Accounting for spectral variability in hyperspectral unmixing using beta endmember distribution


Hyperspectral imaging is widely used in the field of remote sensing (Goetz, et al., 1985; Green, et al., 1998). In a hyperspectral imaging system, sensors collect radiance/reflectance values over an area (or a scene) across hundreds of spectral bands (Goetz, et al., 1985). The hyperspectral image yielded by such system can be represented by a three-dimensional data cube containing those radiance/reflectance values in a range of wavelengths (Landgrebe, 2002). There are two common analysis methods for hyperspectral imagery (Hu, et al., 1999): endmember estimation and hyperspectral unmixing. Endmember estimation aims at finding pure individual spectral signatures of the materials (endmembers) in the scene (Adams, et al., 1986). Hyperspectral unmixing, on the other hand, estimates the proportions of each endmember at every pixel of the image. Each pixel in the image can then be represented by endmember spectra weighted by its corresponding proportions.

In order to increase the accuracy of hyperspectral unmixing, sufficient temporal and spatial spectral variability of endmembers must be taken into consideration (Roberts, et al., 1992; Roberts, et al., 1998; Bateson, et al., 2000). The most common factors contributing to spectral variability include environmental factors, such as atmospheric effects, illumination, moisture conditions, and inherent spectral variability of the material itself, such as the variations in biophysical and biochemical composition in vegetation (Song, 2005). Under such influence, the spectral signature of endmembers may vary from time to time and from pixel to pixel in the scene.

In order to account for such endmember spectral variability, endmembers are regarded as either a set, or a “bundle”, of individual spectra (Roberts, et al., 1998; Bateson, et al., 2000), or as a sample from a full distribution. The application of the Normal Compositional Model with Gaussian-distributed endmembers has been discussed in the literature (Eches, et al., 2010; Zare, et al., 2012). Since the domain of Gaussian distribution is negative infinity to infinity, Gaussian endmembers allow samples outside the interval between zero and one. However, in reality, the reflectance value of endmember spectra usually only vary between zero and one. Beta distributions, on the other hand, are defined only over the interval between zero and one.Therefore, in this thesis, the Beta distribution is considered for endmembers in order to make it more physically realistic. The Beta Compositional Model (Zare, et al., 2013) is considered as the mixing model in this case.

Two approaches based on the Normal Compositional Model (Stein, 2003; Eismann, 2006) and the Beta Compositional Model, quadratic programming (QP) approach and Metropolis-Hastings (MH) sampling approach, are presented in this thesis for hyperspectral unmixing, i.e., finding the proportions of each endmember in a hyperspectral image. QP approach determines the proportion values by minimizing the difference between the mean of Beta approximation to the convex combination of Beta endmember distributions, while MH sampling method takes both the mean and variance into consideration.

Furthermore, in this thesis, unmixing algorithms that incorporate spatial information are proposed under the Beta Compositional Model (BCM-Spatial algorithms). These include algorithms based on Fuzzy Local Information C-Means Clustering Algorithm (FLICM), superpixel methods, and spatial K-means algorithm.

Results indicate that unmixing algorithms based on NCM and BCM are able to successfully perform unmixing on simulated data and real hyperspectral data and can incorporate endmember spectral variability. BCM unmixing does a better job than NCM unmixing on data generated from Beta endmembers than those from Gaussian endmembers. The results from BCM-Spatial unmixing algorithms on hyperspectral image data show that the new algorithms are effective at unmixing.


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X. Du, “Accounting for spectral variability in hyperspectral unmixing using beta endmember distribution,” Master Thesis, Columbia, MO, 2013. 
Title = {Accounting for spectral variability in hyperspectral unmixing using beta endmember distribution},
Author = {Xiaoxiao Du},
School = {Univ. of Missouri},
Year = {2013},
Address = {Columbia, MO},
Month = {Dec.},
Url = {}