898 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE
898IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 43, NO. 4, APRIL 2005Vertex Component Analysis: A Fast Algorithm toUnmix Hyperspectral DataJosé M. P. Nascimento, Student Member, IEEE, and José M. Bioucas Dias, Member, IEEEAbstract—Given a set of mixed spectral (multispectral or hyperspectral) vectors, linear spectral mixture analysis, or linearunmixing, aims at estimating the number of reference substances,also called endmembers, their spectral signatures, and theirabundance fractions. This paper presents a new method forunsupervised endmember extraction from hyperspectral data,termed vertex component analysis (VCA). The algorithm exploitstwo facts: 1) the endmembers are the vertices of a simplex and 2)the affine transformation of a simplex is also a simplex. In a seriesof experiments using simulated and real data, the VCA algorithmcompetes with state-of-the-art methods, with a computationalcomplexity between one and two orders of magnitude lower thanthe best available method.Index Terms—Linear unmixing, simplex, spectral mixturemodel, unmixing hypespectral data, unsupervised endmemberextraction, vertex component analysis (VCA).I. INTRODUCTIONHYPERSPECTRAL remote sensing exploits the electromagnetic (EM) scattering patterns of different materials atspecific wavelengths [1], [2]. Hyperspectral sensors have beendeveloped to sample the scattered portion of the EM spectrumextending from the visible region through the near-infrared andmidinfrared, in hundreds of narrow contiguous bands [3], [4].The number and variety of potential civilian and military applications of hyperspectral remote sensing is enormous [5], [6].Very often, the resolution cell corresponding to a single pixelin an image contains several substances (endmembers) [3]. Inthis situation, the scattered energy is a mixing of the endmemberspectra. A challenging task underlying many hyperspectral imagery applications is then decomposing a mixed pixel into a collection of reflectance spectra, called endmember signatures, andthe corresponding abundance fractions [7]–[9].Depending on the mixing scales at each pixel, the observedmixture is either linear or nonlinear [10], [11]. A linear mixingmodel holds approximately when the mixing scale is macroscopic [12] and there is negligible interaction among distinctendmembers [2], [13]. If, however, the mixing scale is microscopic (or intimate mixtures) [14], [15] and the incident solarManuscript received January 6, 2004; revised December 21, 2004. This workwas supported in part by the Fundação para a Ciência e Tecnologia under theProjects POSI/34071/CPS/2000 and PDCTE/CPS/49967/2003 and in part bythe Departamento de Engenharia de Electrónica e Telecomunicações e de Computadores of the Instituto Superior de Engenharia de Lisboa.J. M. P. Nascimento is with the Instituto Superior de Engenharia de Lisboaand the Instituto de Telecomunicações, 1949-001 Lisbon, Portugal (e-mail:zen@isel.pt).J. M. Bioucas Dias is with the Instituto de Telecomunicações and the InstitutoSuperiror Técnico, 1949-001 Lisbon, Portugal (e-mail: bioucas@lx.it.pt).Digital Object Identifier 10.1109/TGRS.2005.844293radiation is scattered by the scene through multiple bounces involving several endmembers [16], the linear model is no longeraccurate.Linear spectral unmixing has been intensively researchedin the last years [8], [9], [11], [17]–[20]. It considers that amixed pixel is a linear combination of endmember signaturesweighted by the correspondent abundance fractions. Underthis model, and assuming that the number of substances andtheir reflectance spectra are known, hyperspectral unmixing isa linear problem for which many solutions have been proposed(e.g., maximum-likelihood estimation [7], spectral signaturematching [21], spectral angle mapper [22], subspace projectionmethods [23], [24], and constrained least squares [25]).In most cases, the number of substances and their reflectancesare not known and, then, hyperspectral unmixing falls into theclass of blind source separation problems [26]. independentcomponent analysis (ICA) has recently been proposed as a toolto blindly unmix hyperspectral data [27]–[30]. ICA is based onthe assumption of mutually independent sources (abundancefractions), which is not the case of hyperspectral data, sincethe sum of abundance fractions is constant, implying statisticaldependence among them. This dependence compromises ICAapplicability to hyperspectral images as shown in [20] and[31]. In