IEEE Trans Image Process. 2018 May;27(5):2242-2256. doi: 10.1109/TIP.2018.2795744.
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.
在考虑端元变异性的情况下进行高光谱解混通常是通过正常的组成模型来完成的，其中假设每个像素的端元是从单峰高斯分布中采样得到的。然而，在实际应用中，物质的分布通常不是高斯分布。在本文中，我们使用高斯混合模型 (GMM) 来表示端元变异性。我们表明，给定 GMM 的初始前提，混合像素（在线性混合模型下）的分布也是一个 GMM（并且从两个角度进行了证明）。第一个角度源于随机变量变换，并给出了在丰度和 GMM 参数给定的情况下像素的条件密度函数。通过对丰度进行适当的平滑性和稀疏性先验约束，条件密度函数导致了一个标准的最大后验 (MAP) 问题，可以使用广义期望最大化来解决。第二个角度源于在 GMM 中对端元进行边缘化，这为我们在每个像素处求解端元提供了基础。因此，与其他基于分布的方法相比，我们的模型不仅可以估计丰度和分布参数，还可以估计每个像素的独特端元集。我们在几个合成和真实数据集上测试了所提出的 GMM，并通过与当前流行的方法进行比较，展示了其潜力。
