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- 1. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Yoann Altmann1 , Nicolas Dobigeon1 , Steve McLaughlin2 and Jean-Yves Tourneret1 1 University of Toulouse - IRIT/INP-ENSEEIHT Toulouse, FRANCE 2 School of Engineering and Electronics - University of Edinburgh, U.K. IEEE IGARSS 2011, Vancouver, Canada 1 / 34
- 2. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Hyperspectral Imagery Hyperspectral Images same scene observed at diﬀerent wavelengths 2 / 34
- 3. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Hyperspectral Imagery Hyperspectral Images same scene observed at diﬀerent wavelengths Hyperspectral Cube 2 / 34
- 4. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Hyperspectral Imagery Hyperspectral Images same scene observed at diﬀerent wavelengths pixel represented by a vector of hundreds of measurements 3 / 34
- 5. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Hyperspectral Imagery Hyperspectral Images same scene observed at diﬀerent wavelengths pixel represented by a vector of hundreds of measurements Hyperspectral Cube 3 / 34
- 6. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Unmixing problem Unmixing: decomposing a measured pixel into a mixture of pure components Presence of mixed pixels in hyperspectral data... ...even at high spatial resolution! Unmixing steps 1. Endmember extraction: estimating the spectral signatures, or endmembers, (PPI1 , N-FINDR2 , VCA3 , MVES4 ...) 2. Inversion: estimating the abundances (FCLS5 , Bayesian algo.6 ...) (1+2). Joint estimation of endmembers and abundances (DECA7 , BLU8 ...) 1 Boardman et al., in Sum. JPL Airborne Earth Science Workshop, 1995. 2 Winter, in Proc. SPIE, 1999. 3 Nascimento et al., IEEE Trans. Geosci. and Remote Sensing, 2005. 4 Chan et al., IEEE Trans. Signal Process., 2009. 5 Heinz et al., IEEE Trans. Geosci. and Remote Sensing, 2001. 6 Dobigeon et al., IEEE Trans. Signal Process., 2008. 7 Nascimento et al., in Proc IGARSS., 2007. 8 Dobigeon et al., IEEE Trans. Signal Process., 2009. 4 / 34
- 7. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Unmixing problem Two strategies: linear vs non-linear unmixing Linear model → pure materials sitting side-by-side in the scene → 1st-order approximation → most of the research works over the last 2 decades ∼ 1000 entries in IEEEXplore Nonlinear model → to describe intimate mixtures (e.g., sands) → to handle multiple scattering eﬀects 5 / 34
- 8. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Unmixing problem Two strategies: linear vs non-linear unmixing Linear model → pure materials sitting side-by-side in the scene → 1st-order approximation → most of the research works over the last 2 decades ∼ 1000 entries in IEEEXplore Nonlinear model → to describe intimate mixtures (e.g., sands) → to handle multiple scattering eﬀects Contribution of this paper Nonlinear unmixing using radial basis functions to solve the inversion step. 5 / 34
- 9. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Outline Radial basis function network Mixing models Network structure Training: selecting RBF centers using OLS Inversion: constrained abundance estimation Simulation results Synthetic data Real Image data Conclusions 6 / 34
- 10. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Radial basis function network Outline Radial basis function network Mixing models Network structure Training: selecting RBF centers using OLS Inversion: constrained abundance estimation Simulation results Synthetic data Real Image data Conclusions 7 / 34
- 11. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Radial basis function network Mixing models Linear mixing model Reference: IEEE Signal Proc. Magazine, Jan. 2002. Single-path of the detected photons Linear combinations of the contributions of the endmembers 8 / 34
- 12. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Radial basis function network Mixing models Nonlinear mixing model Reference: IEEE Signal Proc. Magazine, Jan. 2002. Possible interactions between the components of the scene Nonlinear terms included in the mixing model 9 / 34
- 13. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Radial basis function network Mixing models General (linear/nonlinear) mixing model Deﬁnition y = fM (a) a = [a1 , . . . , aR ]T abundance vector R number of endmembers M endmember matrix fM (linear/nonlinear) function from RR to RL parameterized by M Remark : linear mixing deﬁned by fM : a → Ma Constraints R positivity : ar ≥ 0, ∀r ∈ 1, ..., R, sum-to-one : ar = 1 (1) r=1 Supervised unmixing The inversion problem can be formulated as −1 ˆ a = fM (y) −1 → knowledge of fM (·) required! 10 / 34
