Spectral mixture analysis Linear mixing model and beyond Nicolas - - PowerPoint PPT Presentation

Spectral mixture analysis

Linear mixing model and beyond Nicolas Dobigeon

University of Toulouse, IRIT/INP-ENSEEIHT Institut Universitaire de France (IUF) http://www.enseeiht.fr/˜dobigeon nicolas.dobigeon@enseeiht.fr

Follow-up of the Dagstuhl Seminar 17411, October 2017

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 1 / 77

Introduction

Hyperspectral Imagery Hyperspectral Images

◮ same scene observed at different wavelengths,

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 2 / 77

Introduction

Hyperspectral Imagery Hyperspectral Images

◮ same scene observed at different wavelengths,

Hyperspectral Cube

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 2 / 77

Introduction

Hyperspectral Imagery Hyperspectral Images

◮ same scene observed at different wavelengths, ◮ pixel represented by a vector of hundreds of measurements.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 3 / 77

Introduction

Hyperspectral Imagery Hyperspectral Images

◮ same scene observed at different wavelengths, ◮ pixel represented by a vector of hundreds of measurements.

Hyperspectral Cube

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 3 / 77

Introduction

Multi-band image enhancement Overcoming the spatial vs. spectral resolution trade-off Panchromatic images (PAN)

◮ no spectral resolution (only 1 band), ◮ very high spatial resolution (∼ 10cm).

Multispectral images (MS)

◮ low spectral resolution (∼ 10 bands), ◮ high spatial resolution (∼ 1m).

Hyperspectral images (HS)

◮ high spectral resolution (∼ 100 bands), ◮ low spatial resolution (∼ 10m).

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 4 / 77

Introduction

Multi-band image enhancement Overcoming the spatial vs. spectral resolution trade-off

Spot HS (20m) Quickbird MS (4m) Ikonos PAN (1m)

Reference: ENSEEIHT lecture by Mathieu Fauvel.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 5 / 77

Introduction

Multi-band image enhancement Overcoming the spatial vs. spectral resolution trade-off

Reference: ENSEEIHT lecture by Mathieu Fauvel.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 6 / 77

Introduction

Multi-band image enhancement Overcoming the spatial vs. spectral resolution trade-off

Reference: ENSEEIHT lecture by Mathieu Fauvel.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 7 / 77

Introduction

Multi-band image enhancement Overcoming the spatial vs. spectral resolution trade-off

One solution: spectral mixture analysis

(aka spectral unmixing) ... how to characterize the spatial distribution

• f the elementary components in the image.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 8 / 77

Introduction

Spectral Mixture Analysis One illustrative example (remote sensing) AVIRIS data

◮ Image: 50 × 50 pixels (Moffett field), L = 224 bands, ◮ 3 materials: vegetation, water, soil.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 9 / 77

Introduction

Spectral Mixture Analysis One illustrative example (remote sensing) AVIRIS data

◮ Image: 50 × 50 pixels (Moffett field), L = 224 bands, ◮ 3 materials: vegetation, water, soil.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 10 / 77

Introduction

Spectral Mixture Analysis Given a mixing model: yp ≈ gθ (ap, m1, . . . , mR)

◮ L = 825 (0.4µm → 2.5µm), ◮ R = 3:

◮ green grass

(solid line),

◮ galvanized steel metal

(dashed line),

◮ bare red brick

(dotted line),

◮ ap = [0.3, 0.6, 0.1]T, ◮ SNR ≈ 20dB.

Problem: inverting gθ (·) Estimation of ap under positivity and additivity constraints and m1, . . . , mR under positivity constraints.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 11 / 77

Introduction

Spectral Mixture Analysis Given a mixing model: yp ≈ gθ (ap, m1, . . . , mR)

◮ L = 825 (0.4µm → 2.5µm), ◮ R = 3:

◮ green grass

(solid line),

◮ galvanized steel metal

(dashed line),

◮ bare red brick

(dotted line),

◮ ap = [0.3, 0.6, 0.1]T, ◮ SNR ≈ 20dB.

Problem: inverting gθ (·) Estimation of ap under positivity and additivity constraints and m1, . . . , mR under positivity constraints.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 11 / 77

Introduction

Spectral Mixture Analysis Given a mixing model: yp ≈ gθ (ap, m1, . . . , mR)

◮ Supervised case: m1, . . . , mR are known,

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 12 / 77

Introduction

Spectral Mixture Analysis Given a mixing model: yp ≈ gθ (ap, m1, . . . , mR)

◮ Supervised case: m1, . . . , mR are known, ◮ Semi-supervised case: m1, . . . , mR are partially unknown

(R unknown, the mr belong to a spectral library fixed beforehand),

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 12 / 77

Introduction

Spectral Mixture Analysis Given a mixing model: yp ≈ gθ (ap, m1, . . . , mR)

◮ Supervised case: m1, . . . , mR are known, ◮ Semi-supervised case: m1, . . . , mR are partially unknown

(R unknown, the mr belong to a spectral library fixed beforehand),

◮ Unsupervised case: m1, . . . , mR are unknown.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 12 / 77

Introduction

Spectral Mixture Analysis Within an unsupervised context (1) endmember extraction step i.e., estimation of the spectral components, (2) inversion i.e., abundance estimation, (1+2) joint estimation of the endmembers and abundances. Preprocessing Huge data volume ⇒ dimensional reduction algorithms

◮ Principal Component Analysis (PCA): projection into a lower

dimensional space spanned by directions of high magnitudes,

◮ Maximum Noise Fraction (MNF): projection maximizing the SNR, ◮ ...

Remark: optional step for some algorithms.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 13 / 77

Introduction

Spectral Mixture Analysis Within an unsupervised context (1) endmember extraction step i.e., estimation of the spectral components, (2) inversion i.e., abundance estimation, (1+2) joint estimation of the endmembers and abundances. Preprocessing Huge data volume ⇒ dimensional reduction algorithms

◮ Principal Component Analysis (PCA): projection into a lower

dimensional space spanned by directions of high magnitudes,

◮ Maximum Noise Fraction (MNF): projection maximizing the SNR, ◮ ...

Remark: optional step for some algorithms.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 13 / 77

Introduction

Outline Introduction Linear unmixing Endmember extraction Inversion Joint approaches Illustrative results Non-linear unmixing Intimate mixing models Bilinear mixing models Post-nonlinear mixing models Robust linear mixing models Illustrative results Linear or nonlinear unmixing? Conclusion

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 14 / 77

Linear unmixing

Outline Introduction Linear unmixing Endmember extraction Inversion Joint approaches Illustrative results Non-linear unmixing Intimate mixing models Bilinear mixing models Post-nonlinear mixing models Robust linear mixing models Illustrative results Linear or nonlinear unmixing? Conclusion

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 15 / 77

Linear unmixing

Linear Mixing Model (LMM) Linear Mixing Model (LMM): yp = R

r=1 mrar,p + np

Reference: IEEE Signal Proc. Magazine, Jan. 2002.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 16 / 77

Linear unmixing

Linear Mixing Model (LMM) Linear Mixing Model (LMM): yp = R

r=1 mrar,p + np

Main assumptions

◮ generally, additive (Gaussian) noise ◮ materials sitting side-by-side (as on a checkerboard)

no interactions between materials

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 17 / 77

Linear unmixing

Linear Mixing Model (LMM)

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 17 / 77

Linear unmixing

Linear Mixing Model (LMM) Linear Mixing Model (LMM): yp = R

r=1 mrar,p + np

Main assumptions

◮ generally, additive (Gaussian) noise ◮ materials sitting side-by-side (as on a checkerboard)

no interactions between materials

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 17 / 77

Linear unmixing

Linear Mixing Model (LMM) Linear Mixing Model (LMM): yp = R

r=1 mrar,p + np

Main assumptions

◮ generally, additive (Gaussian) noise ◮ materials sitting side-by-side (as on a checkerboard)

no interactions between materials

◮ no multiple scattering (e.g., due to relief) ◮ only the materials present in the considered pixel contribute

no contribution from materials in neighboring pixels

◮ a single spectral signature characterizes each individual material

no spectral variability

◮ generally, positivity and additivity (sum-to-one) constraints on ap

positivity: ar,p ≥ 0, r = 1, . . . , R sum-to-one: R

r=1 ar,p = 1

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 17 / 77

Linear unmixing

Representation in the hyperspectral space

m1 m2 m2

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 18 / 77

Linear unmixing

Representation in the signal subspace Hence, the interest of subspace learning methods as preprocessing steps (PCA, MNF , Hysime...)

