[Paper Review] Collaborative sparse regression using spatially correlated supports - Application to hyperspectral unmixing

Yoann Altmann, Marcelo Pereyra+1 more|arXiv (Cornell University)|Sep 29, 2014

Remote-Sensing Image Classification25 references4 citations

TL;DR

This paper proposes a Bayesian collaborative sparse regression method for hyperspectral unmixing that models spatially correlated supports of abundance vectors using a truncated multivariate Ising Markov random field (MRF), enabling accurate detection of material presence/absence patterns per pixel. The method employs an adaptive MCMC algorithm that self-tunes MRF parameters, reducing reliance on cross-validation and improving unmixing accuracy on synthetic and real data with up to 25 endmembers.

ABSTRACT

This paper presents a new Bayesian collaborative sparse regression method for linear unmixing of hyperspectral images. Our contribution is twofold; first, we propose a new Bayesian model for structured sparse regression in which the supports of the sparse abundance vectors are a priori spatially correlated across pixels (i.e., materials are spatially organised rather than randomly distributed at a pixel level). This prior information is encoded in the model through a truncated multivariate Ising Markov random field, which also takes into consideration the facts that pixels cannot be empty (i.e, there is at least one material present in each pixel), and that different materials may exhibit different degrees of spatial regularity. Secondly, we propose an advanced Markov chain Monte Carlo algorithm to estimate the posterior probabilities that materials are present or absent in each pixel, and, conditionally to the maximum marginal a posteriori configuration of the support, compute the MMSE estimates of the abundance vectors. A remarkable property of this algorithm is that it self-adjusts the values of the parameters of the Markov random field, thus relieving practitioners from setting regularisation parameters by cross-validation. The performance of the proposed methodology is finally demonstrated through a series of experiments with synthetic and real data and comparisons with other algorithms from the literature.

Motivation & Objective

• To address the challenge of accurately estimating sparse abundance vectors in hyperspectral unmixing by incorporating spatial correlation in material presence/absence patterns across pixels.
• To overcome limitations of global collaborative methods that only estimate union supports, by modeling local spatial structure in the supports of abundance vectors.
• To develop a Bayesian framework that encodes spatial regularity and non-empty pixel constraints via a truncated multivariate Ising MRF.
• To design an MCMC inference algorithm that self-adjusts MRF parameters, eliminating the need for cross-validation in regularization tuning.
• To demonstrate improved unmixing performance on synthetic and real hyperspectral datasets compared to state-of-the-art methods.

Proposed method

• Models the supports of abundance vectors using a truncated multivariate Ising Markov random field to encode spatial correlation in material presence/absence patterns.
• Incorporates non-empty pixel constraints by ensuring at least one material is present in each pixel through truncation of the MRF prior.
• Uses conjugate priors for abundance vectors and noise, enabling closed-form conditional posteriors for efficient MCMC sampling.
• Employs a Metropolis-within-Gibbs MCMC algorithm to estimate posterior probabilities of material presence/absence per pixel.
• Computes MMSE estimates of abundance vectors conditionally on the maximum marginal a posteriori (MAP) support configuration.
• Self-adjusts the MRF regularisation strength during MCMC sampling, eliminating manual cross-validation for parameter tuning.

Experimental results

Research questions

• RQ1 Can modeling spatial correlation in the supports of abundance vectors improve hyperspectral unmixing accuracy compared to methods that only exploit global collaboration?
• RQ2 How can non-empty pixel constraints and varying degrees of spatial regularity across materials be effectively encoded in a Bayesian sparse regression model?
• RQ3 Can an adaptive MCMC algorithm self-tune MRF parameters without requiring cross-validation, thereby reducing practitioner burden?
• RQ4 To what extent does the proposed method outperform existing collaborative sparse regression techniques on both synthetic and real hyperspectral data?
• RQ5 How does the method perform in scenarios with high spectral similarity between materials, where sparsity alone is insufficient?

Key findings

• The proposed method achieves superior unmixing accuracy on synthetic data compared to state-of-the-art collaborative sparse regression methods, particularly in resolving spatially structured material distributions.
• The adaptive MCMC algorithm successfully self-tunes the MRF regularisation parameters, eliminating the need for cross-validation and improving robustness.
• The method effectively captures spatially correlated supports, reducing false positives in material detection and improving support estimation accuracy.
• On real hyperspectral data, the method demonstrates improved abundance estimation and endmember detection, especially in high-resolution scenes with complex spatial patterns.
• The computational complexity limits direct application to libraries with more than 25 endmembers, but performance can be enhanced via pre-processing with MUSIC-CSR to prune the library.
• The method outperforms existing approaches that regularise abundance values or nonlinear effects, as it explicitly models correlations in the supports rather than the abundance values themselves.