fact, ICA finds the endmember signatures by multiplying the spectral vectors with an unmixing matrix whichminimizes the mutual information among channels. If sourcesare independent, ICA provides the correct unmixing, since theminimum of the mutual information corresponds to and onlyto independent sources. This is no longer true for dependentfractional abundances. Nevertheless, some endmembers may beapproximately unmixed. These aspects are addressed in [31].Under the linear mixing model, the observations from a sceneare in a simplex whose vertices correspond to the endmembers.Several approaches [32]–[34] have exploited this geometric feature of hyperspectral mixtures [33].The minimum volume transform (MVT) algorithm [34] determines the simplex of minimum volume containing the data.The method presented in [35] is also of MVT type, but by introducing the notion of bundles, it takes into account the endmember variability usually present in hyperspectral mixtures.The MVT type approaches are complex from the computational point of view. Usually, these algorithms first find theconvex hull defined by the observed data and then fit a minimumvolume simplex to it. For example, the gift wrapping algorithm[36] computes the convex hull of data points in a -dimensional space with a computational complexity of,whereis the highest integer lower or equal than , and isthe number of samples. The complexity of the method presented0196-2892/ 20.00 2005 IEEE
NASCIMENTO AND DIAS: VERTEX COMPONENT ANALYSISin [35] is even higher, since the temperature of the simulated anlaw [37] tonealing algorithm therein used shall follow aassure convergence (in probability) to the desired solution.Aiming at a lower computational complexity, some algorithms such as the pixel purity index (PPI) [33] and theN-FINDR [38] still find the minimum volume simplex containing the data cloud, but they assume the presence in the dataof at least one pure pixel of each endmember. This is a strongrequisite that may not hold in some datasets. In any case, thesealgorithms find the set of most pure pixels in the data.The PPI algorithm uses the minimum-noise fraction (MNF)[39] as a preprocessing step to reduce dimensionality and toimprove the signal-to-noise ratio (SNR). The algorithm thenprojects every spectral vector onto skewers (large number ofrandom vectors) [33], [40], [41]. The points corresponding toextremes, for each skewer direction, are stored. A cumulativeaccount records the number of times each pixel (i.e., a givenspectral vector) is found to be an extreme. The pixels with thehighest scores are the purest ones.The N-FINDR algorithm [38] is based on the fact that inspectral dimensions, the -volume defined by a simplex formedby the purest pixels is larger than any other volume defined byany other combination of pixels. This algorithm finds the set ofpixels defining the largest volume by inflating a simplex insidethe data.ORASIS [42], [43] is a hyperspectral framework developedby the Naval Research Laboratory consisting of several algorithms organized in six modules: exemplar selector, adaptativelearner, demixer, knowledge base/spectral library, and spatialpostprocessor. The first step consists in flat fielding the spectra.Next, the exemplar selection module is used to select spectralvectors that best represent the smaller convex cone containingthe data. The other pixels are rejected when the spectral angledistance is less than a given threshold. The procedure finds thebasis for a subspace of a lower dimension using a modifiedGram-Schmidt orthogonalization. The selected vectors are thenprojected onto this subspace, and a simplex is found by an MVTprocess. ORASIS is oriented to real-time target detection fromuncrewed air vehicles using hyperspectral data [44].In this paper we introduce the vertex component analysis(VCA) algorithm to unmix linear mixtures of endmemberspectra. The algorithm is unsupervised and exploits two facts:1) the endmembers are the vertices of a simplex and 2) theaffine transformation of a simplex is also a simplex. It workswith unprojected and with projected data. As PPI and N-FINDRalgorithms, VCA also assumes the presence of pure pixels inthe data. The algorithm iteratively projects data onto a directionorthogonal to the subspace spanned by the endmembers alreadydetermined. The new endmember signature corresponds tothe extreme of the projection. The algorithm iterates until allendmembers are exhausted. VCA performs much better thanPPI and better than or comparable to