the set $\mathcal{R} = \{\rho_1, \ldots, \rho_m\}$, where $m$ is the number of different materials in the object. Therefore the problem to be solved becomes finding a vector $\vec{x} \in \mathcal{R}^n$ that matches the data best: $$\begin{aligned}
\vec{x}^* = \operatorname*{arg\,min}_{\vec{x} \in \mathcal{R}^n} || \vec{W}\vec{x} - \vec{p} ||. \label{Eq:DiscOptProblem}\end{aligned}$$ Note that this is a minimization problem over a non-empty finite set. Hence, a minimum always exists. Again, it does not need to be unique but we use this notation throughout the paper for simplicity.

In the *multi-channel* setting different properties of the target can be individually interrogated and measured. The information of each property is obtained through a separate channel. An example of channel is an energy level, as in the example in Section \[sec:Intro\]. In Figure \[fig:1b\], the channels are the two energy levels revealing attenuations of the object at different energies. In a more abstract way the object is described as set a of voxels with labels instead of attenuation values, since each material has different attenuation values in different channels. The material labels are values in the set $\mathcal{M} = \{1, 2,\ldots, m\}$. The channel indices are given by $\mathcal{E} = \{E_1, E_2, \ldots, E_C\}$ where the number 