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Joint Bayesian endmember extraction and linear unmixing for hyperspectral imagery

Nicolas Dobigeon, Said Moussaoui, Martial Coulon, Jean-Yves Tourneret, Alfred O. Hero

arXiv:0903.3060v1physics.data-an

TL;DR

The paper addresses the difficult problem of direct parameter estimation in endmember extraction. It presents a Bayesian model with an MCMC algorithm, using Gibbs sampling to handle posterior complexity, and reports performance improvements on simulated and real data compared with other literature algorithms.

  • Problem

    Direct parameter estimation remains a difficult problem in prior endmember extraction approaches.

  • Method

    The paper presents a Bayesian model and an MCMC algorithm that uses Gibbs sampling to generate samples from the complex posterior distribution.

  • Results

    Performance improvements are shown on simulated and real data, with the proposed Bayesian method compared favorably with other algorithms from the literature.

  • Takeaways & Limitations

    The proposed approach provides an unsupervised Bayesian method for endmember extraction and abundance estimation.

Abstract

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This paper studies a fully Bayesian algorithm for endmember extraction and abundance estimation for hyperspectral imagery. Each pixel of the hyperspectral image is decomposed as a linear combination of pure endmember spectra following the linear mixing model. The estimation of the unknown endmember spectra is conducted in a unified manner by generating the posterior distribution of abundances and endmember parameters under a hierarchical Bayesian model. This model assumes conjugate prior distributions for these parameters, accounts for non-negativity and full-additivity constraints, and exploits the fact that the endmember proportions lie on a lower dimensional simplex. A Gibbs sampler is proposed to overcome the complexity of evaluating the resulting posterior distribution. This sampler generates samples distributed according to the posterior distribution and estimates the unknown parameters using these generated samples. The accuracy of the joint Bayesian estimator is illustrated by simulations conducted on synthetic and real AVIRIS images.

2 University of Michigan, Department of EECS, Ann Arbor, MI 48109-2122, USA

Spectral unmixing decomposes hyperspectral pixels into material spectra and their proportions, traditionally separating endmember extraction from abundance estimation. The paper instead proposes jointly estimating both quantities in one Bayesian framework while enforcing the mixing-model constraints.

  • Spectral unmixing decomposes each pixel spectrum into a collection of material spectra, called endmembers, and their corresponding abundances.
  • Traditional linear spectral mixture analysis uses separate endmember extraction and inversion steps to identify materials and estimate their proportions.
  • Many extraction methods locate endmembers as simplex vertices, but some require pure pixels while others depend on initialization or user-selected parameters.
  • The paper jointly estimates endmember spectra and abundances in a single step by casting linear spectral mixture analysis as a blind source separation problem.
  • The Bayesian model enforces non-negativity for endmembers and abundances together with full additivity for abundance coefficients.
  • A hierarchical Bayesian formulation reduces hyperparameter dependence, while Gibbs sampling approximates posterior-based estimators and enables confidence-interval calculation.

II. LINEAR MIXING MODEL AND PROBLEM STATEMENT

The linear mixing model represents each observed hyperspectral pixel as a noisy linear combination of material spectra, with physically motivated abundance and endmember constraints. The paper estimates the abundance and endmember matrices jointly from these observations.

  • Model: Each pixel spectrum is modeled as a linear combination of R material spectra corrupted by additive Gaussian noise.The noise is assumed i.i.d., zero-mean Gaussian with covariance matrix σ2I_L.
  • Noise model: The model assumes independent noise sequences with common variance across spectral bands.The paper notes that the model does not account for noise correlation, although simulations assess robustness to violations of the i.i.d. assumption.
  • Constraints: Abundance vectors satisfy non-negativity and full-additivity constraints, while endmember spectra satisfy non-negativity.These constraints encode the physical interpretation of abundances as material fractions and spectra as nonnegative signals.
  • Problem statement: The paper estimates the abundance matrix A and endmember matrix M from noisy observations Y under these constraints.The matrices collect the observed pixels, endmember spectra, abundances, and noise sequences.

YH, HTA

Even with non-negativity and full-additivity constraints, linear spectral mixing can remain non-unique. The paper illustrates this ambiguity using multiple admissible solutions and formulates the corresponding Gaussian likelihood.