- 14. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Radial basis function network Network structure Radial basis function network (RBFN)9 Proposed inversion procedure Given y, approximating a by the following linear expansion N φ1 (y) −1 a = fM (y) ≈ ˆ φn (y)wn = W . . . n=1 φN (y) ˆ a estimated abundance vector wn = [wn,1 , . . . , wn,R ]T weight vector φn (y) projection of the data vector y onto the nth basis function Gaussian kernels 2 y − cn φn (y) = exp − 2σ 2 cn nth center of the network unique ﬁxed dispersion parameter σ 2 N number of basis functions 9 Guilfoyle et al., IEEE Trans. Geosci. and Remote Sensing, 2001. 11 / 34
- 15. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Radial basis function network Network structure Radial basis function network (RBFN) Radial basis function network for nonlinear unmixing. Estimation of the nonlinear relation relating a to y: training step Estimation of abundance vectors: inversion step 12 / 34
- 16. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Radial basis function network Network structure Radial basis function network (RBFN) Radial basis function network for nonlinear unmixing. Estimation of the nonlinear relation relating a to y: training step Estimation of abundance vectors: inversion step 12 / 34
- 17. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Training: selecting RBF centers using OLS Outline Radial basis function network Mixing models Network structure Training: selecting RBF centers using OLS Inversion: constrained abundance estimation Simulation results Synthetic data Real Image data Conclusions 13 / 34
- 18. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Training: selecting RBF centers using OLS Training step Inputs: training pixels y1 , . . . , yN associated abundance vectors a1 , . . . , aN 14 / 34
- 19. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Training: selecting RBF centers using OLS Training step Inputs: training pixels y1 , . . . , yN associated abundance vectors a1 , . . . , aN Outputs: weights w1 , . . . , wN 15 / 34
- 20. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Training: selecting RBF centers using OLS Training step Learning procedure Given N training pixels y1 , . . . , yN and associated training abundance vectors a1 , . . . , aN , estimating w1 , . . . , wN such that T T T a1 φ (y1 ) w1 . . . A= . = . . . . + E = ΦW + E . T T T aN φ (yN ) wN φ(y) = [φ1 (y), . . . , φN (y)]T projections of y on the N RBFs Φ N × N matrix of projections W N × R matrix of weight vectors related to the N centers E projection error matrix 16 / 34
- 21. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Training: selecting RBF centers using OLS Training step Learning procedure Estimating W of size N × R using least-squares method 2 min A − ΦW F W where Φ is of size N × N . Problem Numerical issues for large values of N (Φ ill-conditioned) Solution Selecting a reduced number of centers out of the N training pixels. 17 / 34
- 22. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Training: selecting RBF centers using OLS Network complexity reduction 18 / 34
- 23. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Training: selecting RBF centers using OLS Network complexity reduction Proposed approach for complexity reduction Selection of the M << N most relevant network centers → How to determine the relevant subspace of span{φ1 , . . . , φN }? 19 / 34
- 24. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Training: selecting RBF centers using OLS Orthogonal least squares10 Orthogonalization A = [φ1 , . . . , φN ] W + E = [q1 , . . . , qN ] Θ + E Q = [q1 , . . . , qN ] N × N orthogonal matrix such that span{φ1 , . . . , φN } = span{q1 , . . . , qN } Θ = [θ1 , . . . , θN ]T new regression coeﬃcients. Decomposition into relevant/unrelevant subspace Sub-matrix decomposition Q = Q1:M QM +1:N Relevant decomposition obtained when the output energy matrix AT A is approximated by N M AT A = T T θm qm qm θm ≈ T T θm qm qm θm m=1 m=1 ⇒ Quantiﬁcation of the contribution of the M ﬁrst orthogonal regressors! 10 Chen et al. IEEE Trans. Neural Network, 1991. 20 / 34
- 25. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Training: selecting RBF centers using OLS Orthogonal least squares Choosing the orthogonal basis Q depends on the orthogonalization of {φ1 , . . . , φN } testing all the orthogonal basis derived from {φ1 , . . . , φN } using permutations and Gram-Schmidt processes ⇒ prohibitive computational cost Proposed alternative sequential and iterative construction of the orthogonal basis in a forward regression manner (i.e., try M = 1, then M = 2, etc...). (approach similar to the orthogonal matching pursuit) stopping rule based on the error reduction ratio M T T m=1 θm qm qm θm F M = AT A F 21 / 34
- 26. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Inversion: constrained abundance estimation Outline Radial basis function network Mixing models Network structure Training: selecting RBF centers using OLS Inversion: constrained abundance estimation Simulation results Synthetic data Real Image data Conclusions 22 / 34