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 19 / 77

Linear unmixing

Representation in the signal subspace Hence, the interest of subspace learning methods as preprocessing steps (PCA, MNF , Hysime...)

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 19 / 77

Linear unmixing Endmember extraction

Outline Introduction Linear unmixing Endmember extraction Inversion Joint approaches Illustrative results Non-linear unmixing Conclusion

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 20 / 77

Linear unmixing Endmember extraction

Endmember extraction (1) Exploiting convex geometry Searching for purest pixels (≈ simplex of max. volume inscribed in the data)

(a) Pixel Purity Index (PPI) (b) N-FINDR

• r successive projections onto orthogonal subspaces (VCA, ORASIS).

Searching for simplex of minimum volume inscribing the data “Minimum Volume Transform” (MVT) algorithms and variants.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 21 / 77

Linear unmixing Inversion

Outline Introduction Linear unmixing Endmember extraction Inversion Joint approaches Illustrative results Non-linear unmixing Conclusion

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 22 / 77

Linear unmixing Inversion

Inversion (2) Constrained inverse problem Constrained optimization Minimizing J(a) = y − Ma2 s.t. ar,p ≥ 0, ∀r = 1, . . . , R (ANC) R

r=1 ar = 1

(ASC) with M = [m1, . . . , mR].

◮ Fully Constrained Least Squares (FCLS) [Heinz et al., 2001],

Remarks

◮ In a semi-supervised context, the materials m1, . . . , mR belong to a

known library S = [s1, . . . , sK] (K ≫ R). The problem is written Minimizing J(a) = y − Sa2 s.t. ANC and ASC → complementary sparsity constraints on a.

◮ Other contextual constraints: spatial regularizations

i.e., to promote smooth or spatially coherent abundance maps

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 23 / 77

Linear unmixing Inversion

Inversion (2) Constrained inverse problem Constrained optimization Minimizing J(a) = y − Ma2 s.t. ar,p ≥ 0, ∀r = 1, . . . , R (ANC) R

r=1 ar = 1

(ASC) with M = [m1, . . . , mR].

◮ Fully Constrained Least Squares (FCLS) [Heinz et al., 2001],

Remarks

◮ In a semi-supervised context, the materials m1, . . . , mR belong to a

known library S = [s1, . . . , sK] (K ≫ R). The problem is written Minimizing J(a) = y − Sa2 s.t. ANC and ASC → complementary sparsity constraints on a.

◮ Other contextual constraints: spatial regularizations

i.e., to promote smooth or spatially coherent abundance maps

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 23 / 77

Linear unmixing Inversion

Inversion (2) Statistical inference problem Unknown parameter vector: θ =

• a, σ2

◮ a = [a1, . . . , aR]T: vector of the R abundance coefficients, ◮ σ2: noise variance,

Bayes paradigm: f(θ|y) ∝ f(y|θ)f(θ) with

◮ Likelihood: f(y|θ) (data-fitting term), ◮ Parameter prior distribution: f(θ) (penalization/regularization).

Choice of the (Bayesian) estimators

◮ maximizing f(θ|y) to reach the maximum a posteriori (MAP) estimator

• ptimization problem (see previously)

◮ computing the mean of f(θ|y) to derive the minimum mean square error

(MMSE) estimator integration problem: use of Markov chain Monte Carlo algorithms (powerful but computationally demanding)

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 24 / 77

Linear unmixing Inversion

Inversion (2) Statistical inference problem Unknown parameter vector: θ =

• a, σ2

◮ a = [a1, . . . , aR]T: vector of the R abundance coefficients, ◮ σ2: noise variance,

Bayes paradigm: f(θ|y) ∝ f(y|θ)f(θ) with

◮ Likelihood: f(y|θ) (data-fitting term), ◮ Parameter prior distribution: f(θ) (penalization/regularization).

Choice of the (Bayesian) estimators

◮ maximizing f(θ|y) to reach the maximum a posteriori (MAP) estimator

• ptimization problem (see previously)

◮ computing the mean of f(θ|y) to derive the minimum mean square error

(MMSE) estimator integration problem: use of Markov chain Monte Carlo algorithms (powerful but computationally demanding)

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 24 / 77

Linear unmixing Inversion

Inversion (2) Computing the Bayesian estimators Maximum a posteriori (MAP) estimator ˆ θMAP = arg max

θ

f (θ|Y) = arg max

θ

f (Y|θ) f (θ). Minimum mean square error (MMSE) estimator ˆ θMMSE = E [θ|Y] =

• θf (θ|Y) dθ

=

• θf (Y|θ) f (θ)dθ
• f (Y|θ) f (θ) dθ

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 25 / 77

Linear unmixing Joint approaches

Outline Introduction Linear unmixing Endmember extraction Inversion Joint approaches Illustrative results Non-linear unmixing Conclusion

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 26 / 77

Linear unmixing Joint approaches

Joint approaches (1+2) For a given pixel p observed in L spectral bands: yp = R

r=1 mrar,p + np

= Map + np Now, consider P pixels: Y = MA + N where Y = [y1, . . . , yP] , M = [m1, . . . , mR] , A = [a1, . . . , aP] , N = [n1, . . . , nP] . Factorize Y ≈ MA under positivity and additivity constraints on A and positivity constraints on M Spectral Mixture Analysis = (constrained) matrix factorization = (constrained) blind source separation

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 27 / 77

Linear unmixing Joint approaches

Joint approaches (1+2) For a given pixel p observed in L spectral bands: yp = R

r=1 mrar,p + np

= Map + np Now, consider P pixels: Y = MA + N where Y = [y1, . . . , yP] , M = [m1, . . . , mR] , A = [a1, . . . , aP] , N = [n1, . . . , nP] . Factorize Y ≈ MA under positivity and additivity constraints on A and positivity constraints on M Spectral Mixture Analysis = (constrained) matrix factorization = (constrained) blind source separation

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 27 / 77

Linear unmixing Joint approaches

Matrix factorization problem

Factorization Y ≈ MA formulated as the minimization problem min

M,A D(Y|MA) =

• p

D(yp|Map) =

• p,ℓ

d

• yℓ,p| [Map]ℓ
• where D(a|b) is a “distance measure”, e.g., D(a|b) = a − b2.

An ill-posed problem! If {M, A} is a solution,

• MP, P−1A
• is a solution1.

⇓ Additional constraints required!

1For all P invertible matrix.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 28 / 77

Linear unmixing Joint approaches

Matrix factorization problem

Factorization Y ≈ MA formulated as the minimization problem min

M,A D(Y|MA) =

• p

D(yp|Map) =

• p,ℓ

d

• yℓ,p| [Map]ℓ
• where D(a|b) is a “distance measure”, e.g., D(a|b) = a − b2.

An ill-posed problem! If {M, A} is a solution,

• MP, P−1A
• is a solution1.

⇓ Additional constraints required!

1For all P invertible matrix.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 28 / 77

Linear unmixing Joint approaches

Matrix factorization problem

Factorization Y ≈ MA formulated as the minimization problem min

M,A D(Y|MA) =

• p

D(yp|Map) =

• p,ℓ

d

• yℓ,p| [Map]ℓ
• where D(a|b) is a “distance measure”, e.g., D(a|b) = a − b2.

An ill-posed problem! If {M, A} is a solution,

• MP, P−1A
• is a solution1.

⇓ Additional constraints required!

1For all P invertible matrix.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 28 / 77

Linear unmixing Joint approaches

Matrix factorization strategies Y ≈ MA ⇔ YT ≈ ATMT

• 1. Principal Component Analysis(PCA)

◮ Searching for orthogonal “principal components” (PCs) mr, ◮ PCs = directions with maximal variance in the data, ◮ Generally used as a dimension reduction procedure.

• 2. Independent Component Analysis (ICA) (of YT)

◮ Maximizing the statistical independence between the sources mr, ◮ Several measures of independence ⇒ several algorithms.

• 3. Nonnegative Matrix Factorization (NMF)

◮ Searching for M et A with positive entries, ◮ Several measures of divergence d (·|·) ⇒ several algorithms.

