Frédéric Schmidt

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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Plan
• Different use of linear unmixing for detection
• blind linear unmixing (what and where?)
• supervised linear unmixing (where?)
F. Schmidt Plan

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Plan
•blind linear unmixing (what and where?)
• supervised linear unmixing (where?)
F. Schmidt Plan

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• Unsupervised automatic detection
• Summary (quick look) of the dataset
• Based on advance signal treatment and
machine learning
• General tool but application to NOMAD/
Exomars TGO
• NO PRECISE QUANTIFICATION
Goal
F. Schmidt Blind Linear Unmixing

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Theory
• Estimation of endmember spectra (source) and
abundances under constraints:
observation
mittance space:
T⇤
(
⌫
) =
T
(
⌫
)
C
(
⌫
)
the spectra into absorbance:
X
(
⌫
) = 1
T⇤
(
⌫
)
is the linear mixture :
X
(
⌫
)
⇡
NS
X
i=1
Si(
⌫
)
.Ai
e source spectra and Ai
the spectral abunda
subject to
positivity
Number of
endmember/source

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Non Negative Matrix
Factorization (NNMF)
• fast
• not robust
Theory
observation
X
(
⌫
) = 1
T⇤
(
⌫
)
p is the linear mixture :
X
(
⌫
)
⇡
NS
X
i=1
Si(
⌫
)
.Ai
he source spectra and Ai
the spectral abund
hysical meaning of Si(
⌫
)
and Ai
is lost but th
4
out of the collection, with the highest abundance of a s
In the following, we will use the continuum estimation
c least square (Eilers and Boelens, 2005), with parameter
= 1 10
2, 10 number of iterations.
2. Non negative matrix factorization
For a collection of spectra, eq. 5 can be written in matri
th i the source index (from 1 to NS
) , j the observation
d k the wavenumber index (from 1 to N⌫
). Thus, one ha
, by minimizing the objective function:
F
=
kX S.Ak2
with k.k, the Frobenius norm (usual L2
norm).
Several algorithms have been proposed to solve this
MU Gillis, N. & Glineur, F. 2012, http://dx.doi.org/10.1162/neco_a_00256
subject to
positivity
Method

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Sparse Non Negative Matrix Factorization (sNNMF)
• fast
• sparsity
• how to determine hyperparameter ?
Theory
observation
X
(
⌫
) = 1
T⇤
(
⌫
)
p is the linear mixture :
X
(
⌫
)
⇡
NS
X
i=1
Si(
⌫
)
.Ai
he source spectra and Ai
the spectral abund
hysical meaning of Si(
⌫
)
and Ai
is lost but th
4
subject to
positivity
Method
proach to reduce the computation time is to select only
the dataset (Moussaoui et al., 2008), but then the stat
chmidt et al., 2010). Thanks to the advances of comp
opose to treat the full dataset. This kind of algorithm is
e formulation is Bayesian, it converge toward an unique
NMF. In order to regularize the problem of eq. 6, on
nalization term to enforce sparsity on A (only few non
im and Park, 2007) :
F
=
kX S.Ak2
+
kAk
1
With k.k
1
, the L1
norm. The ﬁrst term is called data a
ual squared diﬀerence). The second is called regularizat
Li, Y., Ngom, A.: The non-negative matrix factorization toolbox for biological data mining.
Source Code Biol. Med. 8(1), 10 (2013)

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• each component as gamma
• MCMC approach
• no tuning parameter
• slow computation
Method
Bayesian Prior Source Separation
BPSS Moussaoui, S.; et al. 2008, http://dx.doi.org/10.1016/j.neucom.2007.07.034
Schmidt, F.; et al., 2010,http://dx.doi.org/10.1109/TGRS.2010.2062190
N OF NON-NEGATIVE MIXTURE OF NON-NEGATIVE SOURCES
By assuming the mutual independence of the source s
and the mixing coefﬁcients, their associated prior densit
then expressed by
where the vectors and
contain the parameters of the Gamma distributions.
C. Posterior Density and Estimation Issues
T⇤
(
⌫
) =
T
(
⌫
)
C
(
⌫
)
(3)
ert the spectra into absorbance:
X
(
⌫
) = 1
T⇤
(
⌫
)
(4)
tep is the linear mixture :
X
(
⌫
)
⇡
NS
X
i=1
Si(
⌫
)
.Ai
(5)
the source spectra and Ai
the spectral abundance. In this de-
physical meaning of Si(
⌫
)
and Ai
is lost but the apparent SNR
4

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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
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
d
, (2)
hdj
| λd
∼ T N 0, λ−1
d
, 0, ∞ or hdj
|λd
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

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Results on Mars
atmospheric data
• Methane (life) on Mars ?
• PH3 (life) on Venus ?

