View
190
Download
0
Category
Tags:
Preview:
Citation preview
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Nonlinear unmixing of hyperspectral imagesusing radial basis functionsand orthogonal least squares
Yoann Altmann1, Nicolas Dobigeon1,Steve McLaughlin2 and Jean-Yves Tourneret1
1University of Toulouse - IRIT/INP-ENSEEIHT Toulouse, FRANCE2School of Engineering and Electronics - University of Edinburgh, U.K.
IEEE IGARSS 2011, Vancouver, Canada
1 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Hyperspectral Imagery
Hyperspectral Images
I same scene observed at different wavelengths
I pixel represented by a vector of hundreds of measurements
Hyperspectral Cube
2 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Hyperspectral Imagery
Hyperspectral Images
I same scene observed at different wavelengths
I pixel represented by a vector of hundreds of measurements
Hyperspectral Cube
2 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Hyperspectral Imagery
Hyperspectral Images
I same scene observed at different wavelengths
I pixel represented by a vector of hundreds of measurements
Hyperspectral Cube
3 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Hyperspectral Imagery
Hyperspectral Images
I same scene observed at different wavelengths
I pixel represented by a vector of hundreds of measurements
Hyperspectral Cube
3 / 34
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 componentsPresence of mixed pixels in hyperspectral data...
...even at high spatial resolution!
Unmixing steps
1. Endmember extraction: estimating the spectral signatures, orendmembers, (PPI1, N-FINDR2, VCA3, MVES4...)
2. Inversion: estimating the abundances (FCLS5, Bayesian algo.6...)
(1+2). Joint estimation of endmembers and abundances (DECA7, BLU8...)
1Boardman et al., in Sum. JPL Airborne Earth Science Workshop, 1995.2Winter, in Proc. SPIE, 1999.3Nascimento et al., IEEE Trans. Geosci. and Remote Sensing, 2005.4Chan et al., IEEE Trans. Signal Process., 2009.5Heinz et al., IEEE Trans. Geosci. and Remote Sensing, 2001.6Dobigeon et al., IEEE Trans. Signal Process., 2008.7Nascimento et al., in Proc IGARSS., 2007.8Dobigeon et al., IEEE Trans. Signal Process., 2009.
4 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Unmixing problem
Two strategies: linear vs non-linear unmixing
I 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
I Nonlinear model→ to describe intimate mixtures (e.g., sands)→ to handle multiple scattering effects
Contribution of this paperNonlinear unmixing using radial basis functions to solve the inversion step.
5 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Unmixing problem
Two strategies: linear vs non-linear unmixing
I 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
I Nonlinear model→ to describe intimate mixtures (e.g., sands)→ to handle multiple scattering effects
Contribution of this paperNonlinear unmixing using radial basis functions to solve the inversion step.
5 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Outline
Radial basis function networkMixing modelsNetwork structure
Training: selecting RBF centers using OLS
Inversion: constrained abundance estimation
Simulation resultsSynthetic dataReal Image data
Conclusions
6 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Radial basis function network
Outline
Radial basis function networkMixing modelsNetwork structure
Training: selecting RBF centers using OLS
Inversion: constrained abundance estimation
Simulation resultsSynthetic dataReal Image data
Conclusions
7 / 34
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.
I Single-path of the detected photons
I Linear combinations of the contributions of the endmembers
8 / 34
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.
I Possible interactions between the components of the scene
I Nonlinear terms included in the mixing model
9 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Radial basis function network
Mixing models
General (linear/nonlinear) mixing model
Definitiony = fM (a)
I a = [a1, . . . , aR]T abundance vector
I R number of endmembers
I M endmember matrix
I fM (linear/nonlinear) function from RR to RL parameterized by MRemark : linear mixing defined by fM : a 7→Ma
Constraints
positivity : ar ≥ 0, ∀r ∈ 1, ..., R, sum-to-one :R∑
r=1
ar = 1 (1)
Supervised unmixingThe inversion problem can be formulated as
a = f−1M (y)
→ knowledge of f−1M (·) required!
10 / 34
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 procedureGiven y, approximating a by the following linear expansion
a = f−1M (y) ≈
N∑n=1
φn(y)wn = W
φ1(y)...
