Spectral Unmixing: A Derivation of the Extended Linear Mixing Model From the Hapke Model

Abstract

In hyperspectral imaging, spectral unmixing aims at decomposing the image into a set of reference spectral signatures corresponding to the materials present in the observed scene and their relative proportions in every pixel. While a linear mixing model was used for a long time, the complex nature of the physical mixing processes led to shift the community's attention towards nonlinear models or algorithms accounting for the variability of the endmembers. Such intra-class variations are due to local changes in the physico-chemical composition of the materials, and to illumination changes. In the physical remote sensing community, a popular model accounting for illumination variability is the radiative transfer model proposed by Hapke. It is however too complex to be directly used in hyperspectral unmixing in a tractable way. Instead, the Extended Linear Mixing Model (ELMM) allows to easily unmix hyperspectral data accounting for changing illumination conditions and to address nonlinear effects to some extent. In this letter, we show that the ELMM can be obtained from the Hapke model by successive simplifiying physical assumptions, whose validity we experimentally examine, thus demonstrating its relevance to handle illumination induced variability in the unmixing problem.Index Terms-Hyperspectral image unmixing, spectral variability, Hapke model, extended linear mixing model. L. Drumetz is with IMT Atlantique, Lab-STICC, UBL, Technopôle Brest-

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