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Discrete tomography deals with the reconstruction of images from a (usually small) set of X-ray projections. This is achieved by modeling the tomographic problem as a linear system of equations and then applying a suitable discrete reconstruction algorithm based on iterations. In this paper we adopt the well-known grid model and prove some geometric properties of integer solutions consisting of $p\geq 2$ gray levels. In particular, we show that all gray-scale solutions having the same two-norm belong to a same hypersphere, centered at the uniform image related to the data and having radius ranging in an interval whose bounds are explicitly computed. Moving from a uniqueness theorem for gray-scale images, we compute special sets of directions that guarantee uniqueness of reconstruction and exploit them as the input of the Conjugate Gradient Least Squares algorithm. Then we apply an integer rounding to the resulting output and, basing on previously described geometric parameters, we test the quality of the obtained reconstructions for an increasing number of iterations, which leads to a progressive improvement of the percentage of correctly reconstructed pixels, until perfect reconstruction is achieved. Differently, using sets of directions which are classically employed, but far from being sets of uniqueness, only partial reconstructions are obtained.

For a point of the projective space $PG(n,q)$, its R'edei factor is the linear polynomial in $n+1$ variables, whose coefficients are the point coordinates. The power sum polynomial of a subset $S$ of $PG(n,q)$ is the sum of the $(q-1)$-th powers of the R'edei factors of the points of $S$. The fact that many subsets may share the same power sum polynomial offers a natural connection to discrete tomography. In this paper we deal with the two-dimensional case and show that the notion of ghost, whose employment enables to find all solutions of the tomographic problem, can be rephrased in the finite geometry context, where subsets with null power sum polynomial are called ghosts as well. In the latter case, one can add ghosts still preserving the power sum polynomial by means of the multiset sum (modulo the field characteristic). We prove some general results on ghosts in $PG(2,q)$ and compute their number in case $q$ is a prime.

The goal of discrete tomography is to reconstruct an image, seen as a finite set of pixels, by knowing its projections along given directions. Uniqueness of reconstruction cannot be guaranteed in general, because of the existence of the switching components. Therefore, instead of considering the uniqueness problem for the whole image, in this paper we focus on local uniqueness, i.e., we seek what pixels have uniquely determined value. Two different kinds of local uniqueness are presented: one related to the structure of the directions and of the grid supporting the image, having as a sub-case the region of uniqueness (ROU), and the other one depending on the available projections. In the case when projections are taken along two lattice directions, both kinds of uniqueness have been characterized in a graph-theoretical reformulation. This paper is intended to be a starting point in the construction of connections between pixels with uniquely determined value and graphs.