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Iterative spectral unmixing(ISU)F. Van Der MeerPublished online: 25 Nov 2010.

To cite this article: F. Van Der Meer (1999) Iterative spectral unmixing(ISU), International Journal of Remote Sensing, 20:17, 3431-3436, DOI:10.1080/014311699211462

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int. j. remote sensing, 1999, vol. 20, no. 17, 3431± 3436

Iterative spectral unmixing (ISU)

F. VAN DER MEERInternational Institute for Aerospace Survey and Earth Sciences ITC,Geology Division, Hengelosestraat 99, 7534 AE Enschede, the Netherlands

(Received 10 November 1998; in ® nal form 15 April 1999)

Abstract. Spectral unmixing techniques strive to ® nd proportions of end-members within a pixel from the observed mixed pixel spectrum and a numberof pure end-member spectraof known composition. The outcomes of suchanalysisare fraction (abundance) images for the selected (pure) end-members and a rootmean square (RMS) error estimate representing the di� erence between theobserved mixed spectrum and the calculated mixed spectrum. The RMS imagecan be used to select additional end-members and re-position existing ones. Thisis now done manually. In this Letter, an automated iterative approach is proposedusing the RMS error image to select additional end-members and re-distributeolder ones in order to increase the accuracyof the spectral unmixing. Optimizationcriteria are proposed to drive the iterative process including minimization of theaverage RMS, minimizing the spread of the RMS values, minimizing the spatialstructure of the RMS image, minimizing the spatial anisotropy of the RMS imageand minimizing the local variance. The preliminary results of the analysis indicatethat considerable improvement to the spectral unmixing results are achieved usingthe iterative spectral unmixing (ISU) approach.

1. Introduction

Radiation as re¯ ected from a heterogeneous ® eld of view on the ground can beconsidered the result of mixing a number of spectrally pure materials. Spectralunmixing techniques (Adams et al. 1985) provide estimates of the fractions of thesecomponents (or end-members). A linear combination of spectral end-members, Reij

(subscript i is the band number for the n bands and subscript j is the end-member)chosen to decompose themixed re¯ ectance spectrumof each pixel, R i , yields fractions,fj , for each end-member by ® nding a solution to

R i = �n

j= 1fj Reij + ei (1)

by inverse modelling minimising the error term (Settle and Drake 1993). Unmixingcan be further constrained by forcing the fractions to be within the range 0< fj < 1and their sum within 0< fj < 1. End-member spectra can be selected from a spectral(® eld or laboratory) library or they can be obtained directly from the image data,the latter option having the advantage that selected end-members are collected undersimilar atmospheric conditions. A root-mean square (RMS) error can be calculated

International Journal of Remote SensingISSN 0143-1161 print/ISSN 1366-5901 online Ñ 1999 Taylor & Francis Ltd

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F. van der Meer3432

from the di� erence of the modelled (Rj k ) and measured (R ¾j k ) pixel spectrum as

RMS= �m

k= 1

S �n

j= 1(Rj k Õ R ¾j k)2 /n

m(2)

where n is the number of spectral bands and m the number of pixels within theimage. The RMS image is often used to select additional end-members located inareas of relative high unmixing errors. In this Letter an automated approach isoutlined for the process of unmixing including using the RMS image for end-memberre-positioning: iterative spectral unmixing (ISU).

2. Iterative spectral unmixing

2.1. RationaleA major problem in a spectral unmixing analysis is the selection of end-members

that best represent the spectral dimensionality and spectral variability of the imagedata. The RMS error image can be used to select additional end-members or tore-positioning end-members in subsequent further analysis. This is labour intensiveand subject to the bias of the person performing the analysis. Furthermore, the questionwhat is the optimal set of end-members for spectrally analysing the data set?’ isdependent on the application for which the fractional cover estimates are needed.Therefore, several optimization criteria can be de® ned that are valid for these applica-tions. In this Letter, an iterative procedure is proposed to reach to an optimumspectral characterization of the image data through optimization criteria applied tothe RMS error estimates assuming the use of derived image end-members. The proced-ure will be outlined step-wise below, in the resulting section an example will be given.