- 27. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Inversion: constrained abundance estimation Constrained abundance estimation Network output T ˆ a = WM φ(y) Network ﬁxed after the training step WM estimated using M relevant regressors Problem Constraints not ensured when unmixing a new pixel! Solution: constrained least-squares method 2 a = arg min φ(y) − WM† a ˜ T a 2 subject to the positivity and sum-to-one constraints for a WM† pseudo-inverse of WM T T problem solved using the FCLS algorithm11 11 Heinz et al., IEEE Trans. Geosci. and Remote Sensing, 2001. 23 / 34
- 28. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Simulation results Outline Radial basis function network Mixing models Network structure Training: selecting RBF centers using OLS Inversion: constrained abundance estimation Simulation results Synthetic data Real Image data Conclusions 24 / 34
- 29. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Simulation results Synthetic data Synthetic data Simulation parameters 3 synthetic images (10 × 10 pixels) composed of R = 3 endmembers (green grass, olive green paint, galvanized steel metal) corrupted by an additive Gaussian noise with SNR 15dB I1 = Linear Mixing Model (LMM) 3 mixing models I2 = Fan’s model (FM)12 I3 = Nascimento’s model (NM)13 Abundances uniformly drawn over the simplex deﬁned by the constraints. Training data 3 training images (50 × 50 = 2500 pixels) composed of the same R = 3 endmembers (SNR 15dB) T1 = LMM 3 mixing models T2 = FM T3 = NM 12 Fan and al., Remote Sensing of Environment, 2009. 13 Nascimento et al., Proc. SPIE, 2009. 25 / 34
- 30. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Simulation results Synthetic data Synthetic data Training step: center selection Initial 2500 centers (left) and selected centers for the images T1 (middle) and T2 (right) with the OLS procedure. M = 11 centers selected for T1 (LMM) M = 13 centers selected for T2 (FM) M = 17 centers selected for T3 (NM) (not displayed here) 26 / 34
- 31. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Simulation results Synthetic data Synthetic data Quality of unmixing Root mean square error N 1 2 RMSE = an − an ˆ NR n=1 ˆ an estimate of the nth abundance vector an RMSE (×10−1 ) without OLS with OLS Model-based algo. RBFN CRBFN RBFN CRBFN I1 (LMM) 0.409 0.407 0.411 0.403 0.395 (FCLS) I2 (FM) 0.391 0.378 0.376 0.393 0.42014 I3 (NM) 0.541 0.532 0.547 0.544 0.689 (FCLS) 14 Fan and al., Remote Sensing of Environment, 2009. 27 / 34
- 32. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Simulation results Real Image data Real data Simulation parameters Real hyperspectral image of 50 × 50 pixels extracted from a larger image acquired in 1997 by AVIRIS (Moﬀett Field, CA, USA) data set reduced from 224 to 189 bands (water absorption bands removed) VCA used to extract the R = 3 approximated endmembers associated to water, soil and vegetation. Real hyperspectral data: Moﬀett ﬁeld acquired by AVIRIS in 1997 (left) and the region of interest shown in true colors (right). 28 / 34
- 33. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Simulation results Real Image data Real data Estimated endmembers Training step Estimated spectra used to generate training data sets of 2500 pixels according to the LMM and FM (SNR ≈ 15 dB) OLS procedure performed to reduce the number of centers (LMM: 21 centers, FM: 23 centers) 29 / 34
- 34. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Simulation results Real Image data Real data Estimated abundance maps (LMM) Top: Constrained RBFN. Bottom: FCLS. Maps similar to maps obtained using dedicated algorithms 30 / 34
- 35. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Simulation results Real Image data Real data Estimated abundance maps (FM) Top: Constrained RBFN. Bottom: Fan-LS. Maps similar to maps obtained using dedicated algorithms 31 / 34
- 36. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Conclusions Outline Radial basis function network Mixing models Network structure Training: selecting RBF centers using OLS Inversion: constrained abundance estimation Simulation results Synthetic data Real Image data Conclusions 32 / 34
- 37. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Conclusions Conclusions Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Radial basis functions introduced to invert the nonlinear relationship between the observation vector and the associated abundance vector. Selection of a reduced number of RBFs centers from training data to reduce the network complexity using an OLS procedure. Modiﬁcation of the algorithm to satisfy positivity and sum-to-one constraints. Perspectives Adaptive update of the weights and centers for unsupervised unmixing. 33 / 34
- 38. Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Conclusions Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares Yoann Altmann1 , Nicolas Dobigeon1 , Steve McLaughlin2 and Jean-Yves Tourneret1 1 University of Toulouse - IRIT/INP-ENSEEIHT Toulouse, FRANCE 2 School of Engineering and Electronics - University of Edinburgh, U.K. IEEE IGARSS 2011, Vancouver, Canada 34 / 34

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