of channels is given by $C$. Again, the attenuations are known beforehand in the sets $\mathcal{R}_{E_1} = \{\rho_{1,1}, \ldots, \rho_{1,m}\}, \mathcal{R}_{E_2} = \{\rho_{2,1}, \ldots, \rho_{2,m}\}, \ldots, \mathcal{R}_{E_C} = \{\rho_{C,1} \ldots, \rho_{C,m}\}$. In this setting, let $\vec{\mathcal{R}} = \cup \mathcal{R}_{E_c}$. The function $\mu: \mathcal{M} \times \mathcal{E} \mapsto \vec{\mathcal{R}}$ maps the label-channel combinations to their attenuation value, so the attenuation of a material with label $s$ at channel $E_c$ is given by $\mu(s,E_c)$. Note that there is not necessarily a one-to-one correspondence between the attenuation values and the material-channel combinations, because some combinations can have the same attenuation value. In this multi-channel case the projection data is given by a vector of projection data vectors at various channels: $$\begin{aligned}
\vec{P} &= (\vec{p}_{E_1}, \ldots, \vec{p}_{E_C}) \in \mathbb{R}^{n\times C}. \label{Eq:ProjData}.\end{aligned}$$ For each channel $E_c$ the reconstruction problem for $\vec{x}_{E_c}$ is given by the following set of linear equations: $$\begin{aligned}
\vec{W}\vec{x}_{E_c} &= \vec{p}_{E_c}, && E_c \in \{E_1, \ldots E_C\}. \label{Eq:InvProblemMC}\end{aligned}$$ For $\vec{y} \in \mathcal{M}^n$, define $\mu(\vec{y}, E_h) = (\mu(\vec{y}_1, E_h), \ldots, \mu(\vec{y}_n, E_h))^\top$ as the vector of voxel attenuation values at channel $E_h$. The multi-channel problem is now defined as follows. Given data vector $\vec{P}$ and projection matrix $\vec{W}$, find a labeling vector $\vec{y}^* \in 