On Some New Results in Discrete Tomography.

Discrete tomography deals with the reconstruction of finite sets from
knowledge about their interaction with certain query sets. The most
prominent example is that of the reconstruction of a finite subset $F$ of
$ mathbb{Z}^d$ from its X-rays (i.e., line sums) in a small positive
integer number $m$ of directions.
Applications of discrete tomography include quality control in
semiconductor industry, image processing, scheduling, and statistical data
security.
The reconstruction task is an ill-posed discrete inverse problem, depicting
(suitable variants of) all three Hadamard criteria for ill-posedness.


After a short introduction to the field of discrete tomography,
the first part of the talk addresses the following questions.
Does discrete tomography have the power of error correction?
Can noise be compensated by taking more X-ray images, and, if
so, what is the quantