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

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 physicochemical phenomena that affect the spectra of the materials led to shift the community's attention toward algorithms accounting for the variability of the endmembers. Such intraclass variations are mainly due to local changes in the 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 the 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 successively simplifying physical assumptions, whose validity we experimentally examine, thus demonstrating its relevance to handle illumination-induced variability in the unmixing problem.