N-FINDR; yet it hasa computational complexity between one and two orders ofmagnitude lower than N-FINDR.The paper is structured as follows. Section II describes thegeometric fundamentals of the proposed method. Sections IIIand IV evaluate the proposed algorithm using simulated and real899data, respectively. Section V ends the paper by presenting someconcluding remarks.II. VERTEX COMPONENT ANALYSIS ALGORITHMAssuming the linear mixing scenario, each observed spectralvector is given by(1)whereis an-vector ( is the number of bands),is the mixing matrix (denotes theth endmember signature and is the number of endmembers( is a scale factorpresent in the covered area),modeling illumination variability due to surface topography),is the abundance vector containingthe fractions of each endmember (the notationstands forvector transposed) and models system additive noise.Owing to physical constraints [19], abundance fractions areand satisfy the so-called positivity connonnegativestraint, where is avector of ones. Each pixelcan be viewed as a vector in an -dimensional Euclidean space,where each channel is assigned to one axis of space. Since thesetis a simplex, then the setis also a simplex.However, even assuming, the observed vector set belongs tothat is a convex cone, owing to scale factor . Fig. 1(a) illustrates a simplex and a cone, projected on a two-dimensional subspace, defined by a mixture of three endmembers. The simplexboundary is a triangle whose vertices correspond to the endmembers shown in Fig. 2. Small and medium dots are simulatedmixed spectra belonging to the simplexand to thecone, respectively.The projective projection of the convex coneonto a properly chosen hyperplane is a simplex with vertices correspondingto the vertices of the simplex . This is illustrated in Fig. 1(b).The simplexisthe projective projection of the convex coneonto the plane, where the choice of assures that there is no observed vectors orthogonal to it.After identifying , the VCA algorithm iteratively projectsdata onto a direction orthogonal to the subspace spanned by theendmembers already determined. The new endmember signature corresponds to the extreme of the projection. Fig. 1(b) illustrates the two iterations of VCA algorithm applied to the simplexdefined by the mixture of two endmembers. In the firstiteration, data are projected onto the first direction . The extreme of the projection corresponds to endmember. In thenext iteration, endmemberis found by projecting data ontodirection , which is orthogonal to. The algorithm iteratesuntil the number of endmembers is exhausted.A. Dimensionality ReductionUnder the linear observation model, spectral vectors are in asubspace of dimension . If, it is worthy to project theobserved spectral vectors onto the subspace signal. This leads
900IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 43, NO. 4, APRIL 2005(a)(b)Fig. 1. (a) Two-dimensional scatterplot of mixtures of the three endmembersshown in Fig. 2. Circles denote pure materials. (b) Illustration of the VCAalgorithm.Fig. 2.Reflectances of carnallite, ammonioalunite, and biotite.to significant savings in computational complexity and to SNRimprovements.Principal component analysis (PCA) [45], maximum-noisefraction (MNF) [46], and singular value decomposition (SVD)[47] are three well-known projection techniques widely used inremote sensing. PCA, also known as Karhunen–Loéve transform, seeks the projection that best represents data in a leastsquares sense; MNF seeks the projection that optimizes SNR;and SVD provides the projection that best represents data in themaximum-power sense. PCA and MNF are equal in the caseFig. 3. Scatterplot (bands 827 nm and 1780 nm) of the threeendmembers mixture. (a) Unprojected data. (b) Projected data using SVD. Solidand dashed lines represent, respectively, simplexes computed from original andestimated endmembers (using VCA).of white noise. SVD and PCA are also equal in the case ofzero-mean data.As discussed before, in the absence of noise, observed veccontained in a subspaceoftors lie in a convex conedimension . The VCA algorithm starts by identifyingbySVD and then projects points inonto a simplexby computing[see Fig. 1(b)]. This simplex is containedin an affine set of dimension. We note that the rational underlying the VCA algorithm is still valid if the observed datasetis projected onto any subspaceof dimension , for, i.e., the projection of the coneontofollowedby a projective projection is also a simplex with the same vertices. Of course, the