RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection Residual Component Analysis of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection Steve McLaughlin (1) Joint work with Y. Altmann (1) , N. Dobigeon (2) , J.-Y. Tourneret (2) and M. Pereyra (3) (1) Heriot-Watt University – School of Engineering of Physical Sciences, U.K. (2) University of Toulouse – IRIT/INP-ENSEEIHT/T¨ ı¿ 1 2 SA, France (2) University of Bristol – School of Mathematics,U.K. MAHI 2014 workshop, Nice, 15/12/2014 1 /44
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RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
Residual Component Analysis ofHyperspectral Images for Joint NonlinearUnmixing and Nonlinearity Detection
Steve McLaughlin(1)
Joint work with Y. Altmann(1), N. Dobigeon(2), J.-Y. Tourneret(2) and M.Pereyra(3)
(1)Heriot-Watt University – School of Engineering of Physical Sciences, U.K.(2)University of Toulouse – IRIT/INP-ENSEEIHT/Tı¿ 1
2 SA, France(2)University of Bristol – School of Mathematics,U.K.
MAHI 2014 workshop, Nice, 15/12/2014
1 /44
RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
Spectral unmixing
Features of interest
I pure spectral signatures:endmembers
I proportions of eachcomponent: abundances
Unsupervised unmixing
I Endmembers andabundances unknown
I Blind source separation
Supervised unmixing
I Endmembers assumed to beknown
I Abundance estimation:inverse problem
2 /44
RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
Linear spectral unmixing
Linear mixing model (LMM)
yn =
R∑r=1
ar,nmr + en = Man + en
I yn nth observed pixel in L bandsI M = [m1, . . . ,mR] endmember matrix containing the spectra of
the R pure components of the sceneI an = [a1,n, . . . , aR,n]T abundance vector of the nth pixel
Abundance constraints
Abundance positivity: ar,n ≥ 0, r = 1, . . . , R n = 1, . . . , N.
Abundance sum-to-one:∑Rr=1 ar,n = 1 n = 1, . . . , N.
3 /44
RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
Nonlinear spectral unmixing
yn = g (M,an) + en
I Possible interactions between the components of the scene
I Nonlinear terms included in the mixing model
I Several models depending on the nature of the scene (intimatemixtures, bilinear models, kernel-based models,...)
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RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
Nonlinear spectral unmixing
Endmember variability
I Changes in observation conditionsI Intrinsic variability of the materials
Endmember Spectra measured using a handheld ASD spectrometer: Asphalt
Thesis of Xiaoxiao Du, University of Missouri-Columbia).
5 /44
RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
Nonlinear spectral unmixing
Nonlinear unmixing to account for general deviations from the LMM
I Deformation of the classical data simplex
PCA-based representation of the real Villelongue image pixels.
6 /44
RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
Outline
Kernel-based mixing models
Residual Component AnalysisBayesian modelResults
Generalised RCABayesian modelResults
Conclusion
7 /44
RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
RCA-based model
Outline
Kernel-based mixing models
Residual Component AnalysisBayesian modelResults
Generalised RCABayesian modelResults
Conclusion
8 /44
RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
RCA-based model
Kernel-based mixing models
Motivations
I Various sources of nonlinear effects
I Several possible parametric models to be considered (modelselection,. . . )
I Kernel-based models general and flexible tools to handle differentnonlinearities
I Often rely on few hyperparameters
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RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
RCA-based model
Kernel-based mixing models
Two main model classes
I Discriminative models (RBF1, Kernel-FCLS2,...)
an ≈ f(M,yn)
I Generative modelsyn ≈ f(M,an)
⇒ Generative models related to the actual acquisition process, moreintuitive
1Y. Altmann et al., “Non- linear unmixing of hyperspectral images using radial basis functions
and orthogonal least squares,” in Proc. IEEE IGARSS , Vancouver, Canada, 2011.2J. Broadwater et al., “Kernel fully constrained least squares abundance estimates,” in Proc.
IEEE IGARSS, Barcelona, Spain, 2007.
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RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
RCA-based model
Gaussian Processes for SU
Unsupervised unmixing problem
yn ≈ f(M,an)
I Blind nonlinear source separation problem: Estimation of M,anand f(·) unknown
I Challenging problem (even if f(·) is known)
Idea: Modeling f(·) by a Gaussian Process (GP)
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RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
RCA-based model
Gaussian Processes for SU
First Idea
I Consider a set of N pixels observed at L spectral bands gatheredin Y
I Abundance matrix: A
I Vectorized observation matrix Y: (NL× 1 vector)
Y: ∼ N (µM,A,KM,A)
I KM,A models correlations between pixels (through A) andspectral bands (through M)
⇒ Computationally intractable without particular structure for KM,A
Solution: Consideration of correlated pixels or bands only
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RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
RCA-based model
Gaussian Processes for SU
Second Idea: GP-LVM
I L spectral bands assumed to be i.i.d.
y:,` ∼ N (µA,KA) , ∀`
I M not explicit in the model
I Choice for KA: Gaussian Kernel3, Polynomial Kernel4
I Kernels possibly too general: regularization required (LLE,ISOMAP,. . . )
I More promising results obtained with the polynomial kernel(more suited for bilinear mixtures)
3Y. Altmann et al.,“Nonlinear unmixing of hyperspectral images using Gaussian processes,” in
Proc. ICASSP, Kyoto, Japan, 2012.4Y. Altmann et al., “Nonlinear spectral unmixing of hyperspectral images using Gaussian
processes,” IEEE Trans. Signal Processing, May 2013.