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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
nadir (long term)a
≤3.00 mrad
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

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NOMAD/ExoMars TGO
• Methane (life) on Mars ?
• detection limits down to 10 ppt = 10 part per
trillion = 10-11 !
• how to discover new unexpected species out
of large dataset ?!

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• Synthetic toy model
• Simulations
• Real data
Results on Mars atmospheric
data

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•Synthetic toy model
• Simulations
• Real data
data

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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

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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
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
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%
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
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

View Slide

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
hetic dataset is represented in Fig. 1.
heck the quality of the estimation, we simply compute the
ient between SCH4
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

View Slide

Toy model
• linear mixture
10 000 spectra
O
206
207
methane SCH4
208
AH2
O
, ACH4
:
209
X
=
SH2
O
.AH2
O +
SCH4
.ACH4 +
n
210
=
G
(0
,
)
211
O = 5
/
6
.
(1
,
10) + 1
/
6
.U
(0
,
1)
, a mixture of
212
213
214
215
of SH2
O
216
, such that the ba
217
of SCH4
218
219
220
7
• plateau at 5-10 endmembers
ﬁgure 5
3 4 5 6 7 8 9
Nb endmember
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Value
Effect of Ns for noise factor 2 with 100 CH
4
spectra
MU mean corr.
psNMF mean corr.
MU frac. corr. > 0.5
psNMF frac. corr. > 0.5
3 4 5 6 7 8 9
Nb endmember
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Value
Effect of Ns for noise factor 3 with 100 CH
4
spectra
MU mean corr.
psNMF mean corr.
Factor above noise level of 2
Nb endmember
Value
100 hidden CH4 spectra
Factor above noise level of 3
Nb endmember
Value

View Slide

Toy model
• linear mixture
10 000 spectra
O
206
207
methane SCH4
208
AH2
O
, ACH4
:
209
X
=
SH2
O
.AH2
O +
SCH4
.ACH4 +
n
210
=
G
(0
,
)
211
O = 5
/
6
.
(1
,
10) + 1
/
6
.U
(0
,
1)
, a mixture of
212
213
214
215
of SH2
O
216
, such that the ba
217
of SCH4
218
219
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.
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
4
spectra
MU mean corr.
psNMF mean corr.
ﬁgure 6
Factor above noise level
Value
Factor above noise level
Value
• 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
4
spectra
MU mean corr.
psNMF mean corr.
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
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

View Slide

•Simulations
• Real data
data

View Slide

Simulation
• non-linear : GCM + radiative transfer + instrument
simulation : 12 486 spectra, 106 occultation, order
136, May-December 2018
• parameters : H2O density + CH4 density + random
addition of CH4 + noise level
• average of 10 noise realizations
Daerden, F.,et al., 2019.Icarus 326, 197–224.
3060 3065 3070 3075 3080
Wavenumber (cm-1)
0
0.01
0.02
0.03
0.04
0.05
0.06
Transmittance
Simulation spectra