• 4. (Fully Constrained) Spectral Mixture Analysis (SMA)

◮ Positivity constraints on mr ⇒ positive “sources” ◮ Positivity and sum-to-one constraints on ap

⇒ mixing coefficients = proportions/concentrations/probabilities.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 29 / 77

Linear unmixing Joint approaches

Matrix factorization strategies Y ≈ MA ⇔ YT ≈ ATMT

• 1. Principal Component Analysis(PCA)

◮ Searching for orthogonal “principal components” (PCs) mr, ◮ PCs = directions with maximal variance in the data, ◮ Generally used as a dimension reduction procedure.

• 2. Independent Component Analysis (ICA) (of YT)

◮ Maximizing the statistical independence between the sources mr, ◮ Several measures of independence ⇒ several algorithms.

• 3. Nonnegative Matrix Factorization (NMF)

◮ Searching for M et A with positive entries, ◮ Several measures of divergence d (·|·) ⇒ several algorithms.

• 4. (Fully Constrained) Spectral Mixture Analysis (SMA)

◮ Positivity constraints on mr ⇒ positive “sources” ◮ Positivity and sum-to-one constraints on ap

⇒ mixing coefficients = proportions/concentrations/probabilities.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 29 / 77

Linear unmixing Joint approaches

Matrix factorization strategies Y ≈ MA ⇔ YT ≈ ATMT

• 1. Principal Component Analysis(PCA)

◮ Searching for orthogonal “principal components” (PCs) mr, ◮ PCs = directions with maximal variance in the data, ◮ Generally used as a dimension reduction procedure.

• 2. Independent Component Analysis (ICA) (of YT)

◮ Maximizing the statistical independence between the sources mr, ◮ Several measures of independence ⇒ several algorithms.

• 3. Nonnegative Matrix Factorization (NMF)

◮ Searching for M et A with positive entries, ◮ Several measures of divergence d (·|·) ⇒ several algorithms.

• 4. (Fully Constrained) Spectral Mixture Analysis (SMA)

◮ Positivity constraints on mr ⇒ positive “sources” ◮ Positivity and sum-to-one constraints on ap

⇒ mixing coefficients = proportions/concentrations/probabilities.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 29 / 77

Linear unmixing Joint approaches

Matrix factorization strategies Y ≈ MA ⇔ YT ≈ ATMT

• 1. Principal Component Analysis(PCA)

◮ Searching for orthogonal “principal components” (PCs) mr, ◮ PCs = directions with maximal variance in the data, ◮ Generally used as a dimension reduction procedure.

• 2. Independent Component Analysis (ICA) (of YT)

◮ Maximizing the statistical independence between the sources mr, ◮ Several measures of independence ⇒ several algorithms.

• 3. Nonnegative Matrix Factorization (NMF)

◮ Searching for M et A with positive entries, ◮ Several measures of divergence d (·|·) ⇒ several algorithms.

• 4. (Fully Constrained) Spectral Mixture Analysis (SMA)

◮ Positivity constraints on mr ⇒ positive “sources” ◮ Positivity and sum-to-one constraints on ap

⇒ mixing coefficients = proportions/concentrations/probabilities.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 29 / 77

Linear unmixing Joint approaches

Geometrical formulation of SMA

SMA = looking for a simplex enclosing the data

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 30 / 77

Linear unmixing Joint approaches

Geometrical formulation of SMA

In practice: non-unique solution + trade-off noise vs. constraints...

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 31 / 77

Linear unmixing Joint approaches

Matrix factorization strategies NMF-based algorithms

◮ Assumption: positivity of the endmember spectra and the abundances, ◮ Various counterparts, to handle additional constraints

sum-to-one, minimum volume, (collaborative) sparsity... Example: MVC-NMF2 Bayesian estimation

◮ Choice of prior ensuring various constraints, ◮ Allows nuisance parameters to be jointly estimated

noise parameters, hyperparameters, classification maps

◮ Difficult statistical estimation → MCMC algorithm.

Example: BLU3 (efficient implementation strategies available4)

2Miao and Qi, IEEE TGRS,2007 3Dobigeon et al., IEEE TSP, 2009. 4Schmidt et al., IEEE TGRS, 2010.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 32 / 77

Linear unmixing Joint approaches

Spectral Mixture Analysis... ... and applications    y1,p . . . yL,p    =   m1 . . . mR      a1,p . . . aR,p    p = 1, . . . , P Multi-band imaging

◮ remote sensing5, planetology6, EELS7,... ◮ evolution parameter: pixel index in the image.

Spectrochemical analysis

◮ Raman8, NIR9,... ◮ evolution parameter: time, temperature,...

Bioinformatics

◮ gene expression analysis10 ◮ evolution parameter: time, subject, treatment,...

5Keshava et al., IEEE SP Mag., 2002. 6Schmidt et al., IEEE TGRS, 2010. 7de la Pe˜

na et al., Ultramicroscopy, 2011. 8Dobigeon et al., SP , 2009. 9Moussaoui, IEEE TSP , 2006. 10Huang et al., PLoS Genetics, 2011.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 33 / 77

Linear unmixing Joint approaches

Spectral Mixture Analysis... ... and applications    y1,p . . . yL,p    =   m1 . . . mR      a1,p . . . aR,p    p = 1, . . . , P Multi-band imaging

◮ remote sensing5, planetology6, EELS7,... ◮ evolution parameter: pixel index in the image.

Spectrochemical analysis

◮ Raman8, NIR9,... ◮ evolution parameter: time, temperature,...

Bioinformatics

◮ gene expression analysis10 ◮ evolution parameter: time, subject, treatment,...

5Keshava et al., IEEE SP Mag., 2002. 6Schmidt et al., IEEE TGRS, 2010. 7de la Pe˜

na et al., Ultramicroscopy, 2011. 8Dobigeon et al., SP , 2009. 9Moussaoui, IEEE TSP , 2006. 10Huang et al., PLoS Genetics, 2011.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 33 / 77

Linear unmixing Joint approaches

Spectral Mixture Analysis... ... and applications    y1,p . . . yL,p    =   m1 . . . mR      a1,p . . . aR,p    p = 1, . . . , P Multi-band imaging

◮ remote sensing5, planetology6, EELS7,... ◮ evolution parameter: pixel index in the image.

Spectrochemical analysis

◮ Raman8, NIR9,... ◮ evolution parameter: time, temperature,...

Bioinformatics

◮ gene expression analysis10 ◮ evolution parameter: time, subject, treatment,...

5Keshava et al., IEEE SP Mag., 2002. 6Schmidt et al., IEEE TGRS, 2010. 7de la Pe˜

na et al., Ultramicroscopy, 2011. 8Dobigeon et al., SP , 2009. 9Moussaoui, IEEE TSP , 2006. 10Huang et al., PLoS Genetics, 2011.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 33 / 77

Linear unmixing Joint approaches

Spectral Mixture Analysis... ... and applications    y1,p . . . yL,p    =   m1 . . . mR      a1,p . . . aR,p    p = 1, . . . , P Multi-band imaging

◮ remote sensing5, planetology6, EELS7,... ◮ evolution parameter: pixel index in the image.

Spectrochemical analysis

◮ Raman8, NIR9,... ◮ evolution parameter: time, temperature,...

Bioinformatics

◮ gene expression analysis10 ◮ evolution parameter: time, subject, treatment,...

