Estimation of Mars surface physical properties from hyperspectral images using the SIR method Caroline Bernard-Michel, Sylvain Douté, Laurent Gardes and Stéphane Girard
Source: ESA
Outline I.
Context • •
II.
Dimension reduction • •
III.
PCA SIR
Regularization and estimation • •
IV. V. VI.
Hyperspectral image data Inverse problem
Zhong et al., 2005 Tikhonov
Validation on simulations Application to the south polar cap of Mars Conclusion and future work
Outline I.
Context • •
II.
Dimension reduction • •
III.
PCA SIR
Regularization and estimation • •
IV. V. VI.
Hyperspectral image data Inverse problem
Zhong et al., 2005 Tikhonov
Validation on simulations Application to the south polar cap of Mars Conclusion and future work
Introduction
Hyperspectral cube
Spectrometer
Spectrum at one pixel
Source: Nasa
Radiative transfer model ate t s l a c i Phys
Ch e com mical p os ition Gr an u
lar ity
tu r x te
e
RADIATIVE TRANSFER MODEL: evaluates direct link between parameters and spectra. Allows the construction of a training data
Inverse problem ate t s l a c i Phys
Ch e com mical p os ition Gr an u
r u t tex
e
lar ity
INVERSE PROBLEM: evaluates the properties of atmospheric and surface materials from the spectra
Aim • To establish functional relationships between: p x ∈ \ – Spectra (p=184) from Mars Express mission – Physical parameter y ∈ \ : proportion of water, proportion of dust, grain size… – Construct f in order to estimate parameters: p
f :\ → \ x →y
Difficulties • Curse of dimensionality (184 wavelengths): dimension of x has to be reduced • Find projection axis a ∈ \ p (here, only the first axis will be retained) • Instead of estimating f such as y = f ( x ) , we will suppose there exists g : \ → \ exists such that: y = g ( < a , x >, ε )
Outline I.
Context • •
II.
Dimension reduction • •
III.
PCA SIR
Regularization and estimation • •
IV. V. VI.
Hyperspectral image data Inverse problem
Zhong et al., 2005 Tikhonov
Validation on simulations Application to the south polar cap of Mars Conclusion and future work
Principal component analysis • Maximizes the variance of the projections of the observations x • Does not take into account y
Sliced inverse regression • • • •
Proposed by Li (1991) Maximizes the between-slice variance of projections −1 PCA of E(Z/Y) with Z = Σ 2 X −1 ∑ Γ with Γ = var( E ( X / Y )) Eigenvectors of ∑ = var( X )
Between-slice variance
Within-slice variance
Application of SIR TRAINING DATA
OBSERVED DATA
Problem • Covariance matrix is ill-conditioned – Bad estimations of the directions – Sensitivity to noise • Can be solved using regularization DATA (x , y) SLICED INVERSE REGRESSION AXIS a < a, x >
< a , x + noise >
Outline I.
Context • •
II.
Dimension reduction • •
III.
PCA SIR
Regularization and estimation • •
IV. V. VI.
Hyperspectral image data Inverse problem
Zhong et al., 2005 Tikhonov
Validation on simulations Application to the south polar cap of Mars Conclusion and future work
Regularization (1) • Usual SIR: – eigenvectors of Σ −1Γ • Regularized SIR: – Zhong et al., 2005: eigenvectors of −1 ( ∑ +λ Id ) Γ – Tikhonov regularization: eigenvectors of −1 2 ( ∑ +λ Id ) Σ Γ
Regularization (2) Usual SIR
¾ ¾ ¾
Regularized SIR (Tikhonov)
Depends on the regularization parameter λ The condition number of the matrix decreases when λ increases The estimation bias increases when λ increases
Estimation • Nearest neighbors (quite long!) • Spline functions (choice of new parameters, boundaries) • Linear interpolation
Choice of the regularization parameter • By minimization of “Normalized RMSE” criterion yˆ − y Residuals sum of square = y− y Total sum of square
Outline I.
Context • •
II.
Dimension reduction • •
III.
PCA SIR
Regularization and estimation • •
IV. V. VI.
Hyperspectral image data Inverse problem
Zhong et al., 2005 Tikhonov
Validation on simulations Application to the south polar cap of Mars Conclusion and future work
Validation (1) Training data
Application of Regularized Sliced Inverse Regression: Determination of an axis a(λ) depending on a regularization parameter λ.
Training data + noise
Test data
Estimation of parameters by linear interpolation
Minimization of Normalized RMSE ESTIMATION Optimal axis a(λ)
Residuals sum of square = y− y Total sum of square
a Γ a between-slice variance = t a ∑ a total variance t
Better when close to 0
Better when close to 1 Tikhonov yˆ − y
Zhong
Nearest neighbor
Weighted nearest neighbor
yˆ − y
yˆ − y
y−y
aΓa t aΣa
y−y
y− y
yˆ − y
Parameter
Variation interval
y− y
aΓa t aΣa
Proportion of dust
[0.0006 0.002]
0.33
0.92
0.33
0.92
0.56
0.53
Proportion of CO2
[0.9960 0.9988]
0.27
0.96
0.23
0.94
0.56
0.54
Proportion of water
[0.0006 0.002]
0.13
1.00
0.12
1.00
0.27
0.28
Grain size of water
[100 400]
0.37
0.92
0.38
0.87
0.40
0.68
Grain size of CO2
[40000 105000]
0.19
0.99
0.18
0.98
0.38
0.82
t
t
• SIR gives better results than nearest neighbor classification • Tikhonov and Zhong regularizations are equivalent • With Tikhonov regularization, minimal normalized RMSE is reached on a larger interval than with Zhong’s.
Outline I.
Context • •
II.
Dimension reduction • •
III.
PCA SIR
Regularization and estimation • •
IV. V. VI.
Hyperspectral image data Inverse problem
Zhong et al., 2005 Tikhonov
Validation on simulations Application to the south polar cap of Mars Conclusion and future work
Application to south polar cap of Mars
Source: “Visions de Mars” /eds: La Martinière
• Model determined by physicists (water + CO2 + dust) • 17753 spectra • 184 wavelengths • Training data simulated by radiative transfer model • 5 parameters to study : proportions of water, dust and CO2, grain sizes of CO2 and water.
Proportion of CO2 Nearest neighbors Regularized Sliced Inverse Regression (Tikhonov)
Weighted nearest neighbors
Proportion of water Nearest neighbors Regularized Sliced Inverse Regression (Tikhonov)
Weighted nearest neighbors
Proportion of Dust
Grain size of water
Grain size of CO2
Outline I.
Context • •
II.
Dimension reduction • •
III.
PCA SIR
Regularization and estimation • •
IV. V. VI.
Hyperspectral image data Inverse problem
Zhong et al., 2005 Tikhonov
Validation on simulations Application to the south polar cap of Mars Conclusion and future work
Conclusion and future work • Good results on simulations • Realistic results on real data • Validation is difficult because of the lack of ground measurements • Choice of the regularization parameter? • Uncertainties? • Comparisons to other methods
Validation on simulations. V. Application to the south polar cap of ... RADIATIVE TRANSFER MODEL: evaluates direct link between parameters and spectra.
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