View Slide

10 2 10 3
0
0.2
0.4
0.6
0.8
1
value
Fraction of the 4 main peaks detected
fraction CH
4
= 100%
fraction CH
4
= 50%
fraction CH
4
= 10%
fraction CH
4
= 5%
fraction CH
4
= 1%
10 2 10 3
0
0.5
1
1.5
2
value
Mean distance to the expected center [in pixel]
10 2 10 3
CH
4
density [in ppt]
0
0.05
0.1
0.15
0.2
0.25
value
Abundance of CH
4
in the source
noise level 0.001
noise level 0.0001
10 2 10 3
0
0.2
0.4
0.6
0.8
1
value
fraction CH
4
= 100%
fraction CH
4
= 50%
fraction CH
4
= 10%
fraction CH
4
= 5%
fraction CH
4
= 1%
10 2 10 3
0
0.5
1
1.5
2
value
10 2 10 3
CH
4
density [in ppt]
0
0.05
0.1
0.15
value
Abundance of CH
4
in the source
noise level 0.001
noise level 0.0001
Figure 7: Results of the psNMF algorithm for NS = 5 on simulation dataset, averaged over 10
ﬁgure 7
Abundance of CH4 in the source
CH4 density [in ppt] CH4 density [in ppt]
Value Value Value
Value
Value
Value
Figure 7: Results of the psNMF algorithm for NS = 5 on simulation dataset, averaged over 10
10 ppm H2O 100 ppm H2O
• detection limit of CH4 100-500 ppt

View Slide

• Simulations
•Real data
data

View Slide

Real data pre-treatment
all SO from order 134
Pipeline
• from version 1p0a (up to
15 January 2020)
• Resampling to correct for
detector temperature
• 0 < Altitude < 50 km
• SNR > 100
365 985 spectra
Neefs, E.; et al, 2015, http://dx.doi.org/10.1364/ao.54.008494
Vandaele, A. C. et al., 2018 http://dx.doi.org/10.1007/s11214-018-0517-2

View Slide

Pipeline
• Continuum removal
(ALS)
• Convert to absorbance
365 985 spectra
Eilers, P. H. & Boelens, H. F., Leiden University Medical Centre Report, 2005

View Slide

Pipeline
• Shift correction
(residual of detector
temperature)
365 985 spectra
F. Schmidt Automatic Detection

View Slide

3015 3020 3025 3030 3035
0
0.01
0.02
0.03
0.04
0.05
0.06
sources (total rms = 0.0012748)
mean = 16%
mean = 83%
• 2 sources
Results order 134
).
ationale, we propose to ﬁrst correct for the continuum
e space:
T⇤
(
⌫
) =
T
(
⌫
)
C
(
⌫
)
(3)
ctra into absorbance:
X
(
⌫
) = 1
T⇤
(
⌫
)
(4)
inear mixture :
X
(
⌫
)
⇡
NS
X
i=1
Si(
⌫
)
.Ai
(5)
spectra and Ai
eaning of Si(
⌫
)
and Ai
sources

View Slide

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Abundance
5
10
15
20
25
30
35
40
45
Altitude
Abundances (total RMS = 0.0012748)
mean = 16%
mean = 83%
10 000 spectra
3015 3020 3025 3030 3035
0
0.01
0.02
0.03
0.04
0.05
0.06
sources (total rms = 0.0012748)
mean = 16%
mean = 83%
abundances
cles, scattering by molecules and particles, and also reflection at
broadband features. Such large features are modeled by a conti
often taken as a polynomial, that is filtered out.
The problem with this continuum removal rationale is that w
depth is large, the SNR is decreased and the noise effect on cont
amplified (see Sup. Mat.).
Instead of using this rationale, we propose to first correct for
C
(
⌫
)
in the transmittance space:
T⇤
(
⌫
) =
T
(
⌫
)
C
(
⌫
)
Then convert the spectra into absorbance:
X
(
⌫
) = 1
T⇤
(
⌫
)
The final step is the linear mixture :
X
(
⌫
)
⇡
NS
X
i=1
Si(
⌫
)
.Ai

View Slide

• 2358 orbits
• 365 985 spectra
3015 3020 3025 3030 3035
Wavenumber (cm-1)
0.005
0.01
0.015
0.02
0.025
Intensity
3015 3020 3025 3030 3035
Wavenumber (cm-1)
0
0.02
0.04
0.06
0.08
0.1
Absorbance
Theoretical absorbance trough NOMAD
CO
2
N
2
O
2
CO
H
2
O
O
3
CH
4
Solar
19
Results
Schmidt, F. et al., 2020, JQRST, under review
).
ationale, we propose to ﬁrst correct for the continuum
e space:
T⇤
(
⌫
) =
T
(
⌫
)
C
(
⌫
)
(3)
ctra into absorbance:
X
(
⌫
) = 1
T⇤
(
⌫
)
(4)
inear mixture :
X
(
⌫
)
⇡
NS
X
i=1
Si(
⌫
)
.Ai
(5)
spectra and Ai
eaning of Si(
⌫
)
and Ai
endmembers (sources)