5Keshava et al., IEEE SP Mag., 2002. 6Schmidt et al., IEEE TGRS, 2010. 7de la Pe˜

na et al., Ultramicroscopy, 2011. 8Dobigeon et al., SP , 2009. 9Moussaoui, IEEE TSP , 2006. 10Huang et al., PLoS Genetics, 2011.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 33 / 77

Linear unmixing Illustrative results

Outline Introduction Linear unmixing Endmember extraction Inversion Joint approaches Illustrative results Non-linear unmixing Conclusion

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 34 / 77

Linear unmixing Illustrative results

Experimental results: AVIRIS data [Dobigeon et al., IEEE Trans. SP, 2010] Simulation parameters

◮ Image: 50 × 50 pixels (Moffett field), L = 224 bands.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 35 / 77

Linear unmixing Illustrative results

Experimental results: AVIRIS data [Dobigeon et al., IEEE Trans. SP, 2010] Simulation parameters

◮ Image: 50 × 50 pixels (Moffett field), L = 224 bands.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 35 / 77

Linear unmixing Illustrative results

Experimental results: EELS data [Dobigeon et al., Ultramiscroscopy, 2012] Simulation parameters

◮ Image: 64 × 64 pixels, L = 1340 energy channels.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 36 / 77

Linear unmixing Illustrative results

Experimental results: EELS data [Dobigeon et al., Ultramiscroscopy, 2012] Simulation parameters

◮ Image: 64 × 64 pixels, L = 1340 energy channels.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 36 / 77

Linear unmixing Illustrative results

Experimental results: EELS data [Dobigeon et al., Ultramiscroscopy, 2012] Simulation parameters

◮ Image: 64 × 64 pixels, L = 1340 energy channels.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 36 / 77

Linear unmixing Illustrative results

Experimental results: Mars data [Schmidt et al., IEEE Trans. GRS, 2010] OMEGA data

◮ L = 184 spectral bands, ≈ 300 × 400 pixels, ◮ 3 materials: CO2, dust, H20.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 37 / 77

Linear unmixing Illustrative results

Experimental results: Mars data [Schmidt et al., IEEE Trans. GRS, 2010] OMEGA data

◮ L = 184 spectral bands, ≈ 300 × 400 pixels, ◮ 3 materials: CO2, dust, H20.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 37 / 77

Linear unmixing Illustrative results

Experimental results: spectrometrics [Dobigeon et al., Signal Processing, 2009] Chemical mixtures

◮ L = 1000 spectral bands, 10 observed mixtures over time, ◮ 3 materials: chemical species (reactant A, intermediate C, product D).

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 38 / 77

Linear unmixing Illustrative results

Experimental results: spectrometrics [Dobigeon et al., Signal Processing, 2009] Chemical mixtures

◮ L = 1000 spectral bands, 10 observed mixtures over time, ◮ 3 materials: chemical species (reactant A, intermediate C, product D).

(a) Reactant A (b) Intermediate C (c) Product D (d) Mixing coefficients

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 38 / 77

Linear unmixing Illustrative results

Experimental results: genetics [Huang et al, PLoS Genetics, 2011] Data · 267 blood samples collected for 17 voluntary subjects along time after Influenza A (H3N2) inoculation. · expressions of more than 12000 genes in each sample.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 39 / 77

Linear unmixing Illustrative results

Experimental results: genetics [Huang et al, PLoS Genetics, 2011] Data · 267 blood samples collected for 17 voluntary subjects along time after Influenza A (H3N2) inoculation. · expressions of more than 12000 genes in each sample. Results · identification of a group of genes involved in the occurrence of symptoms (inflammatory factor)

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 39 / 77

Linear unmixing Illustrative results

Experimental results: genetics [Huang et al, PLoS Genetics, 2011] Data · 267 blood samples collected for 17 voluntary subjects along time after Influenza A (H3N2) inoculation. · expressions of more than 12000 genes in each sample. Results · identification of a group of genes involved in the occurrence of symptoms (inflammatory factor)

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 39 / 77

Linear unmixing Illustrative results

Experimental results: genetics [Huang et al, PLoS Genetics, 2011] Data · 267 blood samples collected for 17 voluntary subjects along time after Influenza A (H3N2) inoculation. · expressions of more than 12000 genes in each sample. Results · identification of a group of genes involved in the occurrence of symptoms (inflammatory factor)

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 39 / 77

Non-linear unmixing

Outline Introduction Linear unmixing Endmember extraction Inversion Joint approaches Illustrative results Non-linear unmixing Intimate mixing models Bilinear mixing models Post-nonlinear mixing models Robust linear mixing models Illustrative results Linear or nonlinear unmixing? Conclusion

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 40 / 77

Non-linear unmixing

Non-linear mixing models Non-linear Mixing Model: yp = gθ (ap, m1, . . . , mR) + np

Reference: IEEE Signal Proc. Magazine, Jan. 2002.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 41 / 77

Non-linear unmixing

Non-linear mixing models

Numerous nonlinear models proposed in the literature

Physics-based models

◮ mainly derived from the radiative transfer theory, ◮ require prior knowledge about external parameters related to the studied

scene (leaf index area, geometry or illumination incidence,...)

◮ not developed for unmixing purpose

unmixing algorithms not easily available. Bilinear, polynomial and robust models

◮ mainly derived from intuitive considerations, ◮ do not require prior knowledge about external parameters, ◮ developed for unmixing purpose

(various) unmixing algorithms available.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 42 / 77

Non-linear unmixing

Non-linear mixing models

Numerous nonlinear models proposed in the literature

Physics-based models

◮ mainly derived from the radiative transfer theory, ◮ require prior knowledge about external parameters related to the studied

scene (leaf index area, geometry or illumination incidence,...)

◮ not developed for unmixing purpose

unmixing algorithms not easily available. Bilinear, polynomial and robust models

◮ mainly derived from intuitive considerations, ◮ do not require prior knowledge about external parameters, ◮ developed for unmixing purpose

(various) unmixing algorithms available.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 42 / 77

Non-linear unmixing Intimate mixing models

Outline Introduction Linear unmixing Non-linear unmixing Intimate mixing models Bilinear mixing models Post-nonlinear mixing models Robust linear mixing models Illustrative results Linear or nonlinear unmixing? Conclusion

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 43 / 77

Non-linear unmixing Intimate mixing models

Intimate mixing models Non-linear Mixing Model: yp = gθ (ap, m1, . . . , mR) + np

Reference: IEEE Signal Proc. Magazine, Sept. 2014.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 44 / 77

Non-linear unmixing Intimate mixing models

Intimate mixtures Assumptions

◮ interactions occur at a small scales ◮ abundances associated with the relative mass fractions of the materials ◮ popular models: those promoted by Hapke (1993)

based on meaningful/interpretable parameters with physical significance Intimate unmixing: not easy... Strongly depend on external parameters related to the acquisition e.g., geometry of the scene

◮ use of neural-networks (NN) to learn the nonlinear function

→ training data needed

◮ kernel-based unmixing

implicitly relies on the Hapke model

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 45 / 77

Non-linear unmixing Bilinear mixing models

Outline Introduction Linear unmixing Non-linear unmixing Intimate mixing models Bilinear mixing models Post-nonlinear mixing models Robust linear mixing models Illustrative results Linear or nonlinear unmixing? Conclusion

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 46 / 77

Non-linear unmixing Bilinear mixing models

Bilinear mixing models Non-linear Mixing Model: yp = gθ (ap, m1, . . . , mR) + np

Reference: IEEE Signal Proc. Magazine, Sept. 2014. ◮ Possible interactions between the components of the scene, ◮ Nonlinear terms included into the mixing model.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 47 / 77

Non-linear unmixing Bilinear mixing models

Bilinear mixing models

yp =

R

• r=1

ar,pmr

• linear term

+

R−1

• i=1

R

• j=i+1

bi,j,pmi ⊙ mj

• nonlinear term

+np where bi,j,p tunes the amount of nonlinearity resulting from the interactions between mi and mj.

◮ several bilinear models proposed in the literature ◮ mainly differ by the definition (and related constraints) of bi,j,p

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 48 / 77

Non-linear unmixing Bilinear mixing models

Bilinear mixing models

yp =

R

• r=1

ar,pmr

• linear term

+

R−1

• i=1

R

• j=i+1

bi,j,pmi ⊙ mj

• nonlinear term

+np where bi,j,p tunes the amount of nonlinearity resulting from the interactions between mi and mj.

◮ several bilinear models proposed in the literature ◮ mainly differ by the definition (and related constraints) of bi,j,p

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 48 / 77

Non-linear unmixing Bilinear mixing models

Nascimento’s model (NM) Definition6,7 y =

R

• r=1

armr

• linear term

+

R−1

• i=1

R

• j=i+1

βi,jmi ⊙ mj

• interaction term

+ n where ⊙ is the termwise (Hadamard) product: mi ⊙ mj =    m1,im1,j . . . mL,imL,j    . Constraints positivity: ar ≥ 0, βi,j ≥ 0, r, i = 1, . . . , R j = i + 1, . . . R sum-to-one: R

r=1 ar + R−1 i=1

R

j=i+1 βi,j = 1 6Nascimento and Bioucas-Dias, in Proc. SPIE, Sept. 2009. 7Somers et al., Remote Sens. Env., 2009.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 49 / 77

Non-linear unmixing Bilinear mixing models

Nascimento’s model (NM) y =

R

• r=1

armr +

R−1

• i=1

R

• j=i+1

βi,j mi ⊙ mj

• new spectrum

+ n Properties

◮ defined by the abundances ar,p and the nonlinearity coefficients βi,j ◮ interactions between mi and mj adjusted/quantified with βi,j ◮ if ∀i, j, βi,j = 0, reduces to the LMM. ◮ if ∃i, j, βi,j = 0, abundances do not satisfy

r ar,p = 1

◮ can be seen as an LMM with R∗ = 1

2R(R + 1) correlated endmembers

y =

R∗

• r=1

a∗

r m∗ r + n

a∗

r = ar

and m∗

r = mr

r = 1, . . . , R, a∗

r = βi,j

and m∗

r = mi ⊙ mj

R + 1 ≤ r ≤ R∗.

