minimum andmaximum kernel sizes that will be consideredin the iterative process, respectively. These pa-rameters are interrelated, as the number of iter-ations depends on the minimum and maximumkernel sizes under scrutiny.

Firstly, the minimum kernel size Smin is con-sidered. The kernel is moved through all thepixels of the image. The spectrally purest pixeland the spectrally most highly mixed pixel areobtained at each kernel neighborhood by multi-spectral dilation and erosion operations. A Mor-phological Eccentricity Index (MEI) value isassociated with the purest pixel by comparingthe result of the dilation to the result of erosion.The previously described operation is repeatedby using structuring elements of progressivelyincreased size, and the algorithm performs asmany iterations as needed until the maximumkernel size Smax is achieved.

The associated MEI value of selected pixelsat subsequent iterations is updated by means ofnewly obtained values, as a larger spatial con-text is considered, and a final MEI image isgenerated after L iterations.

Once the competitive endmember selectionprocess is finished, endmember extraction is fi-nally accomplished using a spatial-spectral seed-ed region growing procedure in which the MEIimage is used as a reference. Here, a thresholdparameter T is used to control the process (Plazaet al., 2002).

4. Results

In this section we intend to compare thequality of the endmembers extracted by thePPI, N-FINDR and AMEE algorithms when theground truth information is available and ex-pressed in a set of abundance values. In this ex-

Fig. 3a,b. a) Maximum and b) minimum in datacloud.

a b

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periment we consider that most of the imagepixels are a mixture of endmembers, and someof them are pure pixels.

The methodology of comparison is based inthe realization of the following steps:

1. The image endmembers are extracted us-ing one of the proposed algorithms.

2. Then, the abundance of each one of theendmembers in the image is estimated, using a FCLSU procedure (Fully Constrained LinearSpectral Unmixing) (Heinz and Chang, 2000).As a result, an abundance map is obtained foreach identified spectral signature.

3. The obtained abundance values are com-pared, on a pixel-by-pixel basis, with referencevalues estimated by USGS for the minerals in thisarea, using the RMSE error. It is important to em-phasize that the reference abundances were ob-tained using USGS Tetracorder algorithm, whichproduces spectral similarity scores which areconsideredin this study as approximated abun-dance scores. The considered reference maps areavailable from http://speclab.cr.usgs.gov/.

4. The tests described in this section havebeen carried out using the image AVCUP95, tak-

en by AVIRIS, characterized for having groundtruth information in the form of maps that con-tain the abundance of each mineral in each imagepixel.

This image was acquired in 1995 by AVIRISsensor over the mining region named Cuprite, inthe state of Nevada, U.S.A. Figure 4 shows theplacement of the image over an aerial photo-graph of the zone. The scene comprises some ar-eas composed of minerals, as well as abundantbare soils and an interstate road that crosses thezone in a North to South direction.

The image AVCUP95 consists of 400××350 pixels, each one of which contains 50 re-flectance values in the range of 2 to 2.5 nm.This range, placed in the SWIR-II region of thespectrum, is characterized by the manifestationof singularities that allow the discrimination ofa wide range of calcareous minerals (Clark,1999).

Each spectral value in the above mentionedrange is equivalent to 10 times the percentageof the reflectance in a given wavelength. Thisvalues have been obtained as a result of the ap-plication of the ATREM atmospheric correction

Fig. 4. Location of AVCUP95 image over an aerial photograph of Cuprite mining region, Nevada, U.S.A.

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method (Gao et al., 1993), followed by a post-processing with EFFORT method (Boardman,1998) over the image in radiance units, origi-nally captured by the sensor, suppressing the ef-fects caused by the atmosphere g and the illu-mination source f.

The ground truth information for the imageAVCUP95 has been published by the NorthAmerican Institute of Geological Studies, orU.S. Geological Survey (USGS from now on).The availability of this information of groundtruth, varied and with high quality, favouredAVCUP95 as the image used as a reference totest the performance of new algorithms.

When analysing this image, we centre ourstudy in the most characteristic minerals of theregion, that is, alunite, buddingtonite, calciteand kaolinite.

The results regarding the estimation ofabundances of alunite mineral are very precisein the three tested methods. The results ob-tained in the first experiment revealed that theendmember extracted by PPI method for alunitemineral had less spectral similitude with regardto the ground truth than the endmembers ex-tracted by N-FINDR and AMEE. However, thisfact does not seem to affect the abundances es-timation process significantly.

With the aim of establishing a quantitativeevaluation of the qualitative results previouslymentioned, table I shows the RMSE total errorsobtained when comparing each of the estimat-ed abundance maps with their correspondingground truth map.

The results in table I reveal a very similarbehaviour of the three methods, in global terms,for alunite mineral, with an error of 5% in eachof the cases.

Concerning buddingtonite, the calculatederror rises in the three cases, being always su-perior to 10%. This result confirms that thereare certain errors in the determination of thespectral shape of the endmember, probably dueto atmospheric effects.

Finally, the estimation of the abundance inthe case of the calcite and kaolinite minerals isquite precise in all cases, the PPI algorithm be-ing the one that offers the worst results for cal-cite mineral, with a global error of 9%, and N-FINDR offers the worst results for kaolinitemineral, with a global error of 6%.

The quantitative comparison in table I isuseful to obtain a global vision of the precisionof the considered methods when it deals withthe estimation of abundances.

This study does not reflect the changescaused in the calculated error while the abun-dance of the component varies. A very interest-ing question to study in depth is the evaluationof the presented methods in analyzing the mag-nitude of the calculated error when the materialabundance is small and when it is large.

5. Conclusions

Spectral unmixing is only possible when thenumber of endmembers of the image is lowerthan the number of channels, so that the systemof equations to be solved is over-dimensioned;as the system has a larger number of equationswe will be able to determine more endmemberswhen the spectra have more channels.

The best endmembers must be obtained inthe same condition as the unmixing spectrum(from the same image) preventing external fac-tors, such as the illumination or elevation an-gles, affecting the unmixing results.

The endmember extraction methods thatjointly exploit spatial and spectral informationobtain better quality endmembers than thosemethods which only utilize spectral informa-tion.

It is necessary to design tools for the evalu-ation of the quality of endmember extraction al-gorithms using synthetic images designed forthat purpose from a previously known abun-dance map, as well as real images obtained in

Table I. Global RMSE between the estimated abun-dances and the real ones for every mineral.

Mineral AMEE N-FINDR PPI

Alunite 0.05 0.05 0.05Buddingtonite 0.12 0.13 0.15

Calcite 0.07 0.08 0.09Kaolinite 0.05 0.06 0.04

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strictly controlled conditions, where it could bepossible to obtain the abundance map.

Acknowledgements

This work was supported by Regional Gov-ernment Junta of Extremadura by means of theproject 2PR03A061. The authors gratefully ac-knowledge Prof. Pedro Gómez Vilda (Universi-dad Politécnica de Madrid) for his valuablesuggestions and comments on the work de-scribed in this paper.

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