Estimation of Mars surface physical properties from hyperspectral images using the SIR method

Transcription

1 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

2 Outline I. Context Hyperspectral image data Inverse problem II. III. Dimension reduction PCA SIR Regularization and estimation Zhong et al., 2005 Tikhonov Validation on simulations IV. V. Application to the south polar cap of Mars VI. Conclusion and future work

3 Outline I. Context Hyperspectral image data Inverse problem II. III. Dimension reduction PCA SIR Regularization and estimation Zhong et al., 2005 Tikhonov Validation on simulations IV. V. Application to the south polar cap of Mars VI. Conclusion and future work

4 Introduction Hyperspectral cube Spectrometer Spectrum at one pixel Source: Nasa

5 Radiative transfer model Physical state Chemical composition Granularity texture RADIATIVE TRANSFER MODEL: evaluates direct link between parameters and spectra. Allows the construction of a training data

6 Inverse problem Physical state Chemical composition Granularity texture INVERSE PROBLEM: evaluates the properties of atmospheric and surface materials from the spectra

7 Nearest neighbor Weighted Nearest neighbor Usual methods

8 Aim To establish functional relationships between: Spectra (p=184) from Mars Express mission x p y Physical parameter : proportion of water, proportion of dust, grain size Construct f in order to estimate parameters: f : p x y

9 Difficulties Curse of dimensionality (184 wavelengths): dimension of x has to be reduced Find projection axis axis will be retained) a p (here, only the first Instead of estimating f such as y = f ( x), we will suppose there exists g : exists such that: y = g( < a, x>, ε )

10 Outline I. Context Hyperspectral image data Inverse problem II. III. Dimension reduction PCA SIR Regularization and estimation Zhong et al., 2005 Tikhonov Validation on simulations IV. V. Application to the south polar cap of Mars VI. Conclusion and future work

11 Principal component analysis Maximizes the variance of the projections of the observations x Does not take into account y

12 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 Eigenvectors of Γ with Γ= var( EX ( / Y)) = var( X ) Between-slice variance Within-slice variance

13 Application of SIR TRAINING DATA OBSERVED DATA

14 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 >

15 Outline I. Context Hyperspectral image data Inverse problem II. III. Dimension reduction PCA SIR Regularization and estimation Zhong et al., 2005 Tikhonov Validation on simulations IV. V. Application to the south polar cap of Mars VI. Conclusion and future work

16 Regularization (1) Usual SIR: eigenvectors of 1 Σ Γ Regularized SIR: Zhong et al., 2005: eigenvectors of ( ) + λid 1 Tikhonov regularization: eigenvectors of Γ ( 2 ) + λid 1 Σ Γ

17 Regularization (2) Usual SIR 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

18 Estimation Nearest neighbors (quite long!) Spline functions (choice of new parameters, boundaries) Linear interpolation

19 Choice of the regularization parameter By minimization of Normalized RMSE criterion yˆ y y y = Residuals sum of square Total sum of square

20 Outline I. Context Hyperspectral image data Inverse problem II. III. Dimension reduction PCA SIR Regularization and estimation Zhong et al., 2005 Tikhonov Validation on simulations IV. V. Application to the south polar cap of Mars VI. Conclusion and future work

21 Validation (1) Training data Training data + noise Test data Application of Regularized Sliced Inverse Regression: Determination of an axis a(λ) depending on a regularization parameter λ. Estimation of parameters by linear interpolation Minimization of Normalized RMSE ESTIMATION Optimal axis a(λ) VALIDATION

22 Validation (2) Nearest neighbors Regularized Sliced inverse regression (Tikhonov) Weighted nearest neighbors

23 Validation (3) Normalized RMSE criterion: yˆ y y y = Residuals sum of square Total sum of square Better when close to 0 t a Γ a = t a a SIR criterion: between-slice variance total variance Better when close to 1 Parameter Variation interval Tikhonov ŷ y y y t t a a Γ Σ a a ŷ y Zhong y y t t a a Γ Σ a a Nearest neighbor ŷ y y y Weighted nearest neighbor ŷ y y y Proportion of dust [ ] Proportion of CO2 [ ] Proportion of water [ ] Grain size of water [ ] Grain size of CO2 [ ] 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.

24 Outline I. Context Hyperspectral image data Inverse problem II. III. Dimension reduction PCA SIR Regularization and estimation Zhong et al., 2005 Tikhonov Validation on simulations IV. V. Application to the south polar cap of Mars VI. Conclusion and future work

25 Application to south polar cap of Mars Source: Visions de Mars /eds: La Martinière Model determined by physicists (water + CO2 + dust) 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.

26 Proportion of CO2 Regularized Sliced Inverse Regression (Tikhonov) Nearest neighbors Weighted nearest neighbors

27 Proportion of water Regularized Sliced Inverse Regression (Tikhonov) Nearest neighbors Weighted nearest neighbors

28 Proportion of Dust

29 Grain size of water

30 Grain size of CO2

31 Outline I. Context Hyperspectral image data Inverse problem II. III. Dimension reduction PCA SIR Regularization and estimation Zhong et al., 2005 Tikhonov Validation on simulations IV. V. Application to the south polar cap of Mars VI. Conclusion and future work

32 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