SNR decreases as increases.For illustration purposes, a simulated scene was generated according to (1). Three spectral signatures (A—biotite, B—carnallite, and C—ammonioalunite) were selectedfrom the U.S. Geological Survey (USGS) digital spectrallibrary [48] (see Fig. 2); the abundance fractions follow aDirichlet distribution; parameter is set to 1; and the noiseis zero-mean white Gaussian with covariance matrix,where is the identity matrix andleading toa SNRdB. Fig. 3(a)presents a scatterplot of the simulated spectral mixtures withoutprojection (bandsnm andnm). Twotriangles are also plotted whose vertices represent the true endmembers (solid line) and the estimated endmembers (dashedline) by the VCA algorithm, respectively. Fig. 3(b) presents ascatterplot (same bands) of projected data onto the estimatedaffine set of dimension two inferred by SVD. Noise is clearlyreduced, leading to a visible improvement on the VCA results.As referred before, we apply the rescalingto get ridof the topographic modulation factor. As the SNR decreases,this rescaling amplifies noise, being preferable to identify directly the affine space of dimensionby using only PCA.This phenomenon is illustrated in Fig. 4, where data clouds(noiseless and noisy) generated by two signatures are shown.Affines spacesandidentified, respectively, by PCAof dimensionand SVD of dimension followed by projective projection are schematized by straight lines. In the absenceis better identified by projectiveof noise, the direction of(better than); in the presence ofprojection onto
NASCIMENTO AND DIAS: VERTEX COMPONENT ANALYSIS901strong noise, the direction ofis better identified by orthog(better than). As a concluonal projection ontosion, when the SNR is higher than a given threshold SNR ,data is projected ontofollowed by the rescaling;otherwise data are projected onto. Based on experimentalresults, we propose the threshold SNRdB.Since for zero-mean white noise SNR, thenwe conclude that at SNR ,, i.e., theSNR corresponds to the fixed valueof the SNR measured with respect to the signal subspace.B. VCA AlgorithmThe pseudocode for the VCA method is shown in Algorithm1. Symbolsandstand for the th column ofand for the th to th columns of , respectively. Symbolstands for the estimated mixing matrix.Algorithm 1: Vertex Component Analysis (VCA)INPUT p, R [r1 ; r2 ; . . . ; rN ]1: SNRth 15 10 log10 (p) dB2: if SNR SNRth then3: d : p;4: X : UTd R; {Ud obtained by SVD}5: u : mean(X); {u is a 1 2 d vector}6: [Y]:;j : [X]:;j ([X]T:;j u); {projective projection}7: else8: d : p 0 1;T ([R]:;j 0 r); {Ud obtained by PCA}9: [X]:;j : Ud10: c : arg maxj 1.N k[X]:;j k;11: c : [cjcj . . . jc]; {c is a 1 2 N vector}12: Y : Xc13: end if14: A : [eu j0j . . . j0]; {eu [0; . . . ; 0; 1]T and A is a p 2 pauxiliary matrix}15: for i : 1 to pdo16: w : randn(0; Ip ); {w is a zero-mean random Gaussianvector of covariance Ip }17: f : ((I 0 AA# )w) (k(I 0 AA# )wk); {f is a vectororthonormal to the subspace spanned by [A]:;1:i .}18: v : f T Y ;19: k : arg maxj 1;.;N j[v]:;j j; {find the projection extreme.}20: [A]:;i : [Y]:;k ;21: [indice]i : k ; {stores the pixel index.}22: end for23: if SNR SNRth then24: M : Ud [X]:;indice ; {M is a L 2 p estimated mixingmatrix}25: else26: M : Ud [X]:;indice r; {M is a L 2 p estimated mixingmatrix}27: end ifStep 2 tests if the SNR is higher than SNR in order to decidewhether the data are to be projected onto a subspace of dimension or. In the first case the projection matrixisFig. 4.Illustration of the noise effect on the dimensionality reduction.obtained by SVD from, whereand is the number of pixels. In the second case the projection, where is theis obtained by PCA fromsample mean of, for.Steps 4 and 9 assure that the inner product between any vectorand vector is nonnegative, a crucial condition for theVCA algorithm to work correctly. The chosen value of, assures that the colatitude angle between and any vectoris between 0 and 45 , thenavoiding numerical errors which otherwise would occur for angles near 90 .Step 14 initializes the auxiliary matrix , which stores theprojection of the estimated endmembers signatures. Assumethat there exists at least one pure pixel of each endmember inthe input sample[see Fig. 1(b)]. Each time the loop for isexecuted, a vector orthonormal to the space spanned by thecolumns of the auxiliary matrix is randomly generated andis projected onto . Notationstands for the pseudoinversematrix. Since we assume that pure endmembers occupy thevertices of a simplex, then, for,where