Olivier Eches

Enhancing Hyperspectral Image Unmixing With Spatial Correlations

This paper describes a new algorithm for hyperspectral image unmixing. Most unmixing algorithms proposed in the literature do not take into account the possible spatial correlations between the pixels. In this paper, 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 dependencies between 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. For this model, the posterior distributions of the unknown parameters and hyperparameters allow the parameters of interest to be inferred. 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, we consider a Markov chain Monte Carlo method that generates samples asymptotically distributed according to the posterior. The generated samples are then used for parameter and hyperparameter estimation. The accuracy of the proposed algorithms is illustrated on synthetic and real data.

Bayesian Estimation of Linear Mixtures Using the Normal Compositional Model. Application to Hyperspectral Imagery

This paper studies a new Bayesian unmixing algorithm for hyperspectral images. Each pixel of the image is modeled as a linear combination of so-called endmembers. These endmembers are supposed to be random in order to model uncertainties regarding their knowledge. More precisely, we model endmembers as Gaussian vectors whose means have been determined using an endmember extraction algorithm such as the famous N-finder (N-FINDR) or Vertex Component Analysis (VCA) algorithms. This paper proposes to estimate the mixture coefficients (referred to as abundances) using a Bayesian algorithm. Suitable priors are assigned to the abundances in order to satisfy positivity and additivity constraints whereas conjugate priors are chosen for the remaining parameters. A hybrid Gibbs sampler is then constructed to generate abundance and variance samples distributed according to the joint posterior of the abundances and noise variances. The performance of the proposed methodology is evaluated by comparison with other unmixing algorithms on synthetic and real images.

Estimating the Number of Endmembers in Hyperspectral Images Using the Normal Compositional Model and a Hierarchical Bayesian Algorithm

This paper studies a semi-supervised Bayesian unmixing algorithm for hyperspectral images. This algorithm is based on the normal compositional model recently introduced by Eismann and Stein. The normal compositional model assumes that each pixel of the image is modeled as a linear combination of an unknown number of pure materials, called endmembers. However, contrary to the classical linear mixing model, these endmembers are supposed to be random in order to model uncertainties regarding their knowledge. This paper proposes to estimate the mixture coefficients of the Normal Compositional Model (referred to as abundances) as well as their number using a reversible jump Bayesian algorithm. The performance of the proposed methodology is evaluated thanks to simulations conducted on synthetic and real AVIRIS images.

Adaptive Markov Random Fields for Joint Unmixing and Segmentation of Hyperspectral Images

Linear spectral unmixing is a challenging problem in hyperspectral imaging that consists of decomposing an observed pixel into a linear combination of pure spectra (or endmembers) with their corresponding proportions (or abundances). Endmember extraction algorithms can be employed for recovering the spectral signatures while abundances are estimated using an inversion step. Recent works have shown that exploiting spatial dependencies between image pixels can improve spectral unmixing. Markov random fields (MRF) are classically used to model these spatial correlations and partition the image into multiple classes with homogeneous abundances. This paper proposes to define the MRF sites using similarity regions. These regions are built using a self-complementary area filter that stems from the morphological theory. This kind of filter divides the original image into flat zones where the underlying pixels have the same spectral values. Once the MRF has been clearly established, a hierarchical Bayesian algorithm is proposed to estimate the abundances, the class labels, the noise variance, and the corresponding hyperparameters. A hybrid Gibbs sampler is constructed to generate samples according to the corresponding posterior distribution of the unknown parameters and hyperparameters. Simulations conducted on synthetic and real AVIRIS data demonstrate the good performance of the algorithm.

Markov random fields for joint unmixing and segmentation of hyperspectral images

This paper studies a new Bayesian algorithm for the unmixing of hyperspectral images. The proposed Bayesian algorithm is based on the well-known linear mixing model (LMM). Spatial correlations between pixels are introduced using hidden variables, or labels, and modeled via a Potts-Markov random field. We assume that the pure materials (or endmembers) contained in the image are known a priori or have been extracted by using an endmember extraction algorithm. The mixture coefficients (referred to as abundances) of the whole hyperspectral image are then estimated by using a hierarchical Bayesian algorithm. A reparametrization of the abundances is considered to handle the physical constraints associated to these parameters. Appropriate prior distributions are assigned to the other parameters and hyperparameters associated to the proposed model. To alleviate the complexity of the resulting joint distribution, a hybrid Gibbs algorithm is developed, allowing one to generate samples that are asymptotically distributed according to the full posterior distribution of interest. The generated samples are finally used to estimate the unknown model parameters. Simulations on synthetic data illustrate the performance of the proposed method.

