\Vnoise{\nopix}$$ with $$\label{eq:constraints_NLMM}
  \left\{
   \begin{array}{l}
     \abond{\nomat}{\nopix}\geq 0, \quad \forall \nomat,\ \forall \nopix \\
     \nabond{i,j}{\nopix}\geq 0, \quad \forall \nomat,\ \forall i\neq j \\
     \sum_{r=1}^{\nbmat} \abond{\nomat}{\nopix}+
     \sum_{i=1}^{\nbmat-1}\sum_{j=i+1}^{\nbmat} \nabond{i,j}{\nopix}
     = 1, \ \ \forall \nopix.
   \end{array}
  \right.$$

Note that this model, denoted NM for Nascimento’s model in this article, reduces to the LMM when $\nabond{i,j}{\nopix}= 0$, $\forall
i\neq j$. This is an interesting property since the LMM is known to be an admissible first approximation of the actually involved physical processes[^2]. However, in a more general case (i.e., $\nabond{i,j}{\nopix}\neq0$), the abundance coefficients $\left\{\abond{r}{\nopix}\right\}_{r=1}^R$ are not subject to the sum-to-one constraints defined in .

In [@Fan2009], Fan *et al.* have defined the nonlinearity coefficients $\nabond{i,j}{\nopix}$ as the product of the abundances, $\nabond{i,j}{\nopix}\triangleq \abond{i}{\nopix}
\abond{j}{\nopix}$, under the LMM-based constraints in , leading to the so-called Fan’s Model (FM) $$\label{eq:FM}
  \Vpix{\nopix}^{(\textrm{FM})} \triangleq \sum_{\nomat=1}^{\nbmat} \abond{\nomat}{\nopix}
  \Vmat{\nomat} + \sum_{i=1}^{R-1}\sum_{j=i+1}^{R} \abond{i}{\nopix} \abond{j}{\nopix} \Vmat{i}\odot \Vmat{j} +
  \Vnoise{\nopix}.$$ The main motivation for relating the amount of nonlinear interactions (governed by $\nabond{i,j}{\nopix}$) to the amount of linear contribution (governed by $\abond{i}{\nopix}$ and $\abond{j}{\nopix}$) is straightforward: the more a given material is present in the pixel, the more nonlinear interactions may occur. In particular, if a component $\Vmat{i}$ is absent in the $p$th 