values and correspond to and only to pure pixels. Westore the endmember signature corresponding to.The next time loop for is executed, is orthogonal to the spacespanned by the signatures already determined. Since is theprojection of a zero-mean Gaussian independent random vectoronto the orthogonal space spanned by the columns of,then the probability of being null is zero. Notice that theunderling reason for generating a random vector is only to geta non null projection onto the orthogonal space generated bythe columns of . Fig. 1(b) shows the input samples and thechosen pixels, after the projection. Then a secondvector orthonormal to the endmember is generated and thesecond endmember is stored. Finally, steps 24 and 26 computethe columns of matrix , which contain the estimated endmembers signatures in the -dimensional space.
902IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 43, NO. 4, APRIL 2005III. EVALUATION OF THE VCA ALGORITHMIn this section, we compare VCA, PPI, and N-FINDR algorithms. N-FINDR and PPI were coded accordingly to [38]and [33], respectively. Regarding PPI, the number of skewersmust be large [39], [40], [49]–[51]. Based on Monte Carloruns, we concluded that the minimum number of skewersbeyond which there is no unmixing improvements is about1000. All experiments are based on simulated scenes fromwhich we know the signature endmembers and their fractional abundances. Estimated endmembers are the columns of. We also compare estimated abun,(standsdance fractions given byfor pseudoinverse of ) with the true abundance fractions.To evaluate the performance of the three algorithms,andwe compute vectors of angleswith1(2)(3)is the angle between vectorsand( th endwheremember signature estimate) and is the angle between vectorsand(vectors offormed by the th lines of matrices and, respectively). The symmetricKullback distance [52], a relative entropy-based distance, is another error measure used to compare similarity between signatures, namely under the name spectral information divergence(SID) [53]. SID is defined bySIDwheregiven by(4)is the relative entropy ofwith respect toFirst scenario (N 1000, p 3, L 224, 1 3, 1). (a) rmsSID as function of SNR. (b) rmsSAE as function ofSNR. (c) rmsFAAE as function of SNRs.Fig. 5.(5)and.andBased on , , andSIDSIDSID, we estimate the following rms error distances:(6)(7) 20,fractions. Herein we name , , andas rmsSAE, rmsSID,and rmsFAAE, respectively (SAE stands for signature angleerror and FAAE stands for fractional abundance angle error).Mean values in (6)–(8) are approximated by sample meansbased on 100 Monte Carlo runs.In all experiments, the spectral signatures are selectedfrom the USGS digital spectral library [48]. Fig. 2 showsthree of these endmember signatures. Abundance fractionsare generated according to a Dirichlet distribution given by(8)denotes the expectation operator. The first two quanwheretities measure distances betweenand, for;the third is similar to the first, but for the estimated abundancex; yi stands for the inner product x y.1Notation h(9)where,,is theexpected value of the th endmember fraction, anddenotesthe Gamma function. Parameter is Betadistributed,
NASCIMENTO AND DIAS: VERTEX COMPONENT ANALYSISFig. 6. Illustration of the absence of pure pixels (N 1000, p 3, L 224, 1 3, 1). Scatterplot (bands 827 nm and 1780 nm), with fractional abundance smaller than 0.2 rejected.903Fig. 8. Robustness to the topographic modulation (N 1000, p 3, L 1), rmsSEA as function of224, 1 3, SNR 20 dB,the (variance of ).Fig. 9. The rmsSEA as function of the number of pixels in a scene (p 6,L 224, 1 3, SNR 20 dB, 20, 1).enforcing positivity and full additivity constraints, displays awide range of shapes, depending on the parameters.This flexibility underlies its choice in our simulations.The results next presented are organized into five experiments: in the first experiment, the algorithms are evaluatedwith respect to the SNR and to the absence of pure pixels. Asmentioned before, we defineSNR(10)In the case of zero-mean noise with covarianceabundance fractions, one obtainsSNRtrand Dirichlet(11)wherediagFig. 7. Robustness to the absence of pure pixels (N 1000, p 3, L 20, 1). (a) rmsSID as function ofSNR. (b) rmsSAE as function of SNR. (c) rmsFAAE as function of SNR.224, 1 3,i.e.,, whichis also a Dirichlet distribution. The Dirichlet density, besides(12), andis the variance of parameter. For example, assuming abundance fractionsequaly distributed, we have, after some algebra,SNRforand SNRfor.
904IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 43, NO. 4, APRIL 2005In the second experiment, the performance is measured asfunction of the parameter , which models fluctuations on theillumination due to the surface topography. In the third experiment, the number of pixels of the scene varies, in order to illustrate the algorithm performance with the size of the covered area: as the number of pixels increases, the likelihood ofhaving pure pixels also increases, improving the performanceof the unmixing algorithms; in the fourth experiment, the algorithms are evaluated as function of the number of endmemberspresent in the scene; finally, in the fifth experiment, the numberof floating-point operations (flops) is measured, in order to compare the computational complexity of VCA, N-FINDR, and PPIalgorithms.In the first experiment, the hyperspectral scene has 1000pixels and the abundance fractions are Dirichlet distributed with, for; parameter is Beta distributed withandimplyingand.Fig. 5 shows performance results as function of the SNR.As expected, the presence of noise degrades the performanceof all algorithms. In terms of rmsSID, VCA, and N-FINDR algorithms have identical performances, whereas PPI displays theworst result. In terms of rmsSAE and rmsFAAE [Fig. 5(b) and(c)], we can see that when SNR is less than 20 dB VCA algorithm exhibits the best performance. Note that for noiselessscenes, only the VCA algorithm has zero rmsSAE.Fig. 7 shows performance results as function of the SNR inthe absence of pure pixels. Spectral data without pure pixelswas obtained by rejecting pixels with any fractional abundancesmaller than 0.2. Fig. 6 shows the obtained scatter plot. VCA andN-FINDR display similar results, being both better than PPI.Notice that the performance is almost independent of the SNRand is uniformly worse than that displayed with pure pixels anddB in the first experiment. We conclude that thisSNRfamily of algorithms is more affected by the lack of pure pixelsthan by low SNR.For economy of space and also because rmsSID, rmsSAE,and rmsFAAE disclose similar pattern of behavior, we onlypresent the rmsSAE in the remaining experiments.In the second experiment, abundance fractions are generatedas in the first one, SNR is set to 20 dB, and parameter is Betadistributed withand in the interval [2, 28]. This corresponds to varyfrom 0.66 to 0.96 andfrom 0.23 to 0.03.By varying parameter , the severity of topographic modulation is also varied. Fig. 8 illustrates the effect of topographicmodulation on the performance of the three algorithms. Whengrows ( gets smaller) the performance improves. This isexpected, since the simplex identification is more accurate whenthe topographic modulation is smaller. PPI algorithm displaysthe worst performance for. VCA and N-FINDR algorithms have identical performances when takes higher values; otherwise the VCA algorithm has the best performance. VCA is more robust to topographic modulation, sinceit seeks for the extreme projections of the simplex, whereasN-FINDR seeks for the maximum volume, which is more sensitive to fluctuations on .In the third experiment, the number of pixels is varied, theabundance fractions are generated as in the first one, and SNRdB. Fig. 9 shows that VCA and N-FINDR exhibit identicalImpact of the number of endmembers (N 1000, L 224, 20, 1). (a) rmsSEA as functionof the number of endmembers. (b) rmsSEA function of the SNR with p 10.Fig. 10. 