◮ Unmixing procedures similar to procedures assuming the LMM.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 50 / 77

Non-linear unmixing Bilinear mixing models

Fan’s model (FM) Definition8 y =

R

• r=1

armr

• linear term

+

R−1

• i=1

R

• j=i+1

aiajmi ⊙ mj

• interaction term

+ n Constraints positivity: ar ≥ 0, r = 1, . . . , R sum-to-one: R

r=1 ar = 1

◮ Interaction coefficient in NM fixed as bi,j = aiaj, ◮ Absence of material mi ⇒ ai = 0 ⇒ no nonlinearity terms

mi ⊙ m·,

◮ Specific unmixing strategies must be designed.

8Fan et al., Int. J. Remote Sensing, June 2009.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 51 / 77

Non-linear unmixing Bilinear mixing models

Generalized bilinear model (GBM) Definition9 y =

R

• r=1

armr

• linear term

+

R−1

• i=1

R

• j=i+1

γi,jaiajmi ⊙ mj

• interaction term

+ n Constraints

◮ Abundance coefficients

positivity: ar ≥ 0, r = 1, . . . , R sum-to-one: R

r=1 ar = 1

◮ Nonlinearity coefficients

0 ≤ γi,j ≤ 1, i = 1, . . . , R − 1, j = i + 1, . . . , R

9Halimi et al., IEEE Trans. Geosci. and Remote Sensing, 2011.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 52 / 77

Non-linear unmixing Bilinear mixing models

Generalized bilinear model (GBM) y =

R

• r=1

armr +

R−1

• i=1

R

• j=i+1

γi,jaiajmi ⊙ mj + n

◮ Absence of material mi ⇒ ai = 0 ⇒ no nonlinearity terms mi ⊙ m·, ◮ Generalization of existing mixing models

◮ γi,j = 0, ∀i, j, → LMM ◮ γi,j = 1, ∀i, j, → FM

◮ Interactions between mi and mj adjusted/quantified with γi,j. ◮ Specific unmixing strategies must be designed.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 53 / 77

Non-linear unmixing Bilinear mixing models

Bilinear models: summary Nascimento’s model (NM) y =

R

• r=1

armr +

R−1

• i=1

R

• j=i+1

βi,jmi ⊙ mj + n Fan’s model (FM) y =

R

• r=1

armr +

R−1

• i=1

R

• j=i+1

aiajmi ⊙ mj + n Generalized bilinear model (GBM) y =

R

• r=1

armr +

R−1

• i=1

R

• j=i+1

γi,jaiajmi ⊙ mj + n

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 54 / 77

Non-linear unmixing Bilinear mixing models

Bilinear models: summary

Clusters of observations generated according to the LMM, the NM, the FM and the GBM (blue) and the corresponding endmembers (red).

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 55 / 77

Non-linear unmixing Bilinear mixing models

An extended bilinear model: linear-quadratic mixing model Definition10 y =

R

• r=1

armr

• linear term

+

R

• i=1

R

• j=i

βi,jmi ⊙ mj

• interaction term

+ n Constraints positivity: ar ≥ 0, r = 1, . . . , R βi,j ≥ 0, ∀i, j sum-to-one: R

r=1 ar = 1

◮ Presence of quadratic terms mi ⊙ mi, ◮ Specific unmixing strategies must be designed.

10Meganem et al., IEEE Trans. Geosci. and Remote Sensing, 2013.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 56 / 77

Non-linear unmixing Post-nonlinear mixing models

Outline Introduction Linear unmixing Non-linear unmixing Intimate mixing models Bilinear mixing models Post-nonlinear mixing models Robust linear mixing models Illustrative results Linear or nonlinear unmixing? Conclusion

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 57 / 77

Non-linear unmixing Post-nonlinear mixing models

Post-nonlinear mixing models (PNMM) Definition11 y = g R

• r=1

armr

• + n

Constraints positivity: ar ≥ 0, r = 1, . . . , R sum-to-one: R

r=1 ar = 1

◮ General class of nonlinear models studied for source

separation12,13,

◮ g: unknown linear/nonlinear application.

11Altmann et al., IEEE Trans. Image Process., 2013. 12Jutten and Karhunen, in Proc. 4th ICA Workshop, Apr. 2003. 13Babaie-Zadeh et al., in Proc. 3rd ICA Workshop, Jun. 2001.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 58 / 77

Non-linear unmixing Post-nonlinear mixing models

Post-nonlinear mixing models (PNMM) A particular instance: polynomial PNMM

g : RL → RL s →

• s1 + bs2

1, . . . , sL + bs2 L

T

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 59 / 77

Non-linear unmixing Post-nonlinear mixing models

Post-nonlinear mixing models (PNMM) A particular instance: polynomial PNMM

g : RL → RL s →

• s1 + bs2

1, . . . , sL + bs2 L

T

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 59 / 77

Non-linear unmixing Post-nonlinear mixing models

Post-nonlinear mixing models (PNMM) A particular instance: polynomial PNMM Mathematical formulation y = Ma + b (Ma) ⊙ (Ma) + n = Ma + b

R

• i=1

R

• j=1

aiajmi ⊙ mj + n Remarks

◮ derived from a second-order Taylor expansion of the nonlinearity

function: g(s) ≈ s + bs2 + o(s2).

◮ defined by the abundances ar and the nonlinearity parameter b. ◮ nonlinear interactions adjusted/quantified with a unique parameter b. ◮ b = 0 ⇒ LMM, b = 0 ⇒ nonlinear model. ◮ when b = 0, contains bilinear and quadratic terms mi ⊙ mj similar to

• ther nonlinear models (bilinear and quadratic-linear).

◮ specific unmixing algorithms must be designed.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 60 / 77

Non-linear unmixing Robust linear mixing models

Outline Introduction Linear unmixing Non-linear unmixing Intimate mixing models Bilinear mixing models Post-nonlinear mixing models Robust linear mixing models Illustrative results Linear or nonlinear unmixing? Conclusion

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 61 / 77

Non-linear unmixing Robust linear mixing models

Robust linear mixing models (rLMM) Mathematical formulation y = Ma + r + n Remarks

◮ r represents any potential deviation from the classical LMM

nonlinearity, outliers, spectral variability

◮ r not explicitly parametrized by the endmember spectra or their

respective proportions

◮ r = 0 ⇒ LMM, r = 0 ⇒ nonlinear model. ◮ the rLMM’s differ by the additional constraints/regularizations

e.g., spatial sparsity14, spatial-spectral sparsity15, endmember-driven16...

• 14C. Fevotte and N. Dobigeon, IEEE TIP, 2015.
• 15G. Newstadt et al., in IEEE Proc. SSP, 2014.
• 16Y. Altmann, N. Dobigeon et al., IEEE TIP, 2014.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 62 / 77

Non-linear unmixing Illustrative results

Outline Introduction Linear unmixing Non-linear unmixing Intimate mixing models Bilinear mixing models Post-nonlinear mixing models Robust linear mixing models Illustrative results Linear or nonlinear unmixing? Conclusion

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 63 / 77

Non-linear unmixing Illustrative results

Spectral unmixing of the “Moffett Field” image with the generalized bilinear model Linear model GBM

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 64 / 77

Non-linear unmixing Illustrative results

Spectral unmixing of the “Moffett Field” image with the robust LMM Abundance maps Residual maps

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 65 / 77

Non-linear unmixing Linear or nonlinear unmixing?

