Canonical polyadic decomposition of hyperspectral patch tensors

Spectral unmixing (SU) is one of the most important and studied topics in hyperspectral image analysis. By means of spectral unmixing it is possible to decompose a hyperspectral image in its spectral components, the so-called endmembers, and their respective fractional spatial distributions, so-called abundance maps. The Canonical Polyadic (CP) tensor decomposition has proved to be a powerful tool to decompose a tensor data onto a few rank-one terms in a multilinear fashion. Here, we establish the connection between the CP decomposition and the SU problem when the tensor data is built by stacking small patches of the hyperspectral image. It turns out that the CP decomposition of this hyperspectral patch-tensor is equivalent to solving a blind regularized Extended Linear Mixing Model (ELMM).