The reconstruction of an unknown function $f$ from its line sums is the aim discrete tomography. However, two main aspects prevent being easy task. In general, many solutions are allowed due to presence switching functions. Even when uniqueness conditions available, results about NP-hardness algorithms make their implementation inefficient values in certain sets. We show that this not case take...

In this chapter we present an algebraic theory of patterns which can be applied in discrete tomography for any dimension. We use that the difference of two such patterns yields a configuration with vanishing line sums. We show by introducing generating polynomials and applying elementary properties of polyno-mials that such so-called switching configurations form a linear space. We give a basis...

Tomography tries to reconstruct an object from a number of projections in multiple directions. Many application domains can be imagined, but we will focus on high throughput applications, and will therefore try to reduce the number of necessary projections, while being able to generate good quality reconstructions. We apply several forms of Neural Networks, an Artificial Intelligence method. Th...

The discrete tomography of mathematical quasicrystals with icosahedral symmetry is investigated, placing emphasis on reconstruction and uniqueness problems. The work is motivated by the requirement in materials science for the unique reconstruction of the structures of icosahedral quasicrystals from a small number of images produced by quantitative high-resolution transmission electron microscopy.

A new algorithm for reconstructing a two dimensional object from a set of one dimensional projected views is presented that is both computationally exact and experimentally practical. The algorithm has a computational complexity of O(n log2 n) with n = N2 for an N × N image, is robust in the presence of noise and produces no artefacts in the reconstruction process, as is the case with conventio...

Statistical methods of discrete tomographic reconstruction pose new problems both in stochastic modeling to define an optimal reconstruction, and in optimization to find that reconstruction. Multiscale models have succeeded in improving representation of structure of varying scale in imagery, a chronic problem for common Markov random fields. This chapter shows that associated multiscale method...

The paper gives strong instability results for a basic reconstruction problem of discrete tomography, an area that is particularly motivated by demands from material sciences for the reconstruction of crystalline structures from images produced by quantitative high resolution transmission electron microscopy. In particular, we show that even extremely small changes in the data may lead to entir...

In this report, we present a number-theory-based approach for discrete tomography (DT),which is based on parallel projections of rational slopes. Using a well-controlled geometry of X-ray beams, we obtain a system of linear equations with integer coefficients. Assuming that the range of pixel values is a(i, j) = 0, 1, . . . , M − 1, with M being a prime number, we reduce the equations modulo M ...

In earlier papers we have developed an algebraic theory of discrete tomography. In those papers the structure of the functions f : A → {0, 1} and f : A → Z having given line sums in certain directions have been analyzed. Here A was a block in Z with sides parallel to the axes. In the present paper we assume that there is noise in the measurements and (only) that A is an arbitrary or convex fini...