Outline Introduction Linear unmixing Non-linear unmixing Intimate mixing models Bilinear mixing models Post-nonlinear mixing models Robust linear mixing models Illustrative results Linear or nonlinear unmixing? Conclusion

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 66 / 77

Non-linear unmixing Linear or nonlinear unmixing?

To be or not to be... (non-)linear?

◮ PPNMM and rLMM sufficiently flexible to handle linear and nonlinear

mixtures... ... at a price of a higher computational cost

◮ LMM-based endmember extraction algorithms can still be valid in case

• f weak nonlinearities

requirement: pure pixels are still endmembers (i.e., extrema)

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 67 / 77

Non-linear unmixing Linear or nonlinear unmixing?

To be or not to be... (non-)linear? Figure: Linearly mixed pixels.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 67 / 77

Non-linear unmixing Linear or nonlinear unmixing?

To be or not to be... (non-)linear? Figure: Nonlinearly mixed pixels.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 67 / 77

Non-linear unmixing Linear or nonlinear unmixing?

To be or not to be... (non-)linear?

◮ PPNMM and rLMM sufficiently flexible to handle linear and nonlinear

mixtures... ... at a price of a higher computational cost

◮ LMM-based endmember extraction algorithms can still be valid in case

• f weak nonlinearities

requirement: pure pixels are still endmembers (i.e., extrema)

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 67 / 77

Non-linear unmixing Linear or nonlinear unmixing?

To be or not to be... (non-)linear?

◮ PPNMM and rLMM sufficiently flexible to handle linear and nonlinear

mixtures... ... at a price of a higher computational cost

◮ LMM-based endmember extraction algorithms can still be valid in case

• f weak nonlinearities

requirement: pure pixels are still endmembers (i.e., extrema)

◮ some nonlinear detectors can be implemented before unmixing

detecting nonlinearly vs. linearly mixed pixels → yes, we can!

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 67 / 77

Non-linear unmixing Linear or nonlinear unmixing?

To be or not to be... (non-)linear? Figure: Pixels detected as linear (red crosses) and nonlinear (blue dots) for the four sub-images S1 (LMM), S2 (FM), S3 (GBM) and S4 (PPNMM).

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 67 / 77

Conclusion

Outline

Introduction Linear unmixing Endmember extraction Inversion Joint approaches Illustrative results Non-linear unmixing Intimate mixing models Bilinear mixing models Post-nonlinear mixing models Robust linear mixing models Illustrative results Linear or nonlinear unmixing? Conclusion

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 68 / 77

Conclusion

Conclusion Linear unmixing

◮ usually, convenient first approximation ◮ plenty of algorithms, some of them very fast

→ try first! Nonlinear unmixing Are you sure? yes: good luck... no: detect first! A parametric model for the nonlinearity? yes: specific inversion methods should be designed no: approximating the nonlinearities with generic models PNMM, rLMM, RCA, RUSAL, NUSAL...

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 69 / 77

Conclusion

Conclusion Linear unmixing

◮ usually, convenient first approximation ◮ plenty of algorithms, some of them very fast

→ try first! Nonlinear unmixing Are you sure? yes: good luck... no: detect first! A parametric model for the nonlinearity? yes: specific inversion methods should be designed no: approximating the nonlinearities with generic models PNMM, rLMM, RCA, RUSAL, NUSAL...

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 69 / 77

Conclusion

Conclusion Linear unmixing

◮ usually, convenient first approximation ◮ plenty of algorithms, some of them very fast

→ try first! Nonlinear unmixing Are you sure? yes: good luck... no: detect first! A parametric model for the nonlinearity? yes: specific inversion methods should be designed no: approximating the nonlinearities with generic models PNMM, rLMM, RCA, RUSAL, NUSAL...

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 69 / 77

Conclusion

Not discussed today: spectral variability

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 70 / 77

Conclusion

Not discussed today: spectral variability Endmember as sets or bundles

◮ Automated Endmember Bundles (AEB)17

Endmember as distributions

◮ Normal compositional model (NCM)18,19 ◮ Beta compositional model (BCM)20

Endmember as perturbed signatures

◮ Perturbed linear mixing model (PLMM)21 ◮ Also extended to handle variability in multi-temporal/date images22

• 17B. Somers et al., IEEE JSTARS, 2012.
• 18O. Eches et al., IEEE TIP, June 2010.
• 19A. Zare et al., IEEE TGRS, March 2013.
• 20X. Du et al., IEEE JSTARS, 2014.

21P

.-A. Thouvenin et al., IEEE TSP, Feb. 2016

22P

.-A. Thouvenin et al., IEEE TIP, Sept. 2016.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 71 / 77

Conclusion

Not discussed today: multiple image fusion Overcoming the spatial vs. spectral resolution trade-off Panchromatic images (PAN)

◮ no spectral resolution (only 1 band), ◮ very high spatial resolution (∼ 10cm).

Multispectral images (MS)

◮ low spectral resolution (∼ 10 bands), ◮ high spatial resolution (∼ 1m).

Hyperspectral images (HS)

◮ high spectral resolution (∼ 100 bands), ◮ low spatial resolution (∼ 10m).

Objective of the fusion process: get the best of both resolutions.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 72 / 77

Conclusion

Not discussed today: multiple image fusion Overcoming the spatial vs. spectral resolution trade-off Panchromatic images (PAN)

◮ no spectral resolution (only 1 band), ◮ very high spatial resolution (∼ 10cm).

Multispectral images (MS)

◮ low spectral resolution (∼ 10 bands), ◮ high spatial resolution (∼ 1m).

Hyperspectral images (HS)

◮ high spectral resolution (∼ 100 bands), ◮ low spatial resolution (∼ 10m).

Objective of the fusion process: get the best of both resolutions.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 72 / 77

Conclusion

Not discussed today: multiple image fusion [Q. Wei et al., IEEE TIP, Nov. 2015] Figure: (a) Hyperspectral Image (size: 99 × 46 × 224, res.: 20m × 20m) (b) Multispectral Image

(size: 396 × 184 × 4 res.: 5m × 5m) (c) Target (size: 396 × 184 × 224 res.: 5m × 5m)

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 73 / 77

Conclusion

More inputs? 1 J. M. Bioucas-Dias, A. Plaza, N. Dobigeon et al., ”Hyperspectral unmixing overview: geometrical, statistical, and sparse regression-based approaches,” IEEE J. Sel. Topics Applied Earth Observations and Remote Sensing, vol. 5, no. 2, pp. 354-379, April 2012. 2 N. Dobigeon et al., ”Linear and nonlinear unmixing in hyperspectral imaging,” in Resolving spectral mixtures - With application from ultrafast time-resolved spectroscopy to superresolution imaging, C. Ruckebusch,

• Ed. Oxford, U.K.: Elsevier, vol. 30, Data Handling in Science and

Technology Series, 2016, pp. 185-224. 3 N. Dobigeon et al., ”Nonlinear unmixing of hyperspectral images: models and algorithms,” IEEE Signal Processing Magazine, vol. 31, no. 1, pp. 82-94, Jan. 2014. 5 http://dobigeon.perso.enseeiht.fr/applications/app_ hyper.html

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 74 / 77

Conclusion

Acknowledgments Past and current M.Sc./Ph.D. students · Olivier Eches · C´ ecile Bazot · Yoann Altmann · Abderrahim Halimi · Pierre-Antoine Thouvenin Main collaborators · Nathalie Brun, Univ. Paris Sud, France · Jos´ e C. M. Bermudez, Univ. Fed. Santa Cantarina, Brazil · C´ edric F´ evotte, Univ. of Toulouse, France · Alfred O. Hero, Univ. of Michigan, USA · Steve McLaughlin, Heriot-Watt University, Edinburgh, UK · Sa¨ ıd Moussaoui, Ecole Centrale Nantes, France · Fr´ ed´ eric Schmidt, Univ. Paris Sud, France · Jean-Yves Tourneret, Univ. of Toulouse, France

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 75 / 77

Conclusion

(Subjective) bibliography I

• N. Keshava and J. F

. Mustard, “Spectral unmixing,” IEEE Signal Process. Mag.,

• vol. 19, no. 1, pp. 44–57, Jan. 2002.
• J. M. Bioucas-Dias, A. Plaza, N. Dobigeon, M. Parente, Q. Du, P

. Gader, and

• J. Chanussot, “Hyperspectral unmixing overview: Geometrical, statistical, and

sparse regression-based approaches,” IEEE J. Sel. Topics Appl. Earth Observations Remote Sens., vol. 5, no. 2, pp. 354–379, April 2012. W.-K. Ma, J. M. Bioucas-Dias, P . Gader, T.-H. Chan, N. Gillis, A. Plaza,

• A. Ambikapathi, and C.-Y. Chi, “Signal processing perspective on hyperspectral

unmixing,” IEEE Signal Process. Mag., 2013.

