Frédéric Schmidt - Speaker Deck

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Spectral unmixing on Mars GEOPS Frédéric Schmidt Université Paris-Saclay, CNRS, GEOPS, 91405, Orsay, France S3 seminar - Central SUPELEC 25th September 2020 samedi 26 septembre 20

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Probabilistic Sparse Non-negative Matrix Factorization psNMF • each component as truncated normal/exponential • no explicit sparsity • variational Bayesian approach • no tuning parameter (soft prior with low ) Method Hinrich, J. L. & Mørup, M., 2018, http://dx.doi.org/10.1007/978-3-319-93764-9_45 490 J. L. Hinrich and M. Mørup λd ∼ Gamma (αλ, βλ ) , (1) wid | λd ∼ T N 0, λ−1 d , 0, ∞ or wid |λd ∼ Exponential λ−1 d , (2) hdj | λd ∼ T N 0, λ−1 d , 0, ∞ or hdj |λd ∼ Exponential λ−1 d , (3) τ ∼ Gamma (ατ , βτ ) , x j | W, h j, τ ∼ N x j | Wh j, τ−1II . (4) Where the reconstruction error x j − Wh j follows a normal distribution with Bayesian modeling · 1 Introduction Non-negative matrix fact ization [1] has become a pretable part based repr XI×J ≈ W I×DHD×J in w i.e. wid ≥ 0 ∀i, d and hdj in NMF have been propo this includes multiplicativ component-wise updating Unfortunately, NMF does not sufficiently span tions may equally well re λd ∼ Gamma (αλ, βλ ) , (1) wid | λd ∼ T N 0, λ−1 d , 0, ∞ or wid |λd ∼ Exponential λ−1 d , (2) hdj | λd ∼ T N 0, λ−1 d , 0, ∞ or hdj |λd ∼ Exponential λ−1 d , (3) τ ∼ Gamma (ατ , βτ ) , x j | W, h j, τ ∼ N x j | Wh j, τ−1II . (4) Where the reconstruction error x j − Wh j follows a normal distribution with se precision (inverse variance) τ. Each component of W and H (i.e. w d, h d ) res a common prior λd used to infer the scale of a component, see also [14]. te λd represents the precision or rate for the truncated normal and exponential tribution, respectively. This prior can be used for model order selection and commonly called an automatic relevance determination (ARD) prior. The terior distribution of θ = {W, H, τ, {λd }d=1,2,...,D } is inferred from the data while the shape α⋆ and rate β⋆ of the Gamma distributions are fixed. In ence of a priori information, weak priors are specified (c.f. α⋆ = β⋆ = 10−4), reby the distribution of τ and λd are determined primarily from the data. Probabilistic Sparse Non-negative Matrix Factorization obabilistic modeling can be used to automatically infer the sparsity pattern d level, as supported by the data. The Bayesian framework facilitates sparse F. Schmidt Blind Linear Unmixing samedi 26 septembre 20

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NOMAD/ ExoMars TGO Fig. 8. Evolution of the slit angle during a typical occultation (inclination exag- gerated for clarity). on in the UV–visible (Left Panel) and the IR (Right Panel) recorded by the UVIS and SO 1 ppb) for a typical clear Mars atmosphere, considering the Rayleigh scattering and a ond. For the IR spectra: The limits of the different diffraction orders covered are shown transmittances have been added in light green to clearly indicate where its absorption Fig. 6. Examples of simulated transmittances obtained during a typical solar occultation in the UV–visible (Left Panel) and the IR (Right Panel) recorded by the UVIS and SO channels respectively. These spectra contain the absorption of CO2 , H2 O, O3 and CH4 (1 ppb) for a typical clear Mars atmosphere, considering the Rayleigh scattering and a dust loading of tau¼0.2. The colour code indicates the altitudes to which they correspond. For the IR spectra: The limits of the different diffraction orders covered are shown at the top of the Figure and spectra have been artiﬁcially shifted by 0.40 for clarity. CH4 transmittances have been added in light green to clearly indicate where its absorption lines are located. A.C. Vandaele et al. / Planetary and Space Science 119 (2015) 233–249 242 insulation (MLI), and instrument-to-spacecraft mounting hardware (3.7 kg). NOMAD has to survive in the environmental conditions imposed by the spacecraft. The most severe constraints are the Fig. 1. Different observation modes with NOMAD in orbit around Mars (1 nadir, 2 limb, 3 SO). Table 1. Coalignment Contributors Coalignment Contributors Accuracy Limit Knowledge Accuracy Solar LOS to NOMAD mechanical axis 0.15 mrad Nadir LOS to NOMAD mechanical axis 10.0 mrad NOMAD mechanical axis to spacecraft axis 0.20 mrad 0.05 mrad Any NOMAD solar LOS to spacecraft axisa 0.30 mrad aIt is assumed that the ACS instrument, which performs SO measurements at the same time as NOMAD, has a similar coalignment budget of 0.30 mrad, leading to a maximum misalignment between the NOMAD and ACS solar lines of sight of 0.60 mrad. Table 2. Pointing Budget Pointing Error Limit Knowledge Accuracy Absolute pointing error (APE) solar ≤1.23 mrad Relative pointing error (RPE) solar (short + long term)a ≤1.23 mrad Absolute pointing error (APE) nadir ≤3.50 mrad Relative pointing error (RPE) nadir (long term)a ≤3.00 mrad Relative pointing error (RPE) nadir (short term)a ≤0.54 mrad Overall pointing knowledge solar ≤0.45 mrad Overall pointing knowledge nadir ≤0.54 mrad aShort term is 1 s, long term is 60 s. 8496 Vol. 54, No. 28 / October 1 2015 / Applied Optics Research Article Vandaele, A.; et al, Planetary and Space Science, Elsevier BV, 2015, 119, 233-249, http://dx.doi.org/10.1016/j.pss.2015.10.003 Transmittance F. Schmidt Blind Linear Unmixing samedi 26 septembre 20

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-6 -4 -2 0 2 4 BD -3 0 50 100 150 200 frequency Histogram of BD CH4 @ 3067.2 cm-1 3060 3065 3070 3075 -1 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Absorbance Spectra max BD CH4 @ 3067.2 cm-1 Toy model Figure 1: Synthetic dataset containing 104 spectra with various abundances of H2O and 100 containing CH4 at 3- level of the noise. In blue the reference spectra SH2O of H2O (coming from actual data analysis). In red the reference spectra SCH4 of CH4 (from theoretical data). The final synthetic dataset is represented in Fig. 1. 221 In order to check the quality of the estimation, we simply compute the 222 correlation coefficient between SCH4 and the estimated NS sources ˙ S, using: 223 Q = corr nSCH4 , ˙ S:i o (12) The ith source with the maximum correlation is identified to CH4 contri- 224 bution. The value to the maximum correlation is used as metric to assess the 225 quality of the retrieval. 226 4.1.2. Results 227 By plotting the 10000 samples of the dataset, one is able to identify easily 228 the H2 O bands. Nevertheless, we cannot observe the target CH4 in the average 229 spectrum, even at 3- level, because it is lost in the baseline changes. 230 • linear mixture • 100 spectra of CH4 at 3-sigma 10 000 spectra example, we simulate a linear mixture of NO = 10 observations ⌫ = 320 spectels (see fig. 1) similar to order 136 of NOMAD- rum is a mixture of a spectra of water vapor SH2 O (coming source estimated from real data using psNMF) and theoretical rom Villanueva et al. (2018), with corresponding abundances X = SH2 O .AH2 O + SCH4 .ACH4 + n (11) is assumed to be a Gaussian process with a standard deviation no bias: n = G (0 , ) . All spectra contain pure water vapor with owing AH2 O = 5 / 6 . (1 , 10) + 1 / 6 .U (0 , 1) , a mixture of beta ( ) 5/6 of the sample and an uniform (U) distribution for 1/6 of s process mimics well the water vapor band depth distribution on in section 3.3) of the real dataset (see Fig.2). As the baseline ero, we also mimic baseline correction errors. In addition 100 0000 contain methane with ACH4 = 1 , such that the band depth level. Please note that the model to generate the data is not m-to-one constraint, but fully fulfilling the positivity constraint. d noise and signal level, the noise RMSD is 0.16. 7 F. Schmidt Blind Linear Unmixing samedi 26 septembre 20

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Toy model • linear mixture • 100 spectra of CH4 at 3-sigma 10 000 spectra Figure 1: Synthetic dataset containing 104 spectra with various abundances of H2O and 100 containing CH4 at 3- level of the noise. In blue the reference spectra SH2O of H2O (coming from actual data analysis). In red the reference spectra SCH4 of CH4 (from theoretical data). The final synthetic dataset is represented in Fig. 1. 221 In order to check the quality of the estimation, we simply compute the 222 correlation coefficient between SCH4 and the estimated NS sources ˙ S, using: 223 Q = corr nSCH4 , ˙ S:i o (12) The ith source with the maximum correlation is identified to CH4 contri- 224 bution. The value to the maximum correlation is used as metric to assess the 225 quality of the retrieval. 226 4.1.2. Results 227 By plotting the 10000 samples of the dataset, one is able to identify easily 228 the H2 O bands. Nevertheless, we cannot observe the target CH4 in the average 229 spectrum, even at 3- level, because it is lost in the baseline changes. 230 is able to detect the hidden compounds. example, we simulate a linear mixture of NO = 10 4 observations ⌫ = 320 spectels (see fig. 1) similar to order 136 of NOMAD- trum is a mixture of a spectra of water vapor SH2 O (coming source estimated from real data using psNMF) and theoretical from Villanueva et al. (2018), with corresponding abundances X = SH2 O .AH2 O + SCH4 .ACH4 + n (11) is assumed to be a Gaussian process with a standard deviation no bias: n = G (0 , ) . All spectra contain pure water vapor with owing AH2 O = 5 / 6 . (1 , 10) + 1 / 6 .U (0 , 1) , a mixture of beta ( ) 5/6 of the sample and an uniform (U) distribution for 1/6 of is process mimics well the water vapor band depth distribution ion in section 3.3) of the real dataset (see Fig.2). As the baseline zero, we also mimic baseline correction errors. In addition 100 0000 contain methane with ACH4 = 1 , such that the band depth level. Please note that the model to generate the data is not m-to-one constraint, but fully fulfilling the positivity constraint. ed noise and signal level, the noise RMSD is 0.16. -6 -4 -2 0 2 4 BD 10-3 0 50 100 150 200 frequency Histogram of BD CH4 @ 3067.2 cm-1 3060 3065 3070 3075 wavenumber [cm-1] 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Absorbance Spectra max BD CH4 @ 3067.2 cm-1 Figure 3: (left) Histogram of Band Depth at 3067.2 cm 1 from the dataset containing 100 CH4 at 3- level out of 104 spectra. (right) 100 spectra with the maximum Band Depth at 3067.2 cm 1 specific of CH4. Signal is dominated by water and by noise. No specific signature of CH4 is visible. 3060 3065 3070 3075 wavelength 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 Intensity psNMF : Endmember spectra No1 relevance =36.77% No2 relevance =32.6% No3 relevance =30.24% No4 relevance =0.39% Figure 3: (left) Histogram of Band Depth at 3067.2 cm 1 from the dataset containing 100 CH4 at 3- level out of 104 spectra. (right) 100 spectra with the maximum Band Depth at 3067.2 cm 1 specific of CH4. Signal is dominated by water and by noise. No specific signature of CH4 is visible. 3060 3065 3070 3075 wavelength 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 Intensity psNMF : Endmember spectra No1 relevance =36.77% No2 relevance =32.6% No3 relevance =30.24% No4 relevance =0.39% omial, that is filtered out. this continuum removal rationale is that when the optical R is decreased and the noise effect on continuum removal at.). is rationale, we propose to first correct for the continuum ance space: T⇤ ( ⌫ ) = T ( ⌫ ) C ( ⌫ ) (3) pectra into absorbance: X ( ⌫ ) = 1 T⇤ ( ⌫ ) (4) e linear mixture : X ( ⌫ ) ⇡ NS X i=1 Si( ⌫ ) .Ai (5) • Results : correlation 0.98 F. Schmidt Blind Linear Unmixing samedi 26 septembre 20

