Nonlinear models have recently shown interesting properties for spectral unmixing. Our research studies a generalized bilinear model and a hierarchical Bayesian algorithm for unmixing hyperspectral images. The proposed model is a generalization not only of the accepted linear mixing model but also of a bilinear model that has been recently introduced in the literature. Appropriate priors are chosen for its parameters to satisfy the positivity and sum-to-one constraints for the abundances. The joint posterior distribution of the unknown parameter vector is then derived. Unfortunately, this posterior is too complex to obtain analytical expressions of the standard Bayesian estimators. As a consequence, a Metropolis-within-Gibbs algorithm is proposed, which allows samples distributed according to this posterior to be generated and to estimate the unknown model parameters. The performance of the resulting unmixing strategy shows promising results for both synthetic and real data.

A. Halimi, Y. Altmann, N. Dobigeon and J.-Y. Tourneret, "Nonlinear unmixing of hyperspectral images using a generalized bilinear model," IEEE Trans. Geoscience and Remote Sensing, vol. 49, no. 11, pp. 4153-4162, Nov. 2011.

Nonlinear models have recently shown interesting properties for spectral unmixing. This paper considers a generalized bilinear model recently introduced for unmixing hyperspectral images. Different algorithms are studied to estimate the parameters of this bilinear model. The positivity and sum-to-one constraints for the abundances are ensured by the proposed algorithms. The performance of the resulting unmixing strategy is evaluated via simulations conducted on synthetic and real data

A. Halimi, Y. Altmann, N. Dobigeon, and J.-Y. Tourneret, "Unmixing hyperspectral images using the generalized bilinear model," in Proc. IEEE Int. Geosci. Remote Sens. Symp. (IGARSS), Sendai, Japan, Vancouver, Canada, July 2011, pp. 1886-1889.

This work presents three hyperspectral mixture models jointly with Bayesian algorithms for supervised hyperspectral unmixing. Based on the residual component analysis model, the proposed general formulation assumes the linear model to be corrupted by an additive term whose expression can be adapted to account for nonlinearities (NLs), endmember variability (EV), or mismodeling effects (MEs). The NL effect is introduced by considering a polynomial expression that is related to bilinear models. The proposed new formulation of EV accounts for shape and scale endmember changes while enforcing a smooth spectral/spatial variation. The ME formulation considers the effect of outliers and copes with some types of EV and NL. The known constraints on the parameter of each observation model are modeled via suitable priors. The posterior distribution associated with each Bayesian model is optimized using a coordinate descent algorithm, which allows the computation of the maximum a posteriori estimator of the unknown model parameters. The proposed mixture and Bayesian models and their estimation algorithms are validated on both synthetic and real images showing competitive results regarding the quality of the inferences and the computational complexity, when compared with the state-of-the-art algorithms.

A. Halimi, P. Honeine and J. Bioucas-Dias, "Hyperspectral Unmixing in Presence of Endmember Variability, Nonlinearity or Mismodelling Effects," IEEE Trans. Image Processing, vol. 25, no. 10, Oct. 2016.

This work presents two novel hyperspectral mixture models and associated unmixing algo- rithms. The two models assume a linear mixing model corrupted by an additive term whose expression can be adapted to account for multiple scattering nonlinearities (NL), or mismodelling effects (ME). The NL model generalizes bilinear models by taking into account higher order interaction terms. The ME model accounts for different effects such as endmember variability or the presence of outliers. The abundance and residual parameters of these models are estimated by considering a convex formulation suitable for fast estimation algorithms. This formulation accounts for constraints such as the sum-to-one and non-negativity of the abundances, the non-negativity of the nonlinearity coefficients, the spectral smoothness of the ME terms and the spatial sparseness of the residuals. The resulting convex problem is solved using the alternating direction method of multipliers (ADMM) whose convergence is ensured theoretically. The proposed mixture models and their unmixing algorithms are validated on both synthetic and real images showing competitive results regarding the quality of the inference and the computational complexity when compared to the state-of-the-art algorithms.

A. Halimi, J. M. 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, June 2017.

This work presents two novel hyperspectral mixture models and associated unmixing algo- rithms. The two models assume a linear mixing model corrupted by an additive term whose expression can be adapted to account for multiple scattering nonlinearities (NL), or mismodelling effects (ME). The NL model generalizes bilinear models by taking into account higher order interaction terms. The ME model accounts for different effects such as endmember variability or the presence of outliers. The abundance and residual parameters of these models are estimated by considering a convex formulation suitable for fast estimation algorithms. This formulation accounts for constraints such as the sum-to-one and non-negativity of the abundances, the non-negativity of the nonlinearity coefficients, the spectral smoothness of the ME terms and the spatial sparseness of the residuals. The resulting convex problem is solved using the alternating direction method of multipliers (ADMM) whose convergence is ensured theoretically. The proposed mixture models and their unmixing algorithms are validated on both synthetic and real images showing competitive results regarding the quality of the inference and the computational complexity when compared to the state-of-the-art algorithms.

F. Zhu, A. Halimi, P. Honeine, B. Chen, and N. Zheng, "Correntropy Maximization via ADMM - Application to Robust Hyperspectral Unmixing," IEEE Trans. Geoscience and Remote Sensing, vol. 55, no. 9, Sept. 2017.

This work presents an unsupervised Bayesian algorithm for hyperspectral image unmixing account-ing for endmember variability. The pixels are modeled by a linear combination of endmembers weightedby their corresponding abundances. However, the endmembers are assumed random to take into accounttheir variability in the image. An additive noise is also considered in the proposed model generalizingthe normal compositional model. The proposed algorithm exploits the whole image to benefit fromboth spectral and spatial information. It estimates both the mean and the covariance matrix of eachendmember in the image. This allows the behavior of each material to be analyzed and its variability tobe quantified in the scene. A spatial segmentation is also obtained based on the estimated abundances.In order to estimate the parameters associated with the proposed Bayesian model, we propose to use aHamiltonian Monte Carlo algorithm. The performance of the resulting unmixing strategy is evaluatedvia simulations conducted on both synthetic and real data.

A. Halimi, N. Dobigeon and J.-Y. Tourneret, "Unsupervised Unmixing of Hyperspectral Images Accounting for Endmember Variability," IEEE Trans. Image Processing, vol. 24, no. 12, pp. 4904-4917, 2015.

This work presents an unsupervised Bayesian algorithm for hyperspectral image unmixing account-ing for endmember variability. The pixels are modeled by a linear combination of endmembers weightedby their corresponding abundances. However, the endmembers are assumed random to take into accounttheir variability in the image. An additive noise is also considered in the proposed model generalizingthe normal compositional model. The proposed algorithm exploits the whole image to benefit fromboth spectral and spatial information. It estimates both the mean and the covariance matrix of eachendmember in the image. This allows the behavior of each material to be analyzed and its variability tobe quantified in the scene. A spatial segmentation is also obtained based on the estimated abundances.In order to estimate the parameters associated with the proposed Bayesian model, we propose to use aHamiltonian Monte Carlo algorithm. The performance of the resulting unmixing strategy is evaluatedvia simulations conducted on both synthetic and real data.

A. Halimi, P. Honeine, M. Kharouf, C. Richard and J.-Y. Tourneret, "Estimating the Intrinsic Dimension of Hyperspectral Images Using a Noise Whitened Eigen-Gap Approach," IEEE Trans. Geoscience and Remote Sensing, vol. 54, no7, July 2016.