A Gaussian Mixture Model Representation of Endmember Variability in Hyperspectral Unmixing

Abstract

Hyperspectral unmixing while considering endmember variability is usually performed by the normal compositional model, where the endmembers for each pixel are assumed to be sampled from unimodal Gaussian distributions. However, in real applications, the distribution of a material is often not Gaussian. In this paper, we use Gaussian mixture models (GMM) to represent endmember variability. We show, given the GMM starting premise, that the distribution of the mixed pixel (under the linear mixing model) is also a GMM (and this is shown from two perspectives). The first perspective originates from random variable transformations and gives a conditional density function of the pixels given the abundances and GMM parameters. With proper smoothness and sparsity prior constraints on the abundances, the conditional density function leads to a standard maximum a posteriori (MAP ) problem which can be solved using generalized expectation maximization. The second perspective originates from marginalizing over the endmembers in the GMM, which provides us with a foundation to solve for the endmembers at each pixel. Hence, compared to the other distribution based methods, our model can not only estimate the abundances and distribution parameters, but also the distinct endmember set for each pixel. We tested the proposed GMM on several synthetic and real datasets, and showed its potential by comparing it to current popular methods.

Hyperspectral Unmixing with Gaussian Mixture Model and Low-Rank Representation

“…GMM method [34] is an LMM by considering the endmember variability, use a mixture of Gaussians to approximate any distribution of the endmember and can be classified in the second category mentioned above (endmembers represented using a continuous distribution). GMM model assumes the endmember p(m nj ) follows a mixture of Gaussians, and has the density function as follows:…”

“…More specifically, in the prior of the abundances of p(A), Zhou's model [34] assumes the abundances A have the proper smoothness and sparsity prior constraints. The density function of the abundances A can be generalized as follows:…”

“…When considering the endmember variability, the current popular unmixing methods which model the endmember following the probability distribution have been mentioned above: normal compositional model (NCM) [29], beta compositional model (BCM) [33] and Gaussian mixture model (GMM) [34]. Those methods all ignore the local spatial correlation of HSI.…”

“…However, the true distribution may not be well approximated by either a Gaussian or Beta distribution. In that case, Zhou et al proposed the Gaussian mixture model (GMM) to solve the LMM by considering endmember variability [34]. GMM method assumes that the endmember m nj follows the GMM distribution and noise n n follows the Gaussian distribution: p(n n ) := N (n n |0, D), where D is the noise covariance matrix, and with proper abundances constraint under the Bayesian framework to lead the conditional density function to a standard MAP problem.…”

“…On the other hand, this strategy has great advantages of easily generalizing the Bayesian algorithms, which is introduced in [37,38]. In that vein, in an attempt to achieve better abundance estimation results, Zhou's model (GMM) [34] uses the proper smoothness and sparsity prior constraints on the abundances, while Iordache et al [39] tried to find the minimum difference of the abundance between the adjacent pixels by the total variation (TV) constraint. However, those prior constraint assumptions require a rather strict assumption that the abundance of the local pixel should be piecewise smooth, which means that the mixing pixel and its associated abundance should be similar for the adjacent pixels.…”