Source-linked AI summary
Enhancing hyperspectral image unmixing with spatial correlations
Olivier Eches, Nicolas Dobigeon, Jean-Yves Tourneret
TL;DR
The paper addresses the limited use of spatial correlations in hyperspectral unmixing. It introduces a hierarchical Bayesian model with class-dependent abundance statistics and spatially dependent labels, using MCMC for joint inference. Results on simulated data and real AVIRIS images are reported as very promising, with synthetic experiments showing better performance than non-spatial constrained algorithms.
Problem
Existing unmixing strategies operate pixel by pixel and therefore do not exploit possible spatial correlations between image pixels.
Method
A hierarchical Bayesian model uses class-dependent abundance statistics, Potts-Markov spatial labels, and MCMC to infer abundances, labels, and hyperparameters.
Results
Synthetic-data experiments show the spatial algorithm performs better than the non-spatial constrained Bayesian and FCLS algorithms, while results on simulated and real AVIRIS images are very promising.
Takeaways & Limitations
Spatial correlations can be incorporated into Bayesian hyperspectral unmixing while jointly solving classification and abundance estimation.
Abstract
from arXiv · showhide
This paper describes a new algorithm for hyperspectral image unmixing. Most of the unmixing algorithms proposed in the literature do not take into account the possible spatial correlations between the pixels. In this work, a Bayesian model is introduced to exploit these correlations. The image to be unmixed is assumed to be partitioned into regions (or classes) where the statistical properties of the abundance coefficients are homogeneous. A Markov random field is then proposed to model the spatial dependency of the pixels within any class. Conditionally upon a given class, each pixel is modeled by using the classical linear mixing model with additive white Gaussian noise. This strategy is investigated the well known linear mixing model. For this model, the posterior distributions of the unknown parameters and hyperparameters allow ones to infer the parameters of interest. These parameters include the abundances for each pixel, the means and variances of the abundances for each class, as well as a classification map indicating the classes of all pixels in the image. To overcome the complexity of the posterior distribution of interest, we consider Markov chain Monte Carlo methods that generate samples distributed according to the posterior of interest. The generated samples are then used for parameter and hyperparameter estimation. The accuracy of the proposed algorithms is illustrated on synthetic and real data.
I. INTRODUCTION
Existing hyperspectral unmixing methods largely operate pixel by pixel, overlooking spatial correlations. The paper introduces a hierarchical Bayesian strategy that models class-dependent abundance statistics and spatially dependent hidden labels, with MCMC used for inference.
- Motivation: Pixel-by-pixel inversion methods do not exploit possible spatial correlations between hyperspectral-image pixels.The paper identifies this limitation across constrained optimization and Bayesian inversion strategies.
- Spatial modeling: Spatial dependence is imposed on hidden class labels rather than abundances directly, providing a flexible model for images composed of distinct areas.The Potts-Markov prior encourages neighboring pixels to belong to the same class.
- Model: The proposed model partitions the image into homogeneous classes with shared abundance means and covariances, while hidden labels follow a Potts-Markov field.Conditioned on the labels, class-dependent abundance priors enforce positivity and sum-to-one constraints.
- Inference: The hierarchical Bayesian formulation estimates labels, abundance vectors, and class-specific prior means and variances jointly in a fully unsupervised manner.Non-informative hyperpriors are introduced for the class-dependent abundance statistics.
- Inference: MCMC generates samples from the joint posterior so Bayesian estimators can be approximated despite the posterior’s analytical complexity.The samples support estimators such as MMSE and maximum a posteriori estimates.
A. Unmixing statistical model
The model combines the linear mixing model with class-based spatial dependencies to represent hyperspectral pixels whose abundances are correlated with their neighbors. A Potts-Markov field models pixel classes, while a Gaussian-noise LMM models spectra within each class.
- A. Unmixing statistical model: Each pixel spectrum is modeled as a linear combination of known endmembers corrupted by additive white Gaussian noise.The endmember matrix is known, while abundances and noise characterize each observed spectrum.
- B. Introducing spatial dependencies between abundances: The image is partitioned into K classes whose pixels share abundance first- and second-order statistical moments.A label vector assigns every pixel to one of the classes, whose parameters include abundance means and covariance matrices.