1 3, SNR 30 dB,TABLE ICOMPUTATIONAL COMPLEXITY OF VCA, N-FINDR, AND PPI ALGORITHMSresults, whereas, the PPI algorithm displays the worst result.Note that the behavior of the three algorithms is quasi independent of the number of pixels.In the fourth experiment, we vary the number of signato, the scene has 1000 pixels, andtures fromSNRdB. Fig. 10(a) shows that VCA and N-FINDRperformances are comparable, while PPI displays the worstresult. The rmsSAE increase slightly as the number of endmembers present in the scene increases. It is also plotted the[see Fig. 10(b)].rmsSAE as function of the SNR withCompared with Fig. 5(b), we conclude that when the numberof endmembers increases the performance of the algorithmsslightly decreases.In the fifth and last experiment, the number of flops is measured, in order to compare the computational complexity ofVCA, PPI, and N-FINDR algorithms. Herein, we use the scenarios of the second and third experiments. Table I presents approximated expressions for the number of flops used by eachalgorithm. These expressions do not account for the computational complexities involved in the computations of the samplecovariancenor in the eigendecomposition.The reason is that these operations, compared with the VCA,PPI, and N-FINDR algorithms, have a negligible computationalcost, since the following. The computation ofhas a complexityofflops. However, in practice one does not needto use the complete set of hyperspectral vectors. If thescene is noiseless, onlylinearly independent vectorswould be enough to infer the exact subspace. In thepresence of noise, however, a larger set should be used. Forexample, in a 1000 1000 hyperspectral image, we foundout that only 1000 samples randomly sampled are enoughto find a very good estimate of. Even a sample sizeof 100 leads to good results on this respect. Concerning the eigendecomposition of(or the SVD of), we only need to compute(or ) eigenvectors corresponding to the largest
NASCIMENTO AND DIAS: VERTEX COMPONENT ANALYSISFig. 11.905Computational complexity measured in number of flops.Fig. 13. Band 30 ( 667:3 nm) of a subimage of the AVIRIS CupriteNevada dataset.Fig. 12. The rmsSEA as function of different abundance fractions distributions 20, 1). From left(N 1000, p 3, L 224, SNR 20 dB,to right: (a) 1 2, (b) 1, (c) 2), (d) 3, (e) Dirichlet density( 1 3).Fig. 14.eigenvalues (or single values). For these partialeigendecomposition, we have used the PCA algorithm[45] (or SVD analysis [47]) whose complexity is negligible compared with the remaining operations.The VCA algorithm projects all data ( vectors of size )onto orthogonal directions. N-FINDR computestimes thedeterminant of amatrix, whose complexity is , with[54]. Assuming that, VCA complexity is lower than that of N-FINDR. Concerning PPI, giventhat the number of skewers ( ) is much higher than the usualnumber of endmembers, the PPI complexity is much higher thanthat of VCA. We conclude, then, that the VCA algorithm has always the lowest complexity.Fig. 11 plots the flops for the three algorithms after dataprojection. In Fig. 11(a), the abscissa is the number of endmembers in the scene, whereas in Fig. 11(b), the abscissa is thenumber of pixels. Note that for five endmembers, VCA computational complexity is one order of magnitude lower than thatof the N-FINDR algorithm. When the number of endmembersis higher than 15, the VCA computational complexity is atleast two orders of magnitude lower than PPI and N-FINDRalgorithms.The results presented in this section were based on abundancefractions with symmetric Dirichlet distribution. The same pattern of behavior was, however, found for any other abundancefr