φN (y)
I a estimated abundance vector
I wn = [wn,1, . . . , wn,R]T weight vector
I φn(y) projection of the data vector y onto the nth basis function
I Gaussian kernels
φn(y) = exp
(−‖y − cn‖2
2σ2
)I cn nth center of the networkI unique fixed dispersion parameter σ2
I N number of basis functions
9Guilfoyle et al., IEEE Trans. Geosci. and Remote Sensing, 2001.11 / 34
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.
I Estimation of the nonlinear relation relating a to y: training step
I Estimation of abundance vectors: inversion step
12 / 34
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.
I Estimation of the nonlinear relation relating a to y: training step
I Estimation of abundance vectors: inversion step
12 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Training: selecting RBF centers using OLS
Outline
Radial basis function networkMixing modelsNetwork structure
Training: selecting RBF centers using OLS
Inversion: constrained abundance estimation
Simulation resultsSynthetic dataReal Image data
Conclusions
13 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Training: selecting RBF centers using OLS
Training step
Inputs:
I training pixels y1, . . . ,yN
I associated abundance vectors a1, . . . ,aN
Output:
I weights w1, . . . ,wN
14 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Training: selecting RBF centers using OLS
Training step
Inputs:
I training pixels y1, . . . ,yN
I associated abundance vectors a1, . . . ,aN
Outputs:
I weights w1, . . . ,wN
15 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Training: selecting RBF centers using OLS
Training step
Learning procedureGiven N training pixels y1, . . . ,yN and associated training abundancevectors a1, . . . ,aN , estimating w1, . . . ,wN such that
A =
aT1
...aT
N
=
φT (y1)
...φT (yN )
w
T1
...wT
N
+E = ΦW +E
I φ(y) = [φ1(y), . . . , φN (y)]T projections of y on the N RBFs
I Φ N ×N matrix of projections
I W N ×R matrix of weight vectors related to the N centers
I E projection error matrix
16 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Training: selecting RBF centers using OLS
Training step
Learning procedure
I Estimating W of size N ×R using least-squares method
minW‖A−ΦW ‖2F
where Φ is of size N ×N .
Problem
I Numerical issues for large values of N (Φ ill-conditioned)
Solution
I Selecting a reduced number of centers out of the N training pixels.
17 / 34
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
I Selection of the M << N most relevant network centers
→ How to determine the relevant subspace of span{φ1, . . . ,φN}?
18 / 34
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
I Selection of the M << N most relevant network centers
→ How to determine the relevant subspace of span{φ1, . . . ,φN}?
19 / 34
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
I Q = [q1, . . . , qN ] N ×N orthogonal matrix such that
span{φ1, . . . ,φN} = span{q1, . . . , qN}
I Θ = [θ1, . . . ,θN ]T new regression coefficients.
Decomposition into relevant/unrelevant subspace
I Sub-matrix decomposition Q =[Q1:M QM+1:N
]I Relevant decomposition obtained when the output energy matrixATA is approximated by
ATA =
N∑m=1
θTmq
Tmqmθm ≈
M∑m=1
θTmq
Tmqmθm
⇒ Quantification of the contribution of the M first orthogonal regressors!10Chen et al. IEEE Trans. Neural Network, 1991.
20 / 34
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
I Q depends on the orthogonalization of {φ1, . . . ,φN}I testing all the orthogonal basis derived from {φ1, . . . ,φN} using
permutations and Gram-Schmidt processes⇒ prohibitive computational cost
Proposed alternative
I sequential and iterative construction of the orthogonal basis in aforward regression manner (i.e., try M = 1, then M = 2, etc...).
(approach similar to the orthogonal matching pursuit)
I stopping rule based on the error reduction ratio
εM =
∥∥∥∑Mm=1 θ
Tmq
Tmqmθm
∥∥∥F
‖ATA‖F
21 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Inversion: constrained abundance estimation
Outline
Radial basis function networkMixing modelsNetwork structure
Training: selecting RBF centers using OLS
Inversion: constrained abundance estimation
Simulation resultsSynthetic dataReal Image data
Conclusions
22 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Inversion: constrained abundance estimation
Constrained abundance estimation
Network output
a = W TMφ(y)
I Network fixed after the training step
I WM estimated using M relevant regressors
ProblemConstraints not ensured when unmixing a new pixel!
