  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 and 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. In this letter, we show that the ELMM can be obtained from the
Hapke model by successive simplifiying physical assumptions, thus theoretically
confirming its relevance to handle illumination induced variability in the
unmixing problem.