- C. Markov random fields: Spatial correlations are represented by a Markov random field whose conditional pixel-label distribution depends only on neighboring labels.The paper uses symmetric neighborhood relations and focuses on a four-pixel, first-order neighborhood with vertical and horizontal configurations.
- C. Markov random fields: Sampling Potts-Markov labels is feasible with a Gibbs sampler, but the model’s partition function has no closed-form expression and depends on unknown β.The paper fixes β a priori, while noting that its estimation could also be conducted using existing methods.
- C. Markov random fields: The Potts-Markov granularity coefficient β controls regional homogeneity: small values produce noisy images, whereas large values produce fewer, larger homogeneous regions.For the adopted first-order neighborhood, values β ≥ 2 are unnecessary because realizations become concentrated on single-color images.
D. Abundance Reparametrization
The abundance vectors are reparameterized through logistic coefficients so that their positivity and sum-to-one constraints are satisfied. Within each class, the logistic coefficients retain shared first- and second-order moments.
- D. Abundance Reparametrization: A logistic coefficient vector is assigned to each abundance vector to reparameterize positive abundances that sum to one.The reparameterization rewrites abundances as functions of unconstrained logistic coefficients.
- D. Abundance Reparametrization: The reparameterization has also been applied to hyperspectral unmixing and reported results in a pharmacokinetic problem.The cited prior work introduced the transformation for positive parameters summing to one.
- D. Abundance Reparametrization: The spatial constraints are transferred to logistic coefficients, which share first- and second-order moments within each image class.Under this parameterization, each class is characterized by the corresponding unknown hyperparameters.
III. HIERARCHICAL BAYESIAN MODEL
The hierarchical Bayesian model treats noise variance, pixel classes, and logistic abundance coefficients as unknowns under priors and hyperpriors. The LMM likelihood uses additive white Gaussian noise with a common variance across spectral bands and independent pixels.
- III. HIERARCHICAL BAYESIAN MODEL: The unknown LMM parameters comprise the noise variance, class-label vector, and logistic coefficient matrix used for abundance reparameterization.The noise variance is explicitly assumed unknown in this paper.
- III. HIERARCHICAL BAYESIAN MODEL: Conditioned on logistic coefficients and noise variance, each observed spectrum follows a Gaussian distribution centered at its endmember mixture.The covariance is the noise variance multiplied by the identity matrix over spectral bands.
- III. HIERARCHICAL BAYESIAN MODEL: Because noise sequences are independent across pixels, the likelihood for the full image is formed from the pixel-level likelihoods.The common noise variance is shared across pixels and spectral bands.
- III. HIERARCHICAL BAYESIAN MODEL: The model assigns prior distributions and hyperpriors to unknown parameters, with their dependencies represented by a directed acyclic graph.The DAG identifies fixed parameters separately from the stochastic quantities in the hierarchy.
1) Label prior:
The model uses a Potts-Markov random field for spatially correlated class labels and hierarchical Bayesian priors for class-dependent abundance-related coefficients and noise. Hyperparameters are assigned conjugate or noninformative priors and estimated jointly.
- Label prior:: A Potts-Markov random field with a 1-order neighborhood models spatial dependence among pixel class labels.The neighborhood is defined through the pixels adjacent to each pixel, with a fixed granularity coefficient β.
- Class-conditional coefficients: The logistic coefficient vector for each pixel is Gaussian, with its mean and variance determined by the pixel’s class label.Class-specific means and diagonal covariance matrices describe the coefficient distributions.
- Hierarchical priors: The hierarchical model assumes prior independence across coefficient vectors and class-specific mean vectors and covariance matrices.The resulting priors are organized through the hyperparameters Ψ and Σ.
- Noise model: Noise variance receives a conjugate inverse-gamma prior, while its scale hyperparameter is estimated jointly with the other unknown quantities.The paper fixes ν = 1 and estimates δ within the hierarchical Bayesian algorithm.
- Hyperparameter priors: The remaining hyperparameters use vague conjugate priors or Jeffreys’ priors to represent limited prior knowledge.These choices include priors for class-specific variances, coefficient means, δ, and υ2.
E. Joint distribution
The joint posterior is defined hierarchically but is too complex for closed-form MMSE or MAP estimation. A hybrid Gibbs sampler therefore combines conditional sampling with Metropolis-Hastings steps to approximate Bayesian estimators.