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Toy model • linear mixture 10 000 spectra spanned over N⌫ = 320 spectels (see fig. 1) similar to order 136 of N 205 SO. Each spectrum is a mixture of a spectra of water vapor SH2 O 206 from one actual source estimated from real data using psNMF) and th 207 methane SCH4 from Villanueva et al. (2018), with corresponding ab 208 AH2 O , ACH4 : 209 X = SH2 O .AH2 O + SCH4 .ACH4 + n The noise n is assumed to be a Gaussian process with a standard d 210 of =0.001 and no bias: n = G (0 , ) . All spectra contain pure water va 211 a coefficient following AH2 O = 5 / 6 . (1 , 10) + 1 / 6 .U (0 , 1) , a mixture of 212 distribution for 5/6 of the sample and an uniform (U) distribution f 213 the sample. This process mimics well the water vapor band depth dis 214 (BD, see definition in section 3.3) of the real dataset (see Fig.2). As the 215 of SH2 O is not zero, we also mimic baseline correction errors. In add 216 spectra out of 10000 contain methane with ACH4 = 1 , such that the ba 217 of SCH4 is at 3- level. Please note that the model to generate the da 218 fulfilling the sum-to-one constraint, but fully fulfilling the positivity co 219 Given the defined noise and signal level, the noise RMSD is 0.16. 220 7 MU psNMF BPSS2 Quality Q 0.35±0.12 0.822 ±0.005 0.41±0.06 RMSD relative error 0.1455± 2 . 10 6 0.1461± 5 . 10 6 0.1468± 3 . 10 4 Computation time (s) 13±8 46±9 413±21 Table 1: Results (mean and standard deviation) from 10 realizations of a toy synthetic example with NS = 5 (in agreement with next section on synthetic tests), NO = 10000, N⌫ = 320 and 300 CH4 spectra hidden at a level of 1 std of the noise. Quality is computed as a correlation coefficient (see Eq. 12). RMSD is computed from Eq. 8. Computation time is expressed in second. usual squared difference). The second is called regularization term. 161 lem with this approach, is that hyperparameter is not known an 162 tuned manually. A recent approach has been proposed to solve this 163 the Bayesian framework (Hinrich and Mørup, 2018). The main idea i 164 pass all variables and hyperparameters in a unique problem that i 165 with variational update principle. We will refer this algorithm to 166 sparse NMF (psNMF). This algorithm has the advantage to have 167 computation time and no hyperparameter tuning. It also has a reg 168 term to avoid strong dependence of the initialization on the final so 169 In order to estimate the precision of the reconstruction, we use 170 Mean Square Difference RMSD: 171 RMSD = s⌧⇣X ˙ S. ˙ A⌘2 hXi With h.i, the mean. 172 Once the sources are estimated, we quantify their relevance for 173 dataset. From the total reconstruction ˙ Xkj= ˙ Ski . ˙ Aij , for all i, we c 174 the contribution of source i0, that is to say: ˙ Xi kj= ˙ Ski0 . ˙ Ai0j . Thus, th 175 of source i is defined as: 176 Ri = D ˙ Xi ˙ X E D ˙ XE This definition is convenient since the sum of all Ri is one (th 177 is only present when sources and abundances are positive) and we 178 estimate the % contribution of each source in the final reconstructio 179 dataset containing 104 spectra with various abundances of H2O and 100 level of the noise. In blue the reference spectra SH2O of H2O (coming ysis). In red the reference spectra SCH4 of CH4 (from theoretical data). hetic dataset is represented in Fig. 1. heck the quality of the estimation, we simply compute the ient between SCH4 and the estimated NS sources ˙ S, using: Q = corr nSCH4 , ˙ S:i o (12) e with the maximum correlation is identified to CH4 contri- e to the maximum correlation is used as metric to assess the ieval. e 10000 samples of the dataset, one is able to identify easily Nevertheless, we cannot observe the target CH4 in the average 3- level, because it is lost in the baseline changes. • 300 spectra of CH4 at 1-sigma • average of 10 realizations F. Schmidt Blind Linear Unmixing samedi 26 septembre 20