• M. E. Winter, “N-FINDR: an algorithm for fast autonomous spectral end-member

determination in hyperspectral data,” in Proc. SPIE Imaging Spectrometry V,

• M. R. Descour and S. S. Shen, Eds., vol. 3753, no. 1.

SPIE, 1999, pp. 266–275.

• F. Chaudhry, C.-C. Wu, W. Liu, C.-I Chang, and A. Plaza, “Pixel purity

index-based algorithms for endmember extraction from hyperspectral imagery,” in Recent Advances in Hyperspectral Signal and Image Processing, C.-I Chang, Ed. Trivandrum, Kerala, India: Research Signpost, 2006, ch. 2.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 76 / 77

Conclusion

(Subjective) bibliography II

• J. M. Bioucas-Dias and J. M. P

. Nascimento, “Hyperspectral subspace identification,” IEEE Trans. Geosci. Remote Sens., vol. 46, no. 8, pp. 2435–2445,

• Aug. 2008.
• D. C. Heinz and C. Chang, “Fully constrained least-squares linear spectral

mixture analysis method for material quantification in hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens., vol. 29, no. 3, pp. 529–545, March 2001.

• N. Dobigeon, J.-Y. Tourneret, and C.-I Chang, “Semi-supervised linear spectral

unmixing using a hierarchical Bayesian model for hyperspectral imagery,” IEEE

• Trans. Signal Process., vol. 56, no. 7, pp. 2684–2695, July 2008.
• J. M. Bioucas-Dias, “A variable splitting augmented Lagrangian approach to linear

spectral unmixing,” in Proc. IEEE GRSS Workshop Hyperspectral Image SIgnal Process.: Evolution in Remote Sens. (WHISPERS), Grenoble, France, Aug. 2009.

• L. Miao and H. Qi, “Endmember extraction from highly mixed data using minimum

volume constrained nonnegative matrix factorization,” IEEE Trans. Geosci. Remote Sens., vol. 45, no. 3, pp. 765–776, March 2007.

• N. Dobigeon, S. Moussaoui, M. Coulon, J.-Y. Tourneret, and A. O. Hero, “Joint

Bayesian endmember extraction and linear unmixing for hyperspectral imagery,” IEEE Trans. Signal Process., vol. 57, no. 11, pp. 4355–4368, Nov. 2009.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 77 / 77

Conclusion

(Subjective) bibliography III

• J. M. Bioucas-Dias and M. A. T. Figueiredo, “Alternating direction algorithms for

constrained sparse regression: Application to hyperspectral unmixing,” in Proc. IEEE GRSS Workshop Hyperspectral Image SIgnal Process.: Evolution in Remote Sens. (WHISPERS), vol. 1, 2010, pp. 1–4. M.-D. Iordache, J. M. Bioucas-Dias, and A. Plaza, “Sparse unmixing of hyperspectral data,” IEEE Trans. Geosci. Remote Sens., vol. 49, no. 6, pp. 2014–2039, June 2011.

• N. Dobigeon, S. Moussaoui, J.-Y. Tourneret, and C. Carteret, “Bayesian

separation of spectral sources under non-negativity and full additivity constraints,” Signal Process., vol. 89, no. 12, pp. 2657–2669, Dec. 2009.

• F. Schmidt, M. Guiheneuf, S. Moussaoui, E. Tr´

eguier, A. Schmidt, and

• N. Dobigeon, “Implementation strategies for hyperspectral unmixing using

Bayesian source separation,” IEEE Trans. Geosci. Remote Sens., vol. 48, no. 11,

• pp. 4003–4013, Nov. 2010.
• N. Dobigeon and N. Brun, “Spectral mixture analysis of EELS spectrum-images,”

Ultramicroscopy, vol. 120, pp. 25–34, Sept. 2012.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 78 / 77

Conclusion

(Subjective) bibliography IV

• C. Bazot, N. Dobigeon, J.-Y. Tourneret, A. K. Zaas, G. S. Ginsburg, and A. O.

Hero, “Unsupervised Bayesian linear unmixing of gene expression microarrays,” BMC Bioinformatics, vol. 14, no. 99, March 2013.

• O. Eches, N. Dobigeon, and J. Y. Tourneret, “Enhancing hyperspectral image

unmixing with spatial correlations,” IEEE Trans. Geosci. Remote Sens., vol. 49,

• no. 11, pp. 4239–4247, Nov. 2011.
• N. Dobigeon, J.-Y. Tourneret, C. Richard, J. C. M. Bermudez, S. McLaughlin, and
• A. O. Hero, “Nonlinear unmixing of hyperspectral images: Models and algorithms,”

IEEE Signal Process. Mag., vol. 31, no. 1, pp. 89–94, Jan. 2014.

• R. Heylen, M. Parente, and P

. Gader, “A review of nonlinear hyperspectral unmixing methods,” IEEE J. Sel. Topics Appl. Earth Observations Remote Sens.,

• vol. 7, no. 6, pp. 1844–1868, June 2014.
• N. Dobigeon, L. Tits, B. Somers, Y. Altmann, and P

. Coppin, “A comparison of nonlinear mixing models for vegetated areas using simulated and real hyperspectral data,” IEEE J. Sel. Topics Appl. Earth Observations Remote Sens.,

• vol. 7, no. 6, pp. 1869–1878, June 2014.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 79 / 77

Conclusion

(Subjective) bibliography V

• B. Somers, K. Cools, S. Delalieux, J. Stuckens, D. V. der Zande, W. W.

Verstraeten, and P . Coppin, “Nonlinear hyperspectral mixture analysis for tree cover estimates in orchards,” Remote Sens. Environment, vol. 113, pp. 1183–1193, Feb. 2009.

• J. M. P

. Nascimento and J. M. Bioucas-Dias, “Nonlinear mixture model for hyperspectral unmixing,” in Proc. SPIE Image and Signal Processing for Remote Sensing XV, L. Bruzzone, C. Notarnicola, and F. Posa, Eds., vol. 7477, no. 1. SPIE, 2009, p. 74770I.

• W. Fan, B. Hu, J. Miller, and M. Li, “Comparative study between a new nonlinear

model and common linear model for analysing laboratory simulated-forest hyperspectral data,” IJRS, vol. 30, no. 11, pp. 2951–2962, June 2009.

• A. Halimi, Y. Altmann, N. Dobigeon, and J.-Y. Tourneret, “Nonlinear unmixing of

hyperspectral images using a generalized bilinear model,” IEEE Trans. Geosci. Remote Sens., vol. 49, no. 11, pp. 4153–4162, Nov. 2011.

• Y. Altmann, A. Halimi, N. Dobigeon, and J.-Y. Tourneret, “Supervised nonlinear

spectral unmixing using a post-nonlinear mixing model for hyperspectral imagery,” IEEE Trans. Image Process., vol. 21, no. 6, pp. 3017–3025, June 2012.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 80 / 77

Conclusion

(Subjective) bibliography VI

• Y. Altmann, N. Dobigeon, S. McLaughlin, and J.-Y. Tourneret, “Nonlinear spectral

unmixing of hyperspectral images using Gaussian processes,” IEEE Trans. Signal Process., vol. 61, no. 10, pp. 2442–2453, May 2013.

• Y. Altmann, N. Dobigeon, and J.-Y. Tourneret, “Unsupervised post-nonlinear

unmixing of hyperspectral images using a Hamiltonian Monte Carlo algorithm,” IEEE Trans. Image Process., vol. 23, no. 6, pp. 2663–2675, June 2014.

• M. Babaie-Zadeh, C. Jutten, and K. Nayebi, “Separating convolutive post

non-linear mixtures,” in Proc. Int. Conf. Independent Component Analysis and Blind Source Separation (ICA), San Diego, 2001, pp. 138–143.