- E. Joint distribution: The joint posterior over unknown parameters and hyperparameters is hierarchically defined but too complex for closed-form MMSE or MAP estimators.This motivates posterior sampling rather than direct analytical estimation.
- E. Joint distribution: MCMC generates asymptotically posterior-distributed samples that are then used to approximate Bayesian estimators.The sampler targets the posterior distribution associated with the linear mixing model.
- IV. HYBRID GIBBS SAMPLER: The hybrid Gibbs sampler updates variables according to their conditional distributions, including the label vector and logistic coefficient matrix.The procedure combines Gibbs-style conditional updates with a Metropolis-within-Gibbs construction.
- A. Conditional distribution of the label vector z: Each pixel label is sampled from probabilities proportional to its class-conditional coefficient density times the Potts-Markov prior probability.The label update evaluates each of the K possible classes using neighboring labels and the corresponding class parameters.
- B. Conditional distribution of logistic coefficient matrix T: Because direct sampling of the logistic-coefficient conditional is too difficult, the algorithm uses a Gaussian random-walk Metropolis-Hastings step.The proposal variance is fixed to target an acceptance rate between 0.15 and 0.5.
C. Conditional distributions of the noise variance
The proposed hybrid Gibbs sampler estimates class labels, abundances, and abundance statistics under the linear mixing model. On synthetic data, these estimates agree with the actual class structure and parameter values, while Spatial achieves lower global abundance MSE than non-spatial baselines.
- Synthetic-data estimates: Estimated abundance means for the second class and the reported class means and variances are consistent with the corresponding actual values.Similar histogram results are reported for the other classes, though they are omitted for brevity.
- Synthetic-data estimates: Estimated classes, abundance coefficients, and abundance mean vectors agree with their actual values in the synthetic experiment.The experiment uses a 25 × 25 image with three classes and three mixed components.
- Synthetic-data comparison: Spatial achieves lower global abundance MSE than the non-spatial Bayesian and FCLS algorithms.The comparison uses global mean square error computed for each abundance component.
A. Performance of the proposed algorithm
The proposed spatial algorithm is evaluated on real Moffett Field hyperspectral data using LMM-based classification and abundance estimation. Its maps are similar to Bayesian and FCLS results, remain robust as the number of endmembers changes, and provide posterior uncertainty information at higher computational cost.
- Real-data results: The estimated classification and abundance maps are in good agreement with LMM-based Bayesian and FCLS results on the Moffett image.The proposed algorithm produces classification maps for R = 3, 4, and 5 endmembers.
- Robustness analysis: Classification results remain robust with respect to the number of endmembers for R = 4 and R = 5.The corresponding classification maps are shown for both endmember counts.
- Computational cost: The proposed Spatial algorithm has higher computational cost than FCLS and cNMF because it jointly estimates labels and abundance vectors.The reported timing comparison uses the Moffett image and an unoptimized MATLAB implementation.
- Uncertainty information: The proposed Gibbs sampler provides samples that can be used to determine confidence intervals for the unmixing estimates.This uncertainty information is identified as an additional benefit of the proposed method.
B. Simulation on a larger image
The proposed Bayesian unmixing algorithm was applied to a 190 × 250-pixel Cuprite AVIRIS scene with 14 endmembers and classes. It recovered spatial regions and abundance patterns consistent with known materials and prior results.
- Results: The algorithm identified Cuprite regions similar to those recovered in prior work and estimated both abundance and label maps.The analyzed area contains 190 × 250 pixels, with K = 14 classes and R = 14 extracted endmembers.
- Results: The 6th endmember matches the Montmorillonite spectrum in the USGS library, agreeing with ground truth that Montmorillonite is common in the scene.The comparison is shown against the extracted spectrum and the corresponding USGS library signature.
- Method: The hierarchical Bayesian model uses Markov random fields to model spatial correlations and hidden labels for classes with homogeneous abundance statistics.Abundance constraints are enforced through logistic coefficient reparameterization, and posterior samples are generated with MCMC.
- Conclusion: The proposed algorithm produced promising results on simulated data and real AVIRIS images.The study includes abundance and label-map estimation for the Cuprite scene.
- Conclusion: Future work includes estimating the granularity coefficient used in the Potts-Markov random field instead of fixing it a priori.The current model therefore treats this coefficient as known during inference.