View Slide

• 2358 orbits
• 365 985 spectra
3015 3020 3025 3030 3035
Wavenumber (cm-1)
0.005
0.01
0.015
0.02
0.025
Intensity
3015 3020 3025 3030 3035
Wavenumber (cm-1)
0
0.02
0.04
0.06
0.08
0.1
Absorbance
CO
2
N
2
O
2
CO
H
2
O
O
3
CH
4
Solar
19
H2O
New lines CO2
Simulation
• Detection of H2O
• New lines CO2 magnetic
dipole
• No detection of CH4
Results
Trokhimovskiy, A.; et al., 2020, http://
dx.doi.org/10.1051/0004-6361/202038134
Villanueva, G, 2018,http://dx.doi.org/10.1016/j.jqsrt.2018.05.023

View Slide

2675 2680 2685 2690 2695
Wavenumber (cm-1)
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0.022
Intensity
2675 2680 2685 2690 2695
Wavenumber (cm-1)
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Absorbance
CO
2
N
2
O
2
CO
H
2
O
O
3
CH
4
Solar
Figure 8: Results of the psNMF algorithm for the diﬀraction order 119 for NS = 5. The
18
Figure 8: Results of the psNMF algorithm for the diﬀraction order 119 for NS = 5. The
18
• 821 orbits
• 134 045 spectra
Results
• Detection of H2O and
CO2
H2O
CO2

View Slide

Conclusion
• New fast unsupervised data
quicklook
• Validated on simulation datasets
• Results on real NOMAD data
SO from order 119, 134, 136
• Detection of H2O, CO2
• New lines CO2 magnetic
dipole
Trokhimovskiy, A.; et al., 2020, http://
dx.doi.org/10.1051/0004-6361/202038134
Schmidt, F. et al., 2020, JQSRT, under review
[email protected]@universite-paris-saclay.fr

View Slide

Plan
• blind linear unmixing (what and where?)
•supervised linear unmixing (where?
How much?)
F. Schmidt Plan

View Slide

Instrument
Surface
…!
Surface!
Atmosphere!
target!
11th!
6th!
1st!
…!
"≈70º!
"≈70º!
"≈0º!
"≈30º!
"≈30º!
$
in!
$
out!
Atmosphere
nadir
image
incoming
EPF
outgoing
EPF
Targeted observations
11 multi-angle images
Murchie et al., JGR, 2007
10 off-nadir images (180 m/pxl)
eme±70°, constant inc
1 nadir image (20 m/pxl) credit: http://crism.jhuapl.edu
Hyperspectral image
438+107 bands
(0.36 to 3.92 μm)
CRISM (Compact Reconnaissance Imaging Spectrometer for
Mars in Mars Reconnaissance Orbiter spacecraft)

View Slide

Spectral analysis
1. Classiﬁcation
• Supervised: knowing laboratory spectra
• GOAL: Where are the reference spectra ?
2. Radiative transfer inversion
• Quantitative estimation of surface properties

View Slide

Mathematical problem
• Estimation of abundances, under constraints
d length, (iv) the emergence direction is always t
Thus, based on this model and using the geogra
ture assumption, the radiance factor at location (
at wavelenght λ satisﬁes the following observatio
L(x, y, λ) =
ρa
(λ) + Φ(λ)
P
p=1
αp
(x, y) ρp
(λ) cos [θ(x
where Φ(λ) is the spectral atmospheric tran
θ(x, y) the angle between the solar direction and
face normal (solar incidence angle), P the n
endmembers in the region of coordinates (x, y),
us, based on this model and using the geographic mix-
re assumption, the radiance factor at location (x, y) and
wavelenght λ satisﬁes the following observation model:
L(x, y, λ) =
ρa
(λ) + Φ(λ)
P
p=1
αp
(x, y) ρp
(λ) cos [θ(x, y)] (1)
ere Φ(λ) is the spectral atmospheric transmission,
x, y) the angle between the solar direction and the sur-
e normal (solar incidence angle), P the number of
dmembers in the region of coordinates (x, y), ρp
(λ) the
ectrum of the p-th endmember, αp
(x, y) its weight in
e mixture and ρa
(λ) the radiation that did not arrive
ectly from the area under view. This mixture model can
o be written as:
min ||↵p.⇢p L||
min ||↵p.⇢p L||, ↵p > 0,
P
↵p
= 1
•Property : linearly
dependent
spectra in the
database give one
single solution !
1 1.5 2 2.5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Wavelength (µm)
Reflectance (%)
Endmembers
Clay Montmorillonite Bentonite
Flat 1
Flat 0.0001
Positivity,
sum-to-one

