pixel, then $\abond{i}{\nopix} = 0$ and consequently $\nabond{i,j}{\nopix}=0$, which means that there are no interactions between the material $\Vmat{i}$ and any other materials $\Vmat{j}$ ($j\neq i$). Note however that this bilinear model does not extend the LMM.

To cope with this latter limitation, the generalized bilinear model (GBM) [@Halimi2011] weights the products of abundances $\abond{i}{\nopix} \abond{j}{\nopix}$ by additional free parameters $\gamma_{i,j,p}\in (0,1)$ that tune the amount of nonlinear interactions, leading to $\nabond{i,j}{\nopix} \triangleq
\gamma_{i,j,p}\abond{i}{\nopix} \abond{j}{\nopix} $ and $$\label{eq:GBM}
  \Vpix{\nopix}^{(\textrm{GBM})} \triangleq \sum_{\nomat=1}^{\nbmat} \abond{\nomat}{\nopix}
  \Vmat{\nomat} + \sum_{i=1}^{R-1}\sum_{j=i+1}^{R} \gamma_{i,j,p}\abond{i}{\nopix} \abond{j}{\nopix} \Vmat{i}\odot \Vmat{j} +
  \Vnoise{\nopix}.$$ The GBM has the nice properties of i) generalizing the LMM by enforcing $\gamma_{i,j,p}=0$ ($\forall i,j$), similarly to NM but contrary to FM and ii) having the amount of nonlinear interactions to be proportional to the material abundances, similarly to FM but contrary to NM.

Post-nonlinear mixing model
---------------------------

Inspired by pioneered works in blind source separation [@Taleb1999], Altmann *et al.* have introduced in [@Altmann2012ip] a nonlinear model that relies on a $2$nd-order polynomial expansion of the nonlinearity, $$\label{eq:term_PPNM}
\begin{split}
  \boldsymbol{\mu}\left(\MATmat,\Vabond{\nopix},\Vnabond{\nopix}\right)
&\triangleq b_p \left(\MATmat\Vabond{p}\right) \odot
\left(\MATmat\Vabond{p}\right)\\
&=  b_p \sum_{i=1}^R\sum_{j=1}^R \abond{i}{\nopix} \abond{j}{\nopix} \Vmat{i}\odot \Vmat{j}
\end{split}$$ leading to the following polynomial post-nonlinear mixing model (PPNM) $$\Vpix{\nopix}^{(\textrm{PPNM})} = \MATmat\Vabond{p} + b_p \sum_{i=1}^R\sum_{j=1}^R 