On the unmixing of MEx/OMEGA hyperspectral data

On the unmixing of MEx/OMEGA h yp ersp ectral data Konstan tinos E. The melis a,b , F r ´ ed ´ eric Sc hmidt c,d , Olga Sykioti b , Athanasios A. Rontogiannis b , Konstan tinos D. Koutroumb a s b , Ioannis A. Da glis b a Dep artment of Inf ormatics and T ele c ommun ic ations, University of A thens, Ilissia, 157 84 A thens, Gr e e c e b Institute for Sp ac e Appli c ations and R emote Sensing, Nationa l O bservatory of Athens, 152 36, P. Penteli, Gr e e c e c Univ. Paris-Sud, L ab or atoir e ID ES, UMR 8148, Orsay, F-91405, F r anc e d CNRS, Orsay, F-91405, F r anc e Abstract This article presen ts a comparative study of three different t yp es of estima- tors used for sup ervised linear unmixing of tw o MEx/OMEGA h yp ersp ectral cub es. The algorithms tak e into accoun t t he constrain ts of the abundance fractions, in order to get phys ically interpretable results. Abundance maps sho w that the Bay esian maximum a p osteriori probability (MAP) estimator prop osed in Themelis and Ron tog ia nnis (2008) outp erforms the other tw o sc hemes, offering a compromise b et w een complexit y and estimation p erfor- mance. Th us, t he MAP estimator is a candidate algorithm t o p erform ice and minerals detection on large h yp ersp ectral datasets. Keywor ds : Hyp ersp ectral imagery , sup ervised unmixing, OMEGA data , Mars Express Email addr esses: th emelis @spac e.noa.gr (Konstantinos E. Themelis), freder ic.sc hmidt@u-psud.fr (F r´ ed´ eric Sc hmidt), syki oti@sp ace.n oa.gr (Olga Sykioti), tr onto@ space. noa.gr (A tha na sios A. Ro n to g iannis), kout roum@ space. noa.gr (Konstantinos D. Ko utroumbas), daglis@ space. noa.gr (Ioannis A. Dag lis) Pr eprint submitt e d to Planetary and Sp ac e Scienc e Novemb er 3, 2018 1. In t ro duction The surface of Mars is curren tly b eing imaged with a com binat io n of high sp ectral and spatial resolution. This give s the a bilit y to detect and map c hemical comp onen ts on the Martian surfa ce and atmosphere more ac- curately than b efore. Spectral unmixing (SU) is one of the tec hniques cur- ren tly used for this purp ose, Kesha v a a nd Mustard (2002); Moussaoui et a l. (2008); Sc hmidt et al. (201 0). SU is the pro cedure by whic h the measured sp ectrum of a mixed pixel is decomp osed into a n umber o f constituen t sp ec- tra, called endmem b ers, and the corresp onding fractions, or a bundances, that indicate the prop ortion of the presence of eac h endmem b er in the pixel, Kesha v a and Mustard (2002). Linear SU, whic h a dopts the h yp othesis that the sp ectrum of a mixed pixel is a linear combination of its endmem b ers’ sp ectra, is more commonly used in practice. Based on a ph ysical in terpre- tation, tw o hard constrain ts are imp osed on the abundance fractions of the materials in a pixel; they should b e non-negative and sum to one. Sev eral SU tec hniques for the unmixing of OMEGA (Observ ato ire p our la Min ´ eralogie, l’ Eau, les Glaces et l’ Activit´ e hypersp ectral images), Bibring et al. (2004), hav e b een recen tly prop osed in the bibliography . These tec hniques can b e categorized in to unsup ervised, where a special pro cedure is first ex- ecuted to get the endmem b ers’ sp ectral signatures fro m the image, and su- p ervised, where a priori kno wledge of the image endmem b ers is av a ilable. A recen t example o f an unsup ervised tec hnique is the Ba yes ian source sepa- ration metho d, dev elop ed in Schmidt et al. (2010). This tec hnique is based on a Gibbs sampling sc heme to p erform Bay esian inference. Due to its high computationally complexity , a sp ecial implemen tation strategy is dev elop ed 2 for its application to a complete OMEGA image da t a set, Sc hmidt et al. (2010). Band ratio is the most commonly used sup ervised tec hnique to detect minerals (Bibring et al. (200 4); Bibring et al. (2004); Langevin et a l. (2005)). Ho w eve r some multiple endmem b er linear sp ectral unmixing a lgorithms ha v e b een prop osed suc h as MELSUM, Com b e et al. (2008). MELSUM uses a reference library con