A Gaussian mixture model representation of endmember variability in hyperspectral unmixing
This addresses the problem of inaccurate material distribution modeling in hyperspectral imaging for remote sensing applications, but it is incremental as it builds on existing models with a more flexible representation.
The paper tackled hyperspectral unmixing with endmember variability by representing endmembers using Gaussian mixture models instead of unimodal Gaussians, showing that mixed pixels also follow a GMM and enabling estimation of abundances, distribution parameters, and distinct endmember sets per pixel, with testing on synthetic and real datasets demonstrating potential compared to current methods.
Hyperspectral unmixing while considering endmember variability is usually performed by the normal compositional model (NCM), 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 the 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 the random variable transformation 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, 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.