Unmixing hyperspectral images using a normal compositional model and MCMC methods

This paper studies a new unmixing algorithm for hyperspectral images. Each pixel of the image is modeled as a linear combination of endmembers which are supposed to be random in order to model uncertainties regarding their knowledge. More precisely, endmembers are modeled as Gaussian vectors with known means (resulting from an endmember extraction algorithm such as the famous N-FINDR or VCA algorithm). This paper proposes to estimate the mixture coefficients (referred to as abundances) using a Bayesian algorithm. Suitable priors are assigned to the abundances in order to satisfy positivity and additivity constraints whereas a conjugate prior is chosen for the variance. The computational complexity of the resulting Bayesian estimators is alleviated by constructing an hybrid Gibbs algorithm to generate abundance and variance samples distributed according to the posterior distribution of the unknown parameters. The associated hyperparameter is also generated. The performance of the proposed methodology is evaluated thanks to simulation results conducted on synthetic and real images.

An NCM-based Bayesian algorithm for hyperspectral unmixing

This paper studies a new Bayesian algorithm to unmix hyperspectral images. The algorithm is based on the recent normal compositional model introduced by Eismann. Contrary to the standard linear mixing model, the endmember spectra are assumed to be random signatures with know mean vectors. Appropriate prior distributions are assigned to the abundance coefficients to ensure the usual positivity and sum-to-one constraints. However, the resulting posterior distribution is too complex to obtain a closed form expression for the Bayesian estimators. A Markov chain Monte Carlo algorithm is then proposed to generate samples distributed according to the full posterior distribution. These samples are used to estimate the unknown model parameters. Several simulations are conducted on synthetic and real data to illustrate the performance of the proposed method.

Variational methods for spectral unmixing of hyperspectral images

This paper studies a variational Bayesian unmixing algorithm for hyperspectral images based on the standard linear mixing model. Each pixel of the image is modeled as a linear combination of endmembers whose corresponding fractions or abundances are estimated by a Bayesian algorithm. This approach requires to define prior distributions for the parameters of interest and the related hyperparameters. After defining appropriate priors for the abundances (uniform priors on the interval (0, 1)), the joint posterior distribution of the model parameters and hyperparameters is derived. The complexity of this distribution is handled by using variational methods that allow the joint distribution of the unknown parameters and hyperparameter to be approximated. Simulation results conducted on synthetic and real data show similar performances than those obtained with a previously published unmixing algorithm based on Markov chain Monte Carlo methods, with a significantly reduced computational cost.

Joint spectral classification and unmixing using adaptative pixel neighborhoods

A new spatial unmixing algorithm for hyperspectral images is studied. This algorithm is based on the well-known linear mixing model. The spectral signatures (or endmembers) are assumed to be known while the mixture coefficients (or abundances) are estimated by a Bayesian algorithm. As a pre-processing step, an area filter is employed to partition the image into multiple spectrally consistent connected components or adaptative neighborhoods. Then, spatial correlations are introduced by assigning to the pixels of a given neighbourhood the same hidden labels. More precisely, these pixels are modeled using a new prior distribution taking into account spectral similarity between the neighbors. Abundances are reparametrized by using logistic coefficients to handle the associated physical constraints. Other parameters and hyperparameters are assigned appropriate prior distributions. After computing the joint posterior distribution, a hybrid Gibbs algorithm is employed to generate samples that are asymptotically distributed according to this posterior distribution. The generated samples are finally used to estimate the unknown model parameters. Simulations on synthetic data illustrate the performance of the proposed method.

Unmixing hyperspectral images using Markov random fields

This paper proposes a new spectral unmixing strategy based on the normal compositional model that exploits the spatial correlations between the image pixels. The pure materials (referred to as endmembers) contained in the image are assumed to be available (they can be obtained by using an appropriate endmember extraction algorithm), while the corresponding fractions (referred to as abundances) are estimated by the proposed algorithm. Due to physical constraints, the abundances have to satisfy positivity and sum‐to‐one constraints. The image is divided into homogeneous distinct regions having the same statistical properties for the abundance coefficients. The spatial dependencies within each class are modeled thanks to Potts‐Markov random fields. Within a Bayesian framework, prior distributions for the abundances and the associated hyperparameters are introduced. A reparametrization of the abundance coefficients is proposed to handle the physical constraints (positivity and sum‐to‐one) inherent to hyperspe...