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

## TL;DR

This paper derives the Extended Linear Mixing Model (ELMM) from the Hapke physical model, providing a theoretical basis for its use in hyperspectral unmixing under varying illumination conditions.

## Contribution

It demonstrates that the ELMM can be derived from the Hapke model through simplifying assumptions, linking physical and data-driven unmixing models.

## Key findings

- ELMM can be derived from the Hapke model.
- The derivation confirms ELMM's suitability for illumination variability.
- Provides a physical justification for the ELMM in hyperspectral unmixing.

## 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 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.