2.2. Step 1: Deriving image end-members’Deriving image end-members starts by determining the number of end-members

needed to optimally characterize the image data using the Minimum Noise Fraction(MNF) transformation (Green et al. 1988). The MNF transformation is a two-stepprincipal component transformation where during the ® rst step using the noise covari-ance matrix the noise is decorrelated to have unit variance and no band-to-bandcorrelation. The second principal component transformation results in a data set wherecomponents are ranked in terms of noise equivalent radiance. The MNF transforma-tion can be illustrated mathematically as follows. Consider stochastic variables

Z(x) = [Z1 (x) ¼ Zp (x)] T (3)

with expectation E{Z}=0 and dispersion D{Z}= S. New mutually orthogonalvariables with maximum signal-to-noise ratio can be constructed as

Yi = a i1 Z1 + ¼ a ip Zp = aTi , i= 1, ¼ ,p (4)

Assuming additive noise (i.e., Z (x) = S(x)+ N(x) where S and N are uncorrelatedsignal and noise components, respectively) the response equalswhere and are disper-sion matrices for S and N, respectively. The signal-to-noise ratio (estimated from theratio of mean and variance in a local window) for that needs to be maximised canbe de® ned as (Lee et al. 1990; Nielsen and Larsen, 1994)

VAR{aTi S}

VAR{aTi N}

=aT

i Sa i

aTi SNa i

Õ 1=1li

Õ 1 andaT

i SNa i

aTi Sa i

= li (5)

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Remote Sensing L etters 3433

the factors li are the eigenvalues of SN with respect to S and the factors ai are thecorresponding conjugate eigenvectors. The inherent spectral dimensionality of the dataset can be found by examining the eigenvalues and associated MNF images. Some ofthese MNF images are associated with large eigenvalues and coherent (MNF) eigen-images while the remainder if the MNF bands have near-unity eigenvalues and imagesdominated by noise. Thus the MNF eigenvalues and eigenimages yield the absolutenumber of end-members required to model the image spectral response.

Next, the locations of the end-member pixels need to be determined from theimage data. Here the Pixel Purity Index (PPI; Boardman et al. 1995) based on theapproach developed by Smith et al. (1985) is applied to the MNF transformed data.This approach regards spectra as points in an n-dimensional space (n being thenumber of bands). The body (referred to as simplex) that spans the data points in an=2-dimensional space is a triangle having n+1 facets. The purest end-membersare found at the locations in this space where the sides of the triangle intersect. Thisprinciple can be extended to higher dimensions in a similar way. The PPI approachcompares the pixels in a scene with the best ® tting simplex and records the numberof times a pixel is found at the extreme facets of the simplex. Portraying this as animage of cumulative count values allows to identify the locations in the data spaceof the initial set of spectrally pure end-members.

2.3. Step 2: First iteration of spectral unmixing’Spectral unmixing is performed using the n-end-members derived from the image

data leading to n-fraction images and an initial RMS error image.

2.4. Step 3: Calculating the optimization criterion’An optimization criterion is de® ned and calculated on the RMS image.

Optimization criteria that havebeen de® nedand tested in thepresent analysis include:E Minimizing the mean RMS; this is calculated as the weighted average RMS

value.E Minimizing the spread of the RMS values; this is calculated as the maximum

minus the minimum RMS value.E Minimizing the spatial structure in the RMS values; a measure for the spatial

structure is found in the (semi) variogram. The variogram in remote sensingcan be estimated from p(h) pairs of observations, z v(xl ) and z v(xl+h) withl= 1, 2, ¼ , p(h) as

c(h)= 1/2p (h) �p (h)

l= 1{z v (xl ) Õ z v (xl+ h )}2 (6)

Parameters of a ® tted mathematical function (the variogram model) include arange, a nugget and a sill. The range is the distance at which the curve levelsof to a constant value of semi-variance indicating the spatial scale of the patternin the image. The nugget variance is a non-zero y-intercept of the variogrammodel indicative of variability at a resolution smaller than the image resolutionoften indicating the level of uncorrelated noise in the data. The sill, i.e. thevalue of maximum variance, is equivalent to the variance of the image pixelvalues. Optimizing the RMS by minimizing the spatial structure of the RMSimage strives at reaching a pure nugget variogram of which the level of thenugget variance represents the noise characteristics of the data. A measure forthe spatial structure to be minimized is derived from the ratio of the totalvariance over the non-structurally controlled nugget variance. When no spatial

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F. van der Meer3434

structure exists in a RMS image, the total variance is equal to the nuggetvariance and the ratio is 1. Else the ratio is larger than 1.