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Toy model • linear mixture 10 000 spectra SO. Each spectrum is a mixture of a spectra of water vapor SH2 O 206 from one actual source estimated from real data using psNMF) and th 207 methane SCH4 from Villanueva et al. (2018), with corresponding ab 208 AH2 O , ACH4 : 209 X = SH2 O .AH2 O + SCH4 .ACH4 + n The noise n is assumed to be a Gaussian process with a standard d 210 of =0.001 and no bias: n = G (0 , ) . All spectra contain pure water va 211 a coefficient following AH2 O = 5 / 6 . (1 , 10) + 1 / 6 .U (0 , 1) , a mixture of 212 distribution for 5/6 of the sample and an uniform (U) distribution f 213 the sample. This process mimics well the water vapor band depth dis 214 (BD, see definition in section 3.3) of the real dataset (see Fig.2). As the 215 of SH2 O is not zero, we also mimic baseline correction errors. In add 216 spectra out of 10000 contain methane with ACH4 = 1 , such that the ba 217 of SCH4 is at 3- level. Please note that the model to generate the da 218 fulfilling the sum-to-one constraint, but fully fulfilling the positivity co 219 Given the defined noise and signal level, the noise RMSD is 0.16. 220 7 1 1.5 2 2.5 3 Noise factor 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Value Effect of noise factor with 100 CH 4 spectra MU mean corr. psNMF mean corr. MU frac. corr. > 0.5 psNMF frac. corr. > 0.5 1 1.5 2 2.5 3 Noise factor 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Value Effect of noise factor with 50 CH 4 spectra MU mean corr. psNMF mean corr. MU frac. corr. > 0.5 psNMF frac. corr. > 0.5 figure 6 100 hidden CH4 spectra Factor above noise level Value Factor above noise level Value 50 hidden CH4 spectra • high correlation up to 2-2.5 noise level 1.5 2 2.5 3 Noise factor of noise factor with 100 CH 4 spectra 5 1 1.5 2 2.5 3 Noise factor 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Value Effect of noise factor with 50 CH 4 spectra MU mean corr. psNMF mean corr. MU frac. corr. > 0.5 psNMF frac. corr. > 0.5 of the MU and psNMF algorithm for NS = 5, NO = 10000, N⌫ = 320, as a ctor above noise level. The average Q of 10 realizations of the best estimated 6 hidden CH4 spectra actor above noise level Factor above noise level Value 50 hidden CH4 spectra s of the MU and psNMF algorithm for NS = 5, NO = 10000, N⌫ = 320, as a ctor above noise level. The average Q of 10 realizations of the best estimated F. Schmidt Blind Linear Unmixing samedi 26 septembre 20

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Spectral database • Selection of spectra for Mars Schmidt, F.; Legendre, M. & Le Mouëlic, S. Minerals detection for hyperspectral images using adapted linear unmixing: LinMin Icarus, 2014, 237, 61-74, http://dx.doi.org/10.1016/j.icarus.2014.03.044 fitting. We also showed that the knowledge of the noise covariance matrix, that can be estimated from dark current or using other techniques is important to asses the detection limits. IPLS is shown to be the best numerical algorithm to solve the MINESINE problem. Its fast GPU implementation is particularly relevant for the treatment of hyperspectral images. In the future, this methodology should be applied in various planetary cases in order to study the surface geology. Also a significant