View Slide

1 1.5 2 2.5
0.4
0.45
0.5
0.55
0.6
Wavelength (µm)
Reflectance (%)
Absorption band thickness
Clay Montmorillonite Bentonite (0%)
Absorption band depth

View Slide

1 1.5 2 2.5
0.35
0.4
0.45
0.5
0.55
0.6
0.65
Wavelength (µm)
Reflectance (%)
Continuum shift
Flat 1 (0%)
Flat 1 (1%)
Flat 1 (2%)
Flat 1 (3%)
Flat 1 (4%)
Flat 1 (5%)
Flat 1 (6%)
Flat 1 (7%)
Flat 1 (8%)
Flat 1 (9%)
Flat 1 (10%)
Continuum shift

View Slide

To model the effect of :
• aerosols
• continuum
• surface roughness of the regolith
• grain size, shape roughness
• mixture (opaque, feldspar, ...)
Adding
other
spectra ?
Remove discrepancies between
observed spectra and database

View Slide

1 1.5 2 2.5
0.35
0.4
0.45
0.5
0.55
0.6
Wavelength (µm)
Reflectance (%)
Maximum shift
sin 1/2 (0%)
sin 1/2 (1%)
sin 1/2 (2%)
sin 1/2 (3%)
sin 1/2 (4%)
sin 1/2 (5%)
sin 1/2 (6%)
sin 1/2 (7%)
sin 1/2 (8%)
sin 1/2 (9%)
sin 1/2 (10%)
Maximum shift ?

View Slide

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 signifi-
cant 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

View Slide

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,
Heinz, D. & I-Chang, C., TGRS, 2001, 39, 529-545,

View Slide

Synthetic test 1
• Synthetic spectra of:
• 90% Flat at 0.35 (average Mars)
• 10% random mixture of one/two components
• Radiative transfer :
• DISORT : non-linear
• Martian aerosols from AOT=0 to AOT=100
• Adding instrumental noise from dark current
Lin, Z.; et al.,Improved discrete ordinate solutions in the presence of an anisotropically reﬂecting
lower boundary: Upgrades of the DISORT computational tool Journal of Quantitative Spectroscopy
and Radiative Transfer, 2015, 157, 119 - 134, http://dx.doi.org/http://dx.doi.org/10.1016/j.jqsrt.
2015.02.014
Wolff, M. et al., Wavelength dependence of dust aerosol single scattering albedo as observed by the Compact Reconnaissance Imaging Spectrometer J. Geophys.
Res., 2009, 114, E00D04-, http://dx.doi.org/10.1029/2009JE003350

View Slide

1 1.5 2 2.5
0.34
0.36
0.38
0.4
0.42
0.44
0.46
0.48
Detection of Gypsum (10 %)
Wavelength
Reflectance (offset for clarity)
aot=0 (ab = 9.6 %)
aot=1 (ab = 5.1 %)
aot=5 (ab = 1.1 %)
aot=100 (ab = 0 %)
1
0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
fa
fa
fa
fa
fa
Figure 4: Examples of results using the IPLS algorithm, with renormalizat
More
aerosols
0.7%
37%
100%
0.0%
Atmospheric
transmittance