E Minimizing the directional spatial structure or anisotropy ; an e� ective way todetect anisotropy in the RMS pattern is by calculating its 2D-variogram (i.e.the variogram surface). In this surface the (semi) variance is not only relatedto the mere distance between two points (the lag), but also to the direction.In calculating the variogram value for pairs of points separated by a vector(hx , hy) all pairs are grouped together whose separation is hx Ô Dx, hy Ô Dy.In this case Dx and Dy are half the support size, i.e. (in a rectangular coordinatesystem) the pixel size (Isaaks and Srivastava 1989). The result is a surface withfor each vector a computed (semi) variance. A measure for the directionalspatial structure or anisotropy is the ratio of the longer axis of the anisotropyellipse and the shortest axis of the anisotropy ellipse. When no anisotropyexists in the data, this ratio is 1 (i.e. the spatial structure is isotropic).

E Minimizing the local variance; the local variance can be computed over a(2n+1)Ö (2m+1) window as (Woodcock and Strahler 1987)

s2ij = 1/2n+1)(2m+1) �

i+ n

k= iÕ

n¾ s

j+ m

l= jÕ

m{z (xkl ) Õ u ij }2 (7)

where uij is the mean of the (2n+1) Ö (2m+1) window centred on xij and z(xij )is the value of the pixel locatedat xij in the ith row and the jth column of the image.

2.5. Step 4: Iterative spectral unmixing’The pixel underlying the end-member in which neighbourhood the largest RMS

errors occur as compared to the other end-members is removed and replaced by anend-member on the cluster of pixels of highest error in the RMS image. Spectralunmixing is performed, the optimization criterion (step 3) calculated, if this hasimproved the new end-member set is adopted else the process continues with the oldset of end-members (® gure 1). The process of unmixing continues until the optimizationcriterion is satis® ed or until no signi® cant improvements in this criterion are observed.

3. DiscussionData from the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) from

the Cuprite mining district of Nevada acquired in 1995 were used as a ® rst test ofthe algorithm (Van der Meer and Bakker 1997). The shortwave infrared bands wereselected forming a data cube of 40 bands from 2000 to 2480nm. These were trans-formed to MNF bands from which the spectral dimensionality was derived leadingto 12 end-members. Subsequently, the pixel purity index was calculated and the 12most extreme pixels were marked and their spectra extracted from the AVIRIS datacube. These were used for the ® rst iteration of the spectral unmixing analysis whichwas continued over 25 iterations (® gure 1). The results of the iterative approach areshown in table 1. The degree of the change (improvement) of the various optimizationcriteria is di� erent as can be seen from table 1. The largest improvements (calculatedas the statistic at iteration 0 divided by the statistic at iteration 25) are seen in themean (400%) and local variance (> 300%) criteria while the spread (120%) and thespatial structure (180%) and the directional spatial structure (200%) do not seemto lead to signi® cant improvement. This could be largely induced by the fact thatthese criteria are not applicable to the data set being analysed. It should be notedthat it has not been attempted to investigate the ® nal end-member set in detail. Thusno information is provided on the kind of end-members that we are extracting from

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Remote Sensing L etters 3435

Figure 1. Flow chart explaining the iterative spectral unmixing (ISU) approach.

Table 1. Results of iterative spectral unmixing analysis (between brackets the average RMSafter the iteration is given).