improvement of the mineral detection may be addressed by using spectral database adapted to the context. Acknowledgement We acknowledge support from the “Institut National des Sciences de l’Univers” (INSU), the "Centre National de la Recherche Scientifique" (CNRS) and "Cen- tre National d’Etude Spatiale" (CNES) and through the "Programme National de Planétologie". Appendix Name of the 32 spectra: 1 Inosilicate (Hypersthene OPX PYX02.h >250u) 12 Sulfate; Gypsum 23 Carbonate; Siderite 2 Inosilicate (Diopside CPX CRISM) 13 Sulfate; Jarosite 24 Phyllosilicate (Chlorite) 3 Olivine Fayalite CRISM 14 Sulfate; Kieserite 25 Muscovite GDS116 Tanzania 4 Olivine Forsterite CRISM 15 Epsomite USGS GDS149 26 Alunite GDS83 Na63 5 Phyllosilicate (Clay Montmorillonite Bentonite) 16 Oxide; Goethite 27 Atmospheric Transmission 6 Phyllosilicate (Clay Illite Smectite) 17 Oxide; Hematite 28 H2O grain 1 7 Phyllosilicate (Serpentine Chrysotile Clinochry.) 18 Oxide; Magnetite 29 H2O grain 100 8 Phyllosilicate (Serpentine Lizardite) 19 Ferrihydrite USGS GDS75 Sy F6 30 H2O grain 1000 9 Phyllosilicate (Clay Illite) 20 Maghemite USGS GDS81 Sy (M-3) 31 CO2 grain 100 10 Phyllosilicate (Clay Kaolinite) 21 Carbonate; Calcite 32 CO2 grain 10 000 11 Phyllosilicate (Nontronite) 22 Carbonate; Dolomite Name of the 12 additional spectra: 33 Flat 1 37 cos 1/4 41 cos 1/2 34 Flat 0.0001 38 sin 1/4 42 sin 1/2 35 Slope Increasing 39 -cos 1/4 43 -cos 1/2 36 Slope Decreasing 40 -sin 1/4 44 -sin 1/2 References 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Endmember Wavelength (microns) Reflectance Figure 2: 32 Reference spectra of minerals, ice and atmospheric gas representing major cla of contributions of surface spectra. See Appendix for the names. • 90% is a flat component at reflectance 0.35 in agreement with OMEG studies Vincendon in order to reproduce the low level and flatness of act Martian spectra. The flatness is also representative of the Moon or ot planetary surface. • 10% of a random mixture of two over 32 reference spectra with rand uniform mixing coe cients, noted as A0 . For each endmember i, th is ~30 spectral mixture with non-null mixing coe cient (noted Apositi and ~970 with null mixing coe cient (Afalse). We add synthetic noise, simulating the noise level of a typical OMEG observation after gas correction (covariance matrix from dark current noise ORB41_1, which has an wavelength-average standard deviation of 1.3 10 Other noise statistics can be used, such MNF shift di⌧erence from ENVI so ware, but we point the fact that OMEGA dark current is archived, estimat the minimum noise statistics (excluding spike and other non-linear e⌧ects). On a desktop with Dual Core at 2.53 Ghz with 4Go RAM memory, typical computation time to solve the non-normalized problem of eq. 2 M = 104000 spectra with (1000 mixture of spectra and 104 values of AOT, next section), N = 110, N = 44 are : 3.3 h for BI-ICE, 4.5 min for IPLS, h for Simplex Projection. The estimated mixing coe cients are noted AIP 5 samedi 26 septembre 20

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Algorithm • FCLS is not doing the job • Primal-dual interior-point • GPU implementation Chouzenoux, E.; Legendre, M.; Moussaoui, S. & Idier, J. Fast Constrained Least Squares Spectral Unmixing Using Primal-Dual Interior-Point Optimization Selected Topics in Applied Earth Observations and Remote Sensing, IEEE Journal of, 2014, 7, 59-69, http:// dx.doi.org/10.1109/JSTARS.2013.2266732 ted olution e chan- els. We hannels be dis- ocused ta cube Mars in st were p with 73 µm es from ter cal- express Thus, based on this model and using t ture assumption, the radiance factor a at wavelenght λ satisﬁes the following L(x, y, λ) = ρa (λ) + Φ(λ) P p=1 αp (x, y) ρp (λ where Φ(λ) is