  • Non-uniqueness: The unconstrained blind source separation problem is ill-posed because multiple endmember-abundance pairs can explain the same observations.The paper describes non-uniqueness under transformations involving unitary matrices H.
  • Non-uniqueness: Full-additivity constraints partially address scale indeterminacy by imposing unit ℓ1-norm constraints on abundance vectors.These constraints avoid more complex strategies for directly estimating scale.
  • Non-uniqueness: Uniqueness of the constrained pair {M, A} is not systematically ensured.The paper illustrates the issue with 50 admissible solutions for two endmembers mixed into 2500 pixels.
  • Likelihood: Under the linear mixing model and Gaussian noise assumptions, each observed pixel has a conditionally Gaussian distribution.Independence across pixels yields the likelihood for all observations.

B. Prior model for the endmember spectra

The endmember prior model reduces the spectra to a PCA-derived subspace and estimates their projected vertices with constrained truncated Gaussian priors. This exploits the simplex geometry of mixed pixels.

  • Dimensionality reduction: In the noise-free model, mixed spectra form a convex polytope whose vertices are the endmember spectra.The polytope has dimension R−1 and can be represented in a lower-dimensional subspace without loss of information.
  • Dimensionality reduction: For K = R−1, the projected endmembers form a simplex whose vertices are estimated using a Bayesian approach.This connects the lower-dimensional representation to standard endmember extraction methods.
  • Dimensionality reduction: The paper identifies a K-dimensional subspace using the K principal axes obtained by PCA on the observations.The PCA step can be replaced by alternatives such as the MNF transform.
  • Endmember prior: Non-negativity of reconstructed spectra is enforced by restricting each projected endmember to a set defined by L inequalities.Membership in this set is equivalent to non-negativity of the corresponding reconstructed spectrum.
  • Endmember prior: The projected endmembers receive conjugate multivariate Gaussian priors truncated to their admissible sets.Prior means can use projected spectra identified by an external endmember extraction algorithm, while large variances represent limited prior knowledge.
  • Endmember prior: The prior variances encode confidence in previously identified projected spectra, with large values used when no additional knowledge is available.The projected endmember matrix is assembled from the individual projected spectra.

C. Abundance prior

The abundance prior is uniform over the simplex, reflecting limited prior knowledge while retaining the physical abundance constraints. In this blind source separation setting, it also favors smaller-volume projected polytopes posteriori.

  • Prior specification: Each partial abundance vector c_p has a uniform prior distribution on the simplex.The reparameterization expresses the full abundance vector through partial abundances and is suited to the proposed Gibbs sampler.
  • Prior specification: The uniform simplex prior is equivalent to a Dirichlet distribution D(1, ..., 1) for the full abundance vector.The partial abundance vectors are assumed statistically independent.
  • Interpretation: The uniform prior represents a lack of a priori knowledge about the abundance vector.It also imposes a strong constraint on the size of the simplex to be recovered.
  • Interpretation: Among two equiprobable blind source separation solutions, the uniform prior favors the solution whose projected polytope has the smallest volume.The paper identifies this as a posteriori behavior of the chosen abundance prior.
  • Noise variance prior: A conjugate prior is also selected for the noise variance σ2, with ν fixed and γ treated as a random adjustable hyperparameter.The supplied passage states that ν is fixed to 2.

E. Prior distribution for hyperparameter γ

The model uses a non-informative Jeffreys’ prior for γ and integrates this hyperparameter out of the joint posterior. Because Bayesian estimators remain intractable analytically, a Gibbs sampler generates posterior samples for estimation.

  • A non-informative Jeffreys’ prior for γ reflects the lack of knowledge about this hyperparameter.
  • The hierarchical structure integrates γ out of the joint distribution f(θ, γ|Y), producing a posterior for θ = {C, T, σ2}.
  • Deriving MMSE or MAP Bayesian estimators from the posterior remains intractable, motivating Markov chain Monte Carlo methods.
  • A Gibbs sampler generates samples from the conditional posterior distributions of each parameter and thereby approximates Bayesian estimators.

B. Sampling from f (T|C, σ2, Y)

The sampler handles the projected endmember parameters T through conditional truncated Gaussian distributions. Sampling is difficult because of truncation, so components are sampled conditionally, while σ2 uses an inverse Gamma update.

  • The conditional posterior of each projected endmember parameter is a truncated Gaussian over the feasible subset T_r.
  • Truncation makes direct sampling from the multivariate distribution difficult, motivating component-wise conditional sampling.
  • The sampler generates each t_k,r conditionally on the remaining components using a conditional mean and variance.
  • Samples from the two-sided truncated Gaussian are obtained with an algorithm described in prior work.
  • The noise variance σ2 is sampled from an inverse Gamma conditional posterior using a Gamma variate generator.
  • For initialization, projected spectra identified by a standard endmember extraction algorithm such as N-FINDR provide prior mean estimates.