taining v arious spectral signatures of minerals (used as endmem b ers), a nd is based on the classical sp ectral mixture analysis (SMA) algorithm. In Ka nner et al. (20 07), a mo dified Gaussian mo del ( MG M) has b een exploited to estimate the fractional abundances o f a comp ositionally div erse suite o f pyro xene sp ectra in the martian surface. A w av elet based metho d has also b een applied f or the unmixing of h yp ersp ectral mart ian data in Gendrin et al. (2006); Sc hmidt et al. (2007). In this pap er, w e f o cus on the problem of sup ervised SU. Our main ob- jectiv e is to estimate the abundances of the endmem b ers that are presen t in t w o OMEGA images, sub ject to the non- nega t ivit y and sum-to-one con- strain ts. In the follo wing, three differen t sup ervised unmixing algorithms a re considered, namely the ENVI-SVD metho d, Boardman (198 9), a quadratic programming (QP) tec hnique, Coleman a nd Li (1996), and a recen tly pro- p osed Ba y esian maxim um a p osteriori probabilit y soft-constraint (MAPs) estimator, Themelis and R on togia nnis (20 0 8). These algor it hms are applied on tw o differen t h yp ersp ectral OMEGA data sets, and they are ev aluated through their corresp onding abundance maps. The endmem b er reference sp ectra used for sup ervised unmixing are either extracted from the image itself, or selected from a sp ectral library o f pure minerals. The experimen- tal results show that the MAPs estimator results in abundance v alues that 3 satisfy the constrain ts of t he pro blem and provides a compromise b et w een the p erformance of the QP tec hnique a nd the complexit y of the ENVI-SVD metho d. An earlier v ersion of this pap er w as presen ted at the 2010 Europ ean Planetary Science Cong r ess, Themelis et al. (2010). 2. Linear sp ectral unmixing techniques Before w e presen t the unmixing tec hniques considered in this pap er, a short description is presen ted here on the linear mixing mo del (LMM), Kesha v a and Mustard (200 2), whic h is assumed in all three of them. In a h yp ersp ectral image, eac h pixel is represen t ed b y a L - dimensional v ector y , where L is the num b er o f the av ailable sp ectral bands. The elemen ts of y corresp ond to the reflectance measured at the resp ectiv e sp ectral bands. The LMM assum es that the receiv ed pixel’s sp ectrum is generated b y a linear com bination of endmem b ers’ sp ectra. Supp ose that the sp ectral signatures of p mat erials that may exist in the imag e are av a ila ble. Then, y can b e expresse d b y the following linear r egression mo del: y = Φx + n , (1) where Φ =  φ 1 φ 2 . . . φ p  ∈ R L × p + , is the mixing matrix con ta ining the end- mem b ers’ sp ectra ( L -dimensional ve ctors φ i , i = 1 , 2 , . . . , p ), x is a p × 1 v ector with the corresp onding abundance fractions, and n is a L × 1 additive noise ve ctor. Adopting the linear mo del in (1), three differen t unmixing algorithms are applied to the OMEGA da ta sets: i) a singular v alue decomp osition metho d (ENVI-SVD), Boardman ( 1 989), a v aila ble in the ENVI image pro- cessing soft w are ii) a QP tec hnique, Coleman and Li (1996), a v ailable in the 4 Matlab en vironment, a nd iii) a r ecen tly pro p osed Bay esian MAPs estimator, Themelis and Rontogiannis (2008). These algorithms are briefly describ ed in t he following subsections. 2.1. ENVI-SVD ENVI-SVD is a constrained least squares a ppro ac h t o the unmixing prob- lem. Using the singular v alue decomp osition (SVD) algo rithm, the pseudo- in v erse of the mixing matrix Φ is computed. Then, the abundance fractions are easily estimated b y multiplying the pseudo-in v erse matrix with eac h im- age pixel’s sp ectral vec to r. The adv antage of this metho d is its low compu- tational complexit y , since the pseudo-in vers e matrix is computed only once as a prepro cessing step, and is then applied to all imag e pixel v ectors. As far as the sum-to-one constrain t is concerned, it is imp osed to the problem using an extra (w eighted ) equation to the linear system of equations (1) . How ev er, ENVI-SVD do es not take in to accoun t the non-negativity of the abundances, whic h can result in negativ e abundance v a lues that hav e no phy sical meaning. 