the spectral atmosp θ(x, y) the angle between the solar dir face normal (solar incidence angle), endmembers in the region of coordina spectrum of the p-th endmember, αp ted chan- ls. We annels be dis- ocused a cube ars in t were p with 73 µm s from er cal- xpress etween nd the n pho- ugh its at wavelenght λ satisﬁes the following observation m L(x, y, λ) = ρa (λ) + Φ(λ) P p=1 αp (x, y) ρp (λ) cos [θ(x, y) where Φ(λ) is the spectral atmospheric transm θ(x, y) the angle between the solar direction and th face normal (solar incidence angle), P the num endmembers in the region of coordinates (x, y), ρp ( spectrum of the p-th endmember, αp (x, y) its wei the mixture and ρa (λ) the radiation that did not directly from the area under view. This mixture mod also be written as: L(x, y, λ) = P α′ p (x, y) · ρ′ p (λ) + E(x, y, λ) min ||↵p.⇢p L|| min ||↵p.⇢p L||, ↵p > 0, P ↵p = 1 Legendre, M.; Capriotti, L.; Schmidt, F.; Moussaoui, S. & Schmidt, A. GPU implementation issues for fast unmixing of hyperspectral images EGU General Assembly Conference Abstracts, 2013, 15, 11686, Schmidt, F.; Legendre, M. & Le Mouëlic, S. Minerals detection for hyperspectral images using adapted linear unmixing: LinMin Icarus, 2014, 237, 61-74, http://dx.doi.org/10.1016/j.icarus.2014.03.044 Heinz, D. & I-Chang, C., TGRS, 2001, 39, 529-545, samedi 26 septembre 20

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Ben Mhenni, R.; Bourguignon, S.; Ninin, J. & Schmidt, F. Spectral Unmixing with Sparsity and Structuring Constraints Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS), 2018 9th Workshop, held 23-26 September 2018, Amsterdam, The Netherlands, 2018, 1-4 FCLS GE `0 Backward Abundance 100 200 300 400 0 0.1 0.2 0.3 0.4 100 200 300 400 0 0.1 0.2 0.3 0.4 100 200 300 400 0 0.1 0.2 0.3 0.4 100 200 300 400 0 0.1 0.2 0.3 0.4 Reflectance 1 1.5 2 2.5 0 0.1 0.2 1 1.5 2 2.5 0 0.1 0.2 1 1.5 2 2.5 0 0.1 0.2 1 1.5 2 2.5 0 0.1 0.2 (µm) (µm) (µm) (µm) Fig. 1. Example of result, K = 4 spectra, SNR = 40 dB. Top: estimated (black squares) and true (red stars) abundances. Bottom: true (solid red line) and estimated (dashed black line) endmembers weighted by their abundances, and noise (blue solid line). k b a ˚ a k 2 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 Fig. 2. Error on estimated abundances for FCLS (+), GE (4), `0 (o), SA+GE ( ⇧ ), and backward [12] (x). lowed for MIP resolution (1000 s). SNR Method K=3 K=4 K=5 K=6 K=7 55 dB GE 1.3 2.2 3.3 5.9 9.8 `0 1.5 3 11.8 124(3) 468(9) `0 +GE 1.6 3.1 7.1 82(1) 320(7) SA+GE 2.5 11.5 29 76 264(5) 40 dB GE 1.3 2.2 2.1 3.2 2.9 `0 9.2 107(1) 574(13) 963(27) 1000(30) `0 +GE 8.1 75 515(11) 929(25) 982(28) SA+GE 282(6) 426(8) 746(19) 936(26) 997(29) Table 1. Computing times (s) for optimization of MIP prob- lems, averaged over 30 instances. In parentheses: number of Abu 100 200 300 400 0 0.1 0.2 100 200 300 400 0 0.1 0.2 100 200 0 0.1 0.2 Reflectance 1 1.5 2 2.5 0 0.1 0.2 1 1.5 2 2.5 0 0.1 0.2 1 1.5 0 0.1 0.2 (µm) (µm) (µm Fig. 1. Example of result, K = 4 spectra, SNR = 40 dB. Top: estimated (black s true (solid red line) and estimated (dashed black line) endmembers weighted b k b a ˚ a k 2 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 Fig. 2. Error on estimated abundances for FCLS (+), GE (4), `0 (o), SA+GE ( ⇧ ), and backward [12] (x). lowed for MIP resolution (1000 s). Table 1 gives average computing times obtained on SNR M 55 dB ` S 40 dB ` S Table 1. Com lems, averag instances for •SA : significant abundance •GE : group exclusivity •Backward : J. B. Greer, “Sparse demixing of hyperspectral images,” IEEE Transactions on Image Processing, vol. 21, no. 1, pp. 219–228, Jan 2012. Average of 50 realizations samedi 26 septembre 20

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