the pixels with nonlinear mixing models, the abundance vector $\Vabond{\nopix}$ and the nonlinearity parameter vector $\Vnabond{\nopix}$ associated with each pixel $\Vpix{\nopix}$ are estimated by solving the following constrained optimization problem $$\label{eq:criterion_NLMM}
  \left(\hatVabond{\nopix}, \hatVnabond{\nopix}\right) =
  \operatornamewithlimits{argmin}_{\Vabond{\nopix},\Vnabond{\nopix}}
  \left\|\Vpix{\nopix} - \MATmat\Vabond{\nopix} -
  \boldsymbol{\mu}\left(\MATmat,\Vabond{\nopix},\Vnabond{\nopix}\right)\right\|_2^2.$$ Depending on the considered model, the set of constraints imposed to the abundance vector $\Vabond{\nopix}$ and the possible nonlinear coefficient vector $\Vnabond{\nopix}$ may differ. For the FM, GBM and PPNM, the abundance vector $\Vabond{\nopix}$ should satisfy the LMM-based constraints , while for the NM, this constraint is applied to the joint vector $\left[\Vabond{\nopix},\Vnabond{\nopix}\right]$. Similarly, the nonlinear coefficient vector $\Vnabond{\nopix}$ for the GBM and PPNM should satisfy constraints that depend on the considered model and the nonlinearity component $\boldsymbol{\mu}\left(\MATmat,\Vabond{\nopix},\Vnabond{\nopix}\right)$ in or depends also on the considered nonlinear model.

For the experimental results reported in Section \[sec:results\], the FCLS algorithm is used to solve the NM-based unmixing problem since NM can be interpreted as a linear mixture of an extended set of endmembers, as shown in [@Nascimento2009spie]. The FM parameters are estimated with the algorithm detailed in [@Fan2009], based on a first-order Taylor series expansion of the nonlinearity $\boldsymbol{\mu}\left(\MATmat,\Vabond{\nopix},\Vnabond{\nopix}\right)$. Finally, the gradient descent and the subgradient descend algorithms developed in [@Halimi2011igarss] and 