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RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
RCA-based model
Gaussian Processes for SU
Conclusion on GP-based unsupervised unmixing
I GPs flexible enough to handle correlations between pixels andbetween spectral bands
I Interesting results although the problem is highly ill-posed
I Model complexity possible prohibitive for large data sets
I Results highly dependent on the kernel choice (withoutadditional regularization)
I Limited to one mixing model per image...
⇒ GPs more adapted for more constrained problems, e.g., supervisedunmixing
14 /44
RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
RCA-based model
Outline
Kernel-based mixing models
Residual Component AnalysisBayesian modelResults
Generalised RCABayesian modelResults
Conclusion
15 /44
RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
RCA-based model
Residual Component Analysis (RCA)
RCA-based model
yn = Man + φn + en, n = 1, . . . , N
I φn: additive nonlinear effects in the nth pixel
I en ∼ N (0,Σ0) with Σ0 diagonal
Contraints
ar,n ≥ 0, ∀r, n and
(R∑r=1
ar,n = 1 ∀n
)
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RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
RCA-based model
Residual Component Analysis (RCA)
On the sum-to-one constraint
I Widely acknowledged for linear SU
I Accurate when illumination changes occur?
I Abundance interpretation in nonlinear mixture?I Relief: projected areas?I Multilayer scene: visible surfaces, volume?
⇒ Both cases investigated: with/without enforced sum-to-one
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RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
RCA-based model
Residual Component Analysis (RCA)
General nonlinear model
yn = Man + φn + en, n = 1, . . . , N
I φn = 0→ LMM
I φn =∑Ri=1
∑Rj=1 βi,j,nmi �mj → polynomial models5 6 7
(� denotes the Hadamard (termwise) product)
I φn = ψ (M)→ K-Hype8
I ...
5J. M. P. Nascimento and J. M. Bioucas-Dias, “Nonlinear mixture model for hyperspectral
unmixing,” Proc. SPIE Image and Signal Processing for Remote Sensing XV, 2009.6A. Halimi et al., “Nonlinear unmixing of hyperspectral images using a generalized bilinear
model,” IEEE Trans. Geoscience and Remote Sensing, vol. 49, no. 11, pp. 4153-4162, Nov. 2011.7Y. Altmann et al., “Supervised nonlinear spectral unmixing using a post-nonlinear mixing model
for hyperspectral imagery,” IEEE Trans. Image Process., 2012.8J. Chen et al., “Nonlinear unmixing of hyperspectral data based on a
linear-mixture/nonlinear-fluctuation model,” IEEE Trans. Signal Process., 2013.
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RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
RCA-based model
Bayesian model
Modeling nonlinearities
Nonlinearity-based image partition
I K classes {Ck}k=0,...,(K−1) corresponding to K levels ofnonlinearity
I zn: class label of the nth pixel
I yn ∈ Ck ⇔ zn = k
Label priorPotts Markov model for z = [z1, . . . , zN ]T
f(z) =1
G(β)exp[β
∑Nn=1
∑n′∈V(n) δ(zn−zn′ )]
I Spatial dependencies between nonlinearity levels
I β > 0: fixed granularity parameter
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RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
RCA-based model
Bayesian model
Modeling nonlinearities
Linearly mixed pixels (k = 0)
f(φn|zn = 0) =
L∏`=1
δ(φ`,n)
Nonlinearly mixed pixels (k 6= 0)
φn|M, zn = k, s2k ∼ N
(0L, s
2kKM
)I [KM]i,j =
(mTi,:mj,:
)2, i, j ∈ {1, . . . , L}
I s2k: controls the nonlinearity level of the kth class→ included in the Bayesian model: f(s2
k) non-informativeinverse-gamma distribution
I Tractable marginalization of φn
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RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
RCA-based model
Bayesian model
Bayesian model
Other model elements
I Likelihood: Gaussian distribution (zero-mean coloured noise)
I Other model parameters/hyperparameters: classical priordistributions
Joint posterior distributionAfter marginalization of Φ
f(θ|Y,M) ∝ f(Y|M,θ)f(θ)
with θ ={A, z,σ2, s2
}⇒ MCMC method (Metropolis-within-Gibbs) to sample from theposterior distribution...
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RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
RCA-based model
Results
Real data
Data set
I 180× 250 pixels image composed of R = 5 endmembers (L = 160)
I Endmembers extracted from the data using VCA
I K = 5 levels of nonlinearity, β = 1.6
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RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection
RCA-based model
Results
Real data
Estimated Abundances (with sum-to-one constraint)
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RCA of Hyperspectral Images for Joint Nonlinear Unmixing and Nonlinearity Detection