Solution: constrained least-squares method
a = arg mina
∥∥∥φ(y)−W T†M a
∥∥∥2
2
subject to the positivity and sum-to-one constraints for a
I W T†M pseudo-inverse of W T
M
I problem solved using the FCLS algorithm11
11Heinz et al., IEEE Trans. Geosci. and Remote Sensing, 2001.23 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Simulation results
Outline
Radial basis function networkMixing modelsNetwork structure
Training: selecting RBF centers using OLS
Inversion: constrained abundance estimation
Simulation resultsSynthetic dataReal Image data
Conclusions
24 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Simulation results
Synthetic data
Synthetic data
Simulation parameters
I 3 synthetic images (10× 10 pixels) composed of R = 3 endmembers(green grass, olive green paint, galvanized steel metal) corrupted by anadditive Gaussian noise with SNR ' 15dB
I 3 mixing modelsI1 = Linear Mixing Model (LMM)I2 = Fan’s model (FM)12
I3 = Nascimento’s model (NM)13
I Abundances uniformly drawn over the simplex defined by theconstraints.
Training data
I 3 training images (50× 50 = 2500 pixels) composed of the same R = 3endmembers (SNR ' 15dB)
I 3 mixing modelsT1 = LMMT2 = FMT3 = NM
12Fan and al., Remote Sensing of Environment, 2009.13Nascimento et al., Proc. SPIE, 2009.
25 / 34
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.
I M = 11 centers selected for T1 (LMM)
I M = 13 centers selected for T2 (FM)
I M = 17 centers selected for T3 (NM) (not displayed here)
26 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Simulation results
Synthetic data
Synthetic data
Quality of unmixing
I Root mean square error
RMSE =
√√√√ 1
NR
N∑n=1
‖an − an‖2
an estimate of the nth abundance vector an
RMSE (×10−1)
without OLS with OLSModel-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)
14Fan and al., Remote Sensing of Environment, 2009.27 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Simulation results
Real Image data
Real data
Simulation parameters
I Real hyperspectral image of 50× 50 pixels extracted from a largerimage acquired in 1997 by AVIRIS (Moffett Field, CA, USA) data setreduced from 224 to 189 bands (water absorption bands removed)
I VCA used to extract the R = 3 approximated endmembers associatedto water, soil and vegetation.
Real hyperspectral data: Moffett field acquired by AVIRIS in 1997 (left) and theregion of interest shown in true colors (right).
28 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Simulation results
Real Image data
Real data
Estimated endmembers
Training step
I Estimated spectra used to generate training data sets of 2500 pixelsaccording to the LMM and FM (SNR ≈ 15 dB)
I OLS procedure performed to reduce the number of centers (LMM: 21centers, FM: 23 centers)
29 / 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.
I Maps similar to maps obtained using dedicated algorithms
30 / 34
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.
I Maps similar to maps obtained using dedicated algorithms
31 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Conclusions
Outline
Radial basis function networkMixing modelsNetwork structure
Training: selecting RBF centers using OLS
Inversion: constrained abundance estimation
Simulation resultsSynthetic dataReal Image data
Conclusions
32 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Conclusions
Conclusions
Nonlinear unmixing of hyperspectral images using radial basis functions andorthogonal least squares
I Radial basis functions introduced to invert the nonlinear relationshipbetween the observation vector and the associated abundance vector.
I Selection of a reduced number of RBFs centers from training data toreduce the network complexity using an OLS procedure.
I Modification of the algorithm to satisfy positivity and sum-to-oneconstraints.
Perspectives
I Adaptive update of the weights and centers for unsupervised unmixing.
33 / 34
Nonlinear unmixing of hyperspectral images using radial basis functions and orthogonal least squares
Conclusions
Nonlinear unmixing of hyperspectral imagesusing radial basis functionsand orthogonal least squares
Yoann Altmann1, Nicolas Dobigeon1,Steve McLaughlin2 and Jean-Yves Tourneret1
1University of Toulouse - IRIT/INP-ENSEEIHT Toulouse, FRANCE2School of Engineering and Electronics - University of Edinburgh, U.K.
IEEE IGARSS 2011, Vancouver, Canada
34 / 34
Recommended