An analysis of the nonlinear spectral mixing of didymium and soda-lime glass beads using hyperspectral imagery (HSI) microscopy

Nonlinear spectral mixing occurs when materials are intimately mixed. Intimate mixing is a common characteristic of granular materials such as soils. A linear spectral unmixing inversion applied to a nonlinear mixture will yield subpixel abundance estimates that do not equal the true values of the mixture's components. These aspects of spectral mixture analysis theory are well documented. Several methods to invert (and model) nonlinear spectral mixtures have been proposed. Examples include Hapke theory, the extended endmember matrix method, and kernel-based methods. There is, however, a relative paucity of real spectral image data sets that contain well characterized intimate mixtures. To address this, special materials were custom fabricated, mechanically mixed to form intimate mixtures, and measured with a hyperspectral imaging (HSI) microscope. The results of analyses of visible/near-infrared (VNIR; 400 nm to 900 nm) HSI microscopy image cubes (in reflectance) of intimate mixtures of the two materials are presented. The materials are spherical beads of didymium glass and soda-lime glass both ranging in particle size from 63 m to 125 m. Mixtures are generated by volume and thoroughly mixed mechanically. Three binary mixtures (and the two endmembers) are constructed and emplaced in the wells of a 96-well sample plate: 0%/100%, 25%/75%, 50%/50%, 80%/20%, and 100%/0% didymium/soda-lime. Analysis methods are linear spectral unmixing (LSU), LSU applied to reflectance converted to single-scattering albedo (SSA) using Hapke theory, and two kernel-based methods. The first kernel method uses a generalized kernel with a gamma parameter that gauges non-linearity, applying the well-known kernel trick to the least squares formulation of the constrained linear model. This method attempts to determine if each pixel in a scene is linear or non-linear, and adapts to compute a mixture model at each pixel accordingly. The second method uses 'K-hype' with a polynomial (quadratic) kernel. LSU applied to the reflectance spectra of the mixtures produced poor abundance estimates regardless of the constraints applied in the inversion. The 'K-hype' kernel-based method also produced poor fraction estimates. The best performers are LSU applied to the reflectance spectra converted to SSA using Hapke theory and the gamma parameter kernel-based method.