898 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 43, NO. 4, APRIL 2005 Vertex Component Analysis: A Fast Algorithm to Unmix Hyperspectral Data José M. P. Nascimento, Student Member, IEEE, and José M. Bioucas Dias, Member, IEEE Abs
IEEE 3 Park Avenue New York, NY 10016-5997 USA 28 December 2012 IEEE Power and Energy Society IEEE Std 81 -2012 (Revision of IEEE Std 81-1983) Authorized licensed use limited to: Australian National University. Downloaded on July 27,2018 at 14:57:43 UTC from IEEE Xplore. Restrictions apply.File Size: 2MBPage Count: 86Explore furtherIEEE 81-2012 - IEEE Guide for Measuring Earth Resistivity .standards.ieee.org81-2012 - IEEE Guide for Measuring Earth Resistivity .ieeexplore.ieee.orgAn Overview Of The IEEE Standard 81 Fall-Of-Potential .www.agiusa.com(PDF) IEEE Std 80-2000 IEEE Guide for Safety in AC .www.academia.eduTesting and Evaluation of Grounding . - IEEE Web Hostingwww.ewh.ieee.orgRecommended to you b
Amanda St. John Doris Coulthurst 803-898-1324 (PTH) Solid Waste Grants, Planning & Regulation Development Manager Jana White 803 -1346 Woody Barnes -898 1345 Celeste Duckett-898 1338 aDonia Martin 803 -898 0037 Alexandra Miller-898 2375 Stefanie Vandiver-898 1354 803 -898 3079 Janae Ricks 803-8
Signal Processing, IEEE Transactions on IEEE Trans. Signal Process. IEEE Trans. Acoust., Speech, Signal Process.*(1975-1990) IEEE Trans. Audio Electroacoust.* (until 1974) Smart Grid, IEEE Transactions on IEEE Trans. Smart Grid Software Engineering, IEEE Transactions on IEEE Trans. Softw. Eng.
2000 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 43, NO. 9, SEPTEMBER 2005 Cloud Statistics Measured With the Infrared Cloud Imager (ICI) Brentha Thurairajah, Member, IEEE, and Joseph A. Shaw, Senior Member, IEEE Abstract—The Infrared Cloud Imager (ICI) is a ground-base
Spec Sheet PER APPLIANCE 500 1000 3000 5000 7000 9000 Max 10 Gbase-T/iSCSI Ports per Appliance 24 24 24 24 24 24 Max 10/25 GbE/iSCSI Ports per Appliance 28 24 24 24 24 24 Max Raw Capacity*** 384 TB 898.56 TB 898.56 TB 898.56 TB 898.56 TB 898.56 TB 349.25 TiB 817.36 TiB 817.36 TiB 817.36 TiB 817.36 TiB 817.36 TiB .
Financial Aid Doni Hatchel (910) 898-9613 hatcheld@montgomery.edu Admissions Jessica Latham (910) 898-9617 lathamj@montgomery.edu Business Office Jessica Blanchard (910) 898-9633 blanchardj@montgomery.edu Counseling Diana Sanchez (910) 898-9619 sanchezd@montgomery.edu Distance Learning Mallory Smith (910) 898-9646 smithm@montgomery.edu Registration
IEEE TRANSACTIONS ON IMAGE PROCESSING, TO APPEAR 1 Quality-Aware Images Zhou Wang, Member, IEEE, Guixing Wu, Student Member, IEEE, Hamid R. Sheikh, Member, IEEE, Eero P. Simoncelli, Senior Member, IEEE, En-Hui Yang, Senior Member, IEEE, and Alan C. Bovik, Fellow, IEEE Abstract— We propose the concept of quality-aware image, in which certain extracted features of the original (high-
Annual Women’s Day Sunday, August 24 Congratulations on a blessed Youth Day!! Enjoy your break during the month of August. Women’s Day Choir Rehearsals July 31, August 7, 14, 19, 21 . Beginners Andrew Ash Chaz Holder Primary Deion Holder Nia Belton Junior William Ash Deondrea Belton Intermediate RaShaune Finch Jaylin Finch Advanced Rayanna Bibbs Tavin Brinkley Adult #2 Jeffry Martin Joseph .
