set of linear equations for $\vec{x}$: $$\begin{aligned}
\vec{W}\vec{x} &= \vec{p} \label{Eq:OptProblem}.\end{aligned}$$ Here $\vec{W}$ is the projection matrix, also called the forward operator [@KakSlaney]. This matrix incorporates the contribution of each voxel to each projection, where element $w_{ij}$ indicates the contribution of voxel $j$ to projection $i$. Applying the operator $\vec{W}$ on a vector $\vec{x}$ results in the forward projection (also called sinogram). Since inverting the matrix $\vec{W}$ is computationally too expensive (or not even possible, for example when the problem is ill-posed) the reconstruction problem is to find a solution $\vec{x}^*$ whose forward projection $\vec{W}\vec{x}^*$ matches the projection data best with respect to some norm $||\cdot||$. $$\begin{aligned}
\vec{x}^* = \operatorname*{arg\,min}_{\vec{x} \in \mathbb{R}^n} || \vec{W}\vec{x} - \vec{p} || \label{Eq:InvProblem}\end{aligned}$$ Since this is a least squares problem over $\mathbb{R}^n$, a solution always exists. For simplicity of notation we also assume that it is unique. A vector that encapsulates noise from real-world examples can also be modeled with . In our experiments with phantom examples in Section \[sec:Experiments\] there is no noise.

In the *discrete tomography problem*, the image to be reconstructed consists of a limited number of materials with homogeneous densities, each having an attenuation which is known beforehand by means of 