• C. Jutten and J. Karhunen, “Advances in nonlinear blind source separation,” in
• Proc. Int. Conf. Independent Component Analysis and Blind Source Separation

(ICA), Nara, Japan, April 2003, pp. 245–256.

• Y. Altmann, N. Dobigeon, and J.-Y. Tourneret, “Nonlinearity detection in

hyperspectral images using a polynomial post-nonlinear mixing model,” IEEE

• Trans. Image Process., vol. 22, no. 4, pp. 1267–1276, April 2013.
• I. Meganem, P

. D´ eliot, X. Briottet, Y. Deville, and S. Hosseini, “Linear-quadratic mixing model for reflectances in urban environments,” IEEE Trans. Geosci. Remote Sens., vol. 52, no. 1, pp. 544–558, Jan. 2014.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 81 / 77

Conclusion

(Subjective) bibliography VII

• C. F´

evotte and N. Dobigeon, “Nonlinear hyperspectral unmixing with robust nonnegative matrix factorization,” IEEE Trans. Image Process., vol. 24, no. 12, pp. 4810–4819, Dec. 2015.

• Y. Altmann, M. Pereyra, and J. M. Bioucas-Dias, “Collaborative sparse regression

using spatially correlated supports – Application to hyperspectral unmixing,” IEEE

• Trans. Image Process., vol. 24, no. 12, pp. 5800–5811, Dec. 2015.
• Y. Altmann, M. Pereyra, and S. McLaughlin, “Bayesian nonlinear hyperspectral

unmixing with spatial residual component analysis,” IEEE Trans. Comput. Imag.,

• vol. 1, no. 3, pp. 174–185, Sept. 2015.
• G. E. Newstadt, A. O. Hero, and J. Simmon, “Robust spectral unmixing for

anomaly detection,” in Proc. IEEE-SP Workshop Stat. and Signal Process. (SSP), Gold Coast, Australia, July 2014.

• Y. Altmann, S. McLaughlin, and A. O. Hero, “Robust linear spectral unmixing

using anomaly detection,” IEEE Trans. Comput. Imag., vol. 1, no. 2, pp. 74–85, June 2015.

• A. Halimi, J. Bioucas-Dias, N. Dobigeon, G. S. Buller, and S. McLaughlin, “Fast

hyperspectral unmixing in presence of nonlinearity or mismodelling effects,” IEEE

• Trans. Computational Imaging, vol. 3, no. 2, pp. 146–159, April 2017.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 82 / 77

Conclusion

(Subjective) bibliography VIII

• A. Zare and K. Ho, “Endmember variability in hyperspectral analysis: Addressing

spectral variability during spectral unmixing,” IEEE Signal Process. Mag., vol. 31,

• no. 1, pp. 95–104, Jan. 2014.
• O. Eches, N. Dobigeon, C. Mailhes, and J.-Y. Tourneret, “Bayesian estimation of

linear mixtures using the normal compositional model. Application to hyperspectral imagery,” IEEE Trans. Image Process., vol. 19, no. 6, pp. 1403–1413, June 2010.

• B. Somers, G. P

. Asner, L. Tits, and P . Coppin, “Endmember variability in spectral mixture analysis: A review,” Remote Sens. Environment, vol. 115, no. 7, pp. 1603–1616, July 2011.

• X. Du, A. Zare, P

. Gader, and D. Dranishnikov, “Spatial and spectral unmixing using the beta compositional model,” IEEE J. Sel. Topics Appl. Earth Observations Remote Sens., vol. 7, no. 6, pp. 1994–2003, June 2014. ——, “Spectral unmixing cluster validity index for multiple sets of endmembers,” IEEE J. Sel. Topics Appl. Earth Observations Remote Sens., vol. 7, no. 6, pp. 1994–2003, 2014.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 83 / 77

Conclusion

(Subjective) bibliography IX

• A. Halimi, N. Dobigeon, and J.-Y. Tourneret, “Unsupervised unmixing of

hyperspectral images accounting for endmember variability,” IEEE Trans. Image Process., vol. 24, no. 12, pp. 4904–4917, Dec. 2015. P .-A. Thouvenin, N. Dobigeon, and J.-Y. Tourneret, “Hyperspectral unmixing with spectral variability using a perturbed linear mixing model,” IEEE Trans. Signal Process., vol. 64, no. 2, pp. 525–538, Feb. 2016. ——, “Online unmixing of multitemporal hyperspectral images accounting for spectral variability,” IEEE Trans. Image Process., vol. 25, no. 9, pp. 3979–3990,

• Sept. 2016.
• L. Loncan, L. B. Almeida, J. M. Bioucas-Dias, X. Briottet, J. Chanussot,
• N. Dobigeon, S. Fabre, W. Liao, G. Licciardi, M. Simoes, J.-Y. Tourneret,
• M. Veganzones, G. Vivone, Q. Wei, and N. Yokoya, “Hyperspectral

pansharpening: a review,” IEEE Geosci. Remote Sens. Mag., vol. 3, no. 3, pp. 27–46, Sept. 2015.

• Y. Zhang, S. De Backer, and P

. Scheunders, “Noise-resistant wavelet-based Bayesian fusion of multispectral and hyperspectral images,” IEEE Trans. Geosci. Remote Sens., vol. 47, no. 11, pp. 3834–3843, Nov. 2009.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 84 / 77

Conclusion

(Subjective) bibliography X

• O. Bern´

e, A. Tielens, P . Pilleri, and C. Joblin, “Non-negative matrix factorization pansharpening of hyperspectral data: An application to mid-infrared astronomy,” in Proc. IEEE GRSS Workshop Hyperspectral Image SIgnal Process.: Evolution in Remote Sens. (WHISPERS), 2010, pp. 1–4.

• N. Yokoya, T. Yairi, and A. Iwasaki, “Coupled nonnegative matrix factorization

unmixing for hyperspectral and multispectral data fusion,” IEEE Trans. Geosci. Remote Sens., vol. 50, no. 2, pp. 528–537, Feb. 2012.

• M. Sim˜
• es, J. Bioucas Dias, L. Almeida, and J. Chanussot, “A convex formulation

for hyperspectral image superresolution via subspace-based regularization,” IEEE

• Trans. Geosci. Remote Sens., vol. 6, no. 53, pp. 3373–3388, June 2015.
• Q. Wei, N. Dobigeon, and J.-Y. Tourneret, “Bayesian fusion of multi-band images,”

IEEE J. Sel. Topics Signal Process., vol. 9, no. 6, pp. 1117–1127, Sept. 2015.

• Q. Wei, J. M. Bioucas-Dias, N. Dobigeon, and J.-Y. Tourneret, “Hyperspectral and

multispectral image fusion based on a sparse representation,” IEEE Trans.

• Geosci. Remote Sens., vol. 53, no. 17, pp. 3658–3668, July 2015.
• Q. Wei, N. Dobigeon, and J.-Y. Tourneret, “Fast fusion of multi-band images

based on solving a Sylvester equation,” IEEE Trans. Image Process., vol. 24,

• no. 11, pp. 4109–4121, Nov. 2015.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 85 / 77

Conclusion

(Subjective) bibliography XI

• Q. Wei, J. M. Bioucas-Dias, N. Dobigeon, J.-Y. Tourneret, M. Chen, and
• S. Godsill, “Multi-band image fusion based on spectral unmixing,” IEEE Trans.
• Geosci. Remote Sens., vol. 54, no. 12, pp. 7236–7249, Dec. 2016.
• V. Ferraris, N. Dobigeon, Q. Wei, and M. Chabert, “Detecting changes between
• ptical images of different spatial and spectral resolutions: a fusion-based

approach,” IEEE Trans. Geosci. Remote Sens., 2017. [Online]. Available: http://arxiv.org/abs/1609.06074/ ——, “Robust fusion of multi-band images with different spatial and spectral resolutions for change detection,” IEEE Trans. Comput. Imag., vol. 3, no. 2, pp. 175–186, April 2017.

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 86 / 77

Conclusion

Spectral mixture analysis

Linear mixing model and beyond Nicolas Dobigeon

University of Toulouse, IRIT/INP-ENSEEIHT Institut Universitaire de France (IUF) http://www.enseeiht.fr/˜dobigeon nicolas.dobigeon@enseeiht.fr

Follow-up of the Dagstuhl Seminar 17411, October 2017

Nicolas Dobigeon Spectral Mixture Analysis – Linear mixing model and beyond 87 / 77