"fig:"){width="35.00000%"}]{} \[0.35\][ ![Elementary example of separation difficulties during segmentation. (a) Distribution of the three materials (blue, yellow, green) in the object. The background is indicated in red. (b) Histogram of attenuation values of pixels at energy $E_1$ (above) and $E_2$ (below). Vertical lines show true material attenuations. (c) Attenuations of the materials (red dots) and computed attenuations by a reconstruction algorithm for each voxel (colors indicate the material they belong to).[]{data-label="fig:1"}](figure4.eps "fig:"){width="35.00000%"}]{}

\[fig:Separation\]

Problem Formulation {#sec:Problems}
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The standard (single-channel) tomography problem can be modeled as a system of linear equations. The image is characterized by a vector of voxel attenuation values $\vec{x} \in \mathbb{R}^n$, where $n$ is the number of voxels. We will work with 2D images, but the problem formulation and methods in this paper can easily be extended to the 3D setting. We will refer to the image pixels as *voxels* to distinguish these from detector pixels. We will often interchangeably speak of voxels and their corresponding indices. The projection values (also called data) are given as the vector $\vec{p} \in \mathbb{R}^l$, where $l$ is the number of projection angles times the number of detector pixels. The reconstruction problem can then be described by solving the following 