A. Endmember spectrum estimation

The proposed method jointly estimates endmember spectra from noisy mixtures using Bayesian posterior sampling. In synthetic experiments, it outperforms N-FINDR and VCA, although those methods have lower computational complexity.

  • The Gibbs sampler jointly estimates endmember spectra and abundances under the Bayesian model.
  • The proposed endmember spectra clearly outperform N-FINDR and VCA in the spectral comparisons.
  • N-FINDR and VCA perform more poorly because they assume pure pixels, whereas the simulated noisy data contain no pure pixels.
  • MSE and SAD comparisons show that the proposed method performs significantly better than the competing extraction algorithms.
  • The proposed approach has greater computational complexity than VCA and N-FINDR, whose complexity is lower.
  • N-FINDR’s combinatorial complexity increases drastically with the number of pixels and endmembers.

B. Abundance estimation

Abundance estimates are formed by averaging posterior samples and agree well with simulated abundance maps. The Bayesian method also supplies uncertainty information and remains effective across noise levels, endmember counts, and dimensionality-reduction procedures.

  • B. Abundance estimation: MMSE abundance estimates are approximated by empirical averages over the post-burn-in Gibbs samples.
  • B. Abundance estimation: The estimated abundance maps are in good agreement with the simulated maps.
  • B. Abundance estimation: Posterior distributions provide confidence intervals for unknown parameters, unlike N-FINDR and VCA.
  • B. Abundance estimation: Global abundance MSE comparisons confirm the accuracy of the proposed Bayesian estimation method against N-FINDR and VCA coupled with FCLS.
  • B. Abundance estimation: The Bayesian procedure remains effective for SNRdB = 5dB and 25dB and for mixtures containing R = 3 or R = 5 endmembers.
  • B. Abundance estimation: MNF and HySime can replace PCA for dimensionality reduction, and the method is reported as robust to violations of the i.i.d. noise assumption.

VI. REAL DATA

The proposed Bayesian LSMA algorithm is applied to a real AVIRIS image from Moffett Field, recovering three endmembers and corresponding abundance maps consistent with the coastal scene. The paper also summarizes the joint Bayesian model, constrained priors, subspace projection, and Gibbs-sampling estimation procedure.

  • Real-data experiment: The algorithm recovered R = 3 endmembers represented across L = 189 spectral bands after removing water-absorption bands.The corresponding abundance maps were also computed for the real image.
  • Real-data experiment: The recovered endmembers correspond to vegetation, water, and soil, while their abundance maps agree with previously reported results.The spectra and maps are shown in Fig. 8.
  • Bayesian model: The model assigns priors that enforce non-negativity and sum-to-one constraints on abundance vectors under the linear mixing model.These constraints encode physically admissible endmember proportions and spectra.
  • Bayesian estimation: Gibbs sampling generates posterior samples for jointly estimating abundances and endmember spectra, using a lower-dimensional subspace identified by dimension reduction.The proposed approach estimates endmember projections rather than directly estimating signatures in the observation space.
  • Evaluation and scope: Synthetic and real-image simulations illustrate the accuracy of the proposed Bayesian method compared with algorithms from the literature.The conclusion also identifies Bayesian treatment of the intrinsic projection-subspace dimension as an open research question.

APPENDIX

The appendix analyzes uniform abundance priors within the constrained Bayesian blind source separation framework. It relates admissible solutions to solution polytopes and shows how posterior preference can select the smallest admissible polytope under suitable assumptions.

  • Uniform abundance priors: The appendix examines uniform priors for abundance vectors within the constrained Bayesian model.The analysis concerns the prior treatment of abundances rather than the real-data experiment.
  • Posterior selection: When competing admissible polytopes are a priori equiprobable, the posterior can favor the solution associated with the smallest polytope.The appendix notes that sufficiently flat priors on the relevant variances make the equiprobability assumption less restrictive.
  • Admissible solutions: Admissible blind source separation solutions are represented by convex polytopes whose vertices are the endmember columns.Observed pixels belong to the corresponding solution polytope in the absence of noise.
  • Reduced-space geometry: Projection onto the reduced subspace produces a simplex whose vertices are the projected endmembers.The projected simplex is denoted ST^(i) in the appendix notation.
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