Mean Spread Spatial Directional spatial LocalIteration (%) (%)* structure** structure*** variance

0 20.7 58 (20.7) 2.0 (20.7) 2.5 (20.7) 10.2 (20.7)5 10.3 55 (19.9) 1.8 (20.1) 2.2 (20.0) 8.1 (15.2)10 7.1 54 (20.0) 1.5 (18.0) 2.0 (19.2) 7.5 (9.7)15 5.8 53 (20.5) 1.3 (17.2) 1.8 (18.0) 5.1 (7.8)20 5.5 50 (19.5) 1.2 (16.5) 1.4 (17.2) 4.4 (7.5)25 5.2 48 (19.0) 1.1 (16.0) 1.3 (15.5) 3.25(7.0)

*Maximum minus minimum RMS.**Ratio total variance/non-structurally controlled variance.***Ratio longest/shortest anisotropy ellipse axis.

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Iterative spectral unmixing (ISU)3436

the data cube. This would be di� cult given the limited ground truth that we haveavailable. In table 1, the average RMS values for the di� erent criteria and subsequentiterations are given. These show that the iterative approach leads to increase of theaccuracy in most cases, however this increase is only signi® cant for the mean andlocal variance-criteria. In case of the other criteria even some increase of averageRMS values could be observed after some iterations.

4. ConclusionsThe RMS image that results from spectral unmixing provides important informa-

tion on the spectral decomposition of the image data that can be used to selectadditional end-members or re-position existing ones. If this is done by an interpreterit creates bias and is labour intensive. Furthermore, the optimal end-member set isalso dependent on the application for which the results of the mixture modelling areto be used. Iterative Spectral Unmixing (ISU) overcomes these problems becauseend-member sets are optimized using a (selectable) criterion. In all cases tested sofar, the iterative approach led to increase of the accuracy. Work is ongoing to changethe ISU algorithm such that it allows a variable number of end-members to be usedfollowing each iteration rather than a ® xed number based on the estimated spectraldimensionality of the data set.

AcknowledgmentsI am grateful for the critical review and constructive remarks from three anony-

mous IJRS referees that helped to improve the contents and clarity of the manuscript.The MNF transformation, PPI analysis and spectral unmixing are an integral partof the ENVI[ software. The ISU is prototyped in IDL[ using the results obtainedfrom the above.

ReferencesAdams, J. B., Smith, M. O. , and Johnson, P. E., 1985, Spectral mixture modelling: a new analysis

of rock and soil types at the Viking Lander 1 site. Journal of Geophysical Research, 91,8098± 8112.

Boardman, J. W., Knuse, F. A., and Green, R. O., 1995, Mapping target signatures via partialunmixing of AVIRIS data. In Proceedings of the Fifth JPL Airborne Earth ScienceWorkshop (Pasadena: JPL Publication 95± 1), pp. 23± 26.

Green, A. A., Berman, M., Switzer, P., and Graig, M. D., 1988, A transformation for orderingmultispectral data in terms of image quality with implications for noise removal. IEEET ransactions on Geoscience and Remote Sensing, 26, 65± 74.

Isaaks, H. S., and Srivastava, R. M., 1989, Introduction to Applied Geostatistics, 1st edition (NewYork: Oxford University Press).

Lee, J. B., Woodhyatt, S., and Berman, M., 1990, Enhancement of high spectral resolutionremote-sensing data by a noise-adjusted principal components transform. IEEET ransactions on Geoscience and Remote Sensing, 28, 295± 304.

Nielsen, A. A., and Larsen, R., 1994, Restoration of GERIS data using the Maximum NoiseFractions Transform. In Proceedings of the First International Airborne Remote SensingConference and Exhibition (ERIM), Strasbourg, France (AnnArbor: ERIMInternational),pp. 557± 568.

Settle, J. J., and Drake, N. A., 1993, Linear mixing and the estimation of ground coverproportions. International Journal of Remote Sensing, 14, 1159± 1177.

Smith, M. O., Johnson, P. E., and Adams, J. B., 1985, Quantitative determination of mineraltypes and abundances fromre¯ ectance spectra usingprincipal component analysis. Journalof Geophysical Research, 90, 797± 804.

Van der Meer, F., and Bakker, W., 1997, Cross Correlogram Spectral Matching (CCSM):application to surface mineralogical mapping using AVIRIS data from Cuprite, Nevada.Remote Sensing of Environment, 61, 371± 382.

Woodcock, C. E., and Strahler, A. H., 1987, The factor of scale in remote sensing. RemoteSensing of Environment, 21, 311± 332.

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