View Slide

Synthetic test 2
• Pure mineral spectra :
• Grain size factor using Shkuratov theory
from x1/1000 to x1000
• Adding instrumental noise from dark
current
Shkuratov, Y.; Starukhina, L.; Hoffmann, H. & Arnold, G. A Model of
Spectral Albedo of Particulate Surfaces: Implications for Optical
Properties of the Moon Icarus, 1999, 137, 235-246, http://
www.sciencedirect.com/science/article/B6WGF-45GMFKB-5T/
2/2b056567d27e74edbaf976c01f89d10f

View Slide

1 1.5 2 2.5
0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
Detection of Gypsum (10 %)
Wavelength
fact=1000 (ab = 0 %)
fact=10 (ab = 5.9 %)
fact=1 (ab = 9.7 %)
fact=1/10 (ab = 5.1 %)
fact=1/1000 (ab = 0.5 %)
Larger
grain
size
Smaller
grain
size

View Slide

Results on Ophir, Valles
Marineris
• Indurated rocks
made of :
• Orthopyroxene
• Olivine
• Kieserite
• ultramaﬁc rock,
altered by aqueous
processes
COMPOSITIONAL+MAPPING
Preliminary results

View Slide

More sophisticated estimates through Mixed Integer Programming (MIP)
y =
S
a ( + errors )
S
; high number of endmembers
Spectral variability [Zare and Ho, 2014, Meyer et al., 2016]
)
add more constraints to the problem
Models based on binary variables encoding the presence of each member in the data
bn
= 1 , an
6= 0
; reformulated as 0  an
 bn
: Mixed Integer Programs
Sparsity: most abundances are zero [Iordache et al., 2011]
Structuration of the dictionary into groups [Meyer et al., 2016, Drumetz et al., 2019]
Minimum values on the nonzero coe cients [never seen . . . ]
3 / 11
.1cm
Collaboration with S. Bourguignon and R. Ben Mhenni

View Slide

Exact
`0
-norm sparsity
Only a small number of elementary spectra are used for representing the mixture
0
0
0
0
0
0
0.2
0.3
0.5
s
1
s
2
s
3
s
4
s
5
s
6
s
7
s
8
s
9
ਚ S a
≈
! a must be sparse:
kak0
= Card(n|an
6= 0)  K
Standard sparse methods perform poorly (`1
-norm, greedy algorithms)
;
Exact
`0
-norm constraint:
min
a2[0,1]N , b2{0,1}N
1
2
ky
S
ak2 s.t.
8
>
<
>
:
0
 a  b
P
N
n=1
bn
 K
P
N
n=1
an
= 1
4 / 11
.1cm

View Slide

To sum up: reformulations as Mixed-integer Programs (MIPs)
`0
-norm sparsity
min
a2[0,1]N , b2{0,1}N
1
2
ky
S
ak2 s.t.
8
>
<
>
:
0
 a  b
P
N
n=1
bn
 K
P
N
n=1
an
= 1
Group Exclusivity
min
a2[0,1]N , b2{0,1}N
1
2
ky
S
ak2 s.t.
8
>
<
>
:
0
 a  b
P
i2Gj
bn
 1
P
N
n=1
an
= 1
Signiﬁcant Abundances
min
a2[0,1]N , b2{0,1}N
1
2
ky
S
ak2 s.t.
(
⌧b  a  b
P
N
n=1
an
= 1
! E cient resolution via numerical MIP solvers (ex. CPLEX)
! These constraints can be mixed
7 / 1
.1cm
SA
GE

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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

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Plan
• Linear mixing is useful to handle complex
dataset !
• for characterizing unexpected spectra
• to pick/detect out of a large dataset
(Interior-Point VS Mixed Integer Program)
• Next generation of algorithm (non-linear?!) ?
New discoveries in planetary science...
F. Schmidt Conclusion

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So many fruitful
collaboration...
• Jennifer Fernando, François Andrieu, Ines Belgacem,
Guillaume Cruz-Mermy, Chiara Marmo...
• Matthieu Kowalski, Nicolas Gac, Alina Meresescu...
• Saïd Moussaoui, Sébastien Bourguignon, Ramzi Ben
Mhenni, Maxime Legendre...
• Christian Jutten, Jocelyn Chanussot, Sylvain Douté...
F. Schmidt Conclusion

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Frédéric Schmidt

Frédéric Schmidt

S³ Seminar

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