2.2. Quadr atic pr o gr amming te chnique This quadratic prog r amming tec hnique is a reflectiv e Newton metho d, whic h minimizes the quadratic function of the least squares erro r of the unimixing problem, sub ject to the sum-to-one and non- negativit y constrain ts imp osed on the abundances. The QP t ec hnique is ba sed on an iterative op- timization sc heme, whic h has to b e rep eated separately for each image pixel v ector. Keeping in mind tha t a h yp ersp ectral image ma y b e comp osed of thousands of pixels, solving a separate optimization problem for eac h pixel 5 adds up to the computational complexit y of the metho d. An implemen ta- tion of the quadratic programming tec hnique is av ailable in the o pt imizatio n to olb ox of Matlab. 2.3. MAPs estimator The MAPs estimator prop osed in Themelis and Rontogiannis ( 2 008) is a Bay esian estimator sp ecifically designed to address the inv erse problem of sup ervised h yp ersp ectral unmixing. In a Ba yes ian framew or k, appro pri- ate prior distributions are assigned to the unkno wn parameters of the esti- mation pro blem, whic h usually reflect the parameters’ natural c hara cteris- tics. The Bay esian approa c h of Benav oli et a l. (2007) is a dopted in whic h a Gaussian distribution is used as a prior for the abundance v ector x and the MAP estimator is then utilized. By exploiting the symmetry of the prob- lem’s con v ex constraints, the para meters of the Gaussian p osterior distribu- tion (i.e., the mean and the cov ariance matrix) can b e expressed in closed forms, Themelis and Rontogiannis (2008). Due to the statistical nat ur e of the Ba y esian estimator, the constrain ts are not explicitly imp osed to the esti- mated parameters. T o alleviate this, t he final step of the algo rithm is a pro- jection of the MAP estimation p oin t on the polytop e of constrain ts, providing its nearest estimate that satisfies the constraints, Themelis and R o n togiannis (2008). This algorithm has substantially low er complexit y than the QP tech - nique, since it relies on the computation of simple closed-form expressions. 3. Discussion The previously described algorithms are applied to t w o differen t OMEGA data cub es: (a) a scene of Mars’ South P olar Cap, a nd (b) a Syrtis Ma jor 6 observ ation. The OMEGA instrumen t is a sp ectrometer on b oard ESA’s Mars Express satellite, whic h provides hypersp ectral images of the Mars sur- face, with a spatial resolution fro m 300m to 4km, 96 w av elength channels in the visible band a nd 256 w a vele ngt h c hannels in the near infrared band, Bibring et al. (2004). OMEGA uses three differen t detectors, with sp ectral resolutions ab out 7 . 5nm in the 0 . 3 5 − 1 . 05 µ m w av elength range (visible and near inf rared c hannel o r VNIR), 14nm b etw een 0 . 94 and 2 . 70 µ m (short wa ve infrared c hannel or SWIR) and an av erage o f 21nm from 2 . 65 to 5 . 2 µ m (long w a ve infrared channel or L WIR), respectiv ely . The tw o hypersp ectral data sets, the reference sp ectra used for each ima g e and the SU results obtained from the application of the three metho ds ar e analytically describ ed in the follo wing sections. 3.1. South Polar Cap image cub e This data set consists of a single h yp ersp ectral data cub e obtained by lo oking tow ards the South P olar Cap of Mars in the lo cal summer (Jan. 2004). The da t a cub e is made up of t w o c hannels: 128 sp ectral planes from 0 . 93 t o 2 . 73 µ m with a resolution of 14nm and 128 sp ectral planes f r o m 2 . 55 to 5 . 11 µ m with a resolution of 21nm. Noisy bands were excluded, and 156 out of the 25 0 initial bands we re finally utilized in the regio n from 0 . 93 to 2 . 98 µ m to av o id the thermal emission sp ectral range. The linear mo del mixing matrix consists o f the follo wing three r eference sp ectra: (a ) CO 2 ice (syn thetic data with grain size = 100 mm), (b) H 2 O ice (syn thetic data with grain size = 10 µ m), and (c) dust, whic h w ere all detected a priori using the W av a nglet metho d of Sc hmidt et al. (2007). These endmem b ers are dis- cussed in the first OMEGA publication, Bibring et a l. (20 0 4), and a re a lso 7 v erified by Sch midt et al. (2010), using the Bay esian p ositiv e source separa- tion metho d of Moussaoui et al. (2008). The resp ectiv e sp ectral signatures of the endmem b ers are sho wn in Fig. 1. The abundance maps for eac h endmem b er resulting aft er the application of the three estimators are display ed in Fig s. 2 - 4. As show n in these figures, the ENVI-SVD abundances do not satisfy the non-negativity constrain t, e.g., regarding the CO 2 endmem b er, the minimum computed abundance v alue is − 9 . 9 × 10 − 2 . Notice also that the abundance v a lues calculated by QP and MAPs are in full agreemen t, t hey share the same scale and a r e quite differen t f rom those obtained b y ENVI-SVD. This sho ws that the MAPs estimates pro vide reliable information ab out the a bundances. It also adds up to the fact that the MAPs estimator ha s almost similar p erformance with the QP a lgorithm in simulation scenarios with syn thetic data, a s sho wn in Themelis and Rontogiannis (2008). 3.2. Syrtis Major image cub e This dat a set consists o f a sin g le h yp ersp ectral data cub e of the Syrtis Ma- jor region, whic h con tains w ell-iden tified areas with ve ry strong signatures o f mafic minerals, Mustard et al. (20 05). The data cub e consists of 1 0 9 spectral bands out of the 128 original wa v elengths of the SWIR detector. The spatial dimensions of the cub e ar e 366 × 1 28 pixels. The O MEGA observ ations hav e b een calibrated for kno wn instrumen t a rtifacts and for atmospheric CO 2 . The cube ha s b een radiometrically corrected using the standard correction pip eline (SO F T06) and the atmospheric gas transmission has b een empir- ically corrected using the v olcano scan metho d, Langevin et a l. (2005). It is w ell kno wn that OMEGA can iden tify p yroxene and olivine; it discrimi- 8 nates b etw een the high-calcium pyro xenes (HCPs, e.g., clinop yroxe nes) and lo w-calcium py roxene s (LCPs, e.g., orthopyro xenes), Bibring et al. (2005). In the scene under inv estigation, we utilized three endmem b ers whic h hav e previously b een iden tified to b e presen t in the image, Mustard et al. (2 0 05), namely , (a) Hypersthene, (b) Diopside (c) F ay alite. These are all lab ora- tory reference sp ectra, whic h hav e also b een used in Sc hmidt et al. (2011) for sup ervised unmixing. It is interes ting t o note that the last tw o endmem- b ers ha ve b een retriev ed using CRISM m ultisp ectral o bserv ations. Their resp ectiv e sp ectral signatures are displa y ed in Fig. 5. Three more arti- fact endmem b er sp ectra are utilized, sp ecifically t wo neutral sp ectral com- p onen ts (flat lines at 10 − 4 and 1), and a slop e line, a s in Sc hmidt et al. (2011); Le Mouelic et al. (2009). The abundance maps obtained from the application of the three metho ds to the Syrtis Ma jor h yp ersp ectral scene are sho wn in Fig s. 6 - 8. Eac h Figure illustrates the corresp onding abundance map of a single endmem b er, a s it is estimated by all three metho ds. As a reference, the abundance maps of all three endmem b ers using the band ratio metho d are also sho wn in Fig . 9. The band ratio and bands depth estimation metho ds are commonly used to detect minerals on O MEGA data Bibring et al. (2005). This metho d is v alid only if a t least tw o wa vele ngth channels can b e iden tified a s affected by only one particular mineral. F ollow ing the metho do logy of Schmidt et al. (2011), w e used four band ratio s: Index(Olivine) = b2 . 39 / b1 . 06 (2) 9 Index(op x) = 1 − b1 . 84 / ((1 . 84 − 1 . 25) ∗ b1 . 25 + (2 . 47 − 1 . 84) ∗ b2 . 47) ∗ (2 . 47 − 1 . 25) (3) Index(cp x) = 1 − b1 . 85 / ((2 . 3 2 − 1 . 8 5) ∗ b2 . 3 2 + (2 . 56 − 2 . 32) ∗ b2 . 56) ∗ (2 . 56 − 1 . 85)) (4) where the w a ve length ba nd “b1 . 84” stands for the band at 1 . 8 4 microns, Orthop yroxe nes (Hyp erstene) a re not ed as “op x” and Clinop yro xenes (Diop- side) are noted as “cp x”. The differences b et we en the three metho ds a re again pro minen t. Bo th the MAPs and the QP metho ds pro vide consisten t results, as far as the endmem b ers F ay alite and Hyp ersthene is concerned. In addition, for b oth F a y alit e and Hyp ersthene, ENVI-SVD returns negativ e abundance v alues and its resulting maps substan tially deviate from the other t w o metho ds. As fa r a s Dio pside is concerned, the MAPs estimator seems to pro duce a sligh tly differen t abundance map in comparison to the other tw o metho ds. How ev er, it can b e easily v erified by comparing Figs. 6 - 8 a nd 9 that the abundance maps of the MAPs estimator a re in b etter agreemen t with those obtained using the band ratio metho d, compared to the other t w o metho ds. Th us, it can b e argued that MAPs provide s more reliable results than ENVI-SVD and QP . 4. Conclusions In this pap er, w e hav e presen ted a comparison of three differen t sup er- vised sp ectral unmixing metho ds (Bay esian MAPs estimator, ENVI-SVD, QP), on the basis o f t wo differen t OMEGA hypersp ectral data sets. As op- p osed to itera t ive algorithms or Mark ov Chain Monte Carlo metho ds, e.g. 10 Sc hmidt et al. (2010), commonly used for the constrained inv erse problem of abundance estimation, t he computational complexit y of t he MAPs estimator is m uc h lo we r . Sp ecifically , fo r the Syrtis Ma jor dataset, the running time of the MAPs alg orithm w as 4 . 7 secs in a roughly optimized Matlab imple- men tation, while the QP needed 26.4 secs ( b oth algor it hms w ere run o n a 2 . 4-Ghz In tel Core 2 CPU). In addition, as v erified b y exp erimen tal results, the p erformance of the tw o metho ds is approximately equal. 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O n the unmixing o f MEx/OMEGA h yp ersp ectral data. In: Europ ean Planetar y Science Congress 2010 . p. 730. Themelis, K. E., Rontogiannis, A. A., August 20 08. A soft constrained MAP estimator fo r sup ervised hy p ersp ectral signal unmixing. In: Pro c. of Euro- p ean Signal Pro cessing Conference, EUSIPCO’08. 14 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Wavelength Reflectance Dust CO 2 ice H 2 O Figure 1: Reference sp ectra of the South P olar Cap OMEGA image. The a v ailable endmem b ers are: (a ) OMEGA typical dust materials with atmo- sphere absorption. (b) syn thetic CO2 ice with grain size of 100 mm, (c) syn thetic H2O ice with gra in size of 100 microns. 15 ENVI−SVD 0.4 0.6 0.8 (a) QP 0.5 0.6 0.70.8 0.9 (b) MAPs 0.5 0.6 0.7 0.8 0.9 (c) Figure 2: Abundance map of the dus t endmem b er, estimated using (a ) ENVI- SVD, (b) QP , a nd (c) MAPs. 16 ENVI−SVD 0 0.2 0.4 (a) QP 0.2 0.4 (b) MAPs 0.2 0.4 (c) Figure 3: Abundance map of the CO 2 endmem b er, estimated using (a ) ENVI- SVD, (b) QP , a nd (c) MAPs. 17 ENVI−SVD 0.1 0.2 0.3 (a) QP 0.1 0.2 0.3 (b) MAPs 0.1 0.2 0.3 (c) Figure 4: Abundance map of the H 2 O endmem b er, estimated using (a ) ENVI- SVD, (b) QP , a nd (c) MAPs. 18 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Wavelength Reflectance hyperst2.spc Hypersthene PYX02.h >250u Silicate (Ino); Diopside CPX CRISM Olivine Fayalite CRISM Figure 5: Reference sp ectra of t he Syrtis Ma jor OMEGA image. 19 ENVI−SVD 0 0.05 0.1 (a) QP 0 0.05 0.1 (b) MAPs 0.08 0.1 0.12 0.14 0.16 0.18 (c) Figure 6: Abundance map of Hyp ersthene, estimated using (a) ENVI-SVD, (b) QP , and (c) MAPs algorithms. 20 ENVI−SVD 0.02 0.04 0.06 0.08 0.1 (a) QP 0.02 0.04 0.06 0.08 0.1 (b) MAPs 0.1 0.15 0.2 (c) Figure 7: Abundance map of Diopside, estimated using (a ) ENVI-SVD, (b) QP , and (c) MAPs algorit hms. 21 ENVI−SVD −0.1 −0.05 0 0.05 0.1 0.15 (a) QP 0 0.05 0.1 0.15 (b) MAPs 0.05 0.1 0.15 0.2 0.25 (c) Figure 8: Abundance map of F ay alite, estimated using (a) ENVI-SVD, (b) QP , and (c) MAPs algorit hms. 22 OPX 0 0.02 0.04 0.06 0.08 0.1 (a) CPX 0 0.05 0.1 (b) OL 0 0.05 0.1 0.15 (c) Figure 9: Abundance maps o f the endmem b ers (a) Hyp ersthene, (b) D iop- side, and (c) F a y alite in the Syrtis Ma jor scene. The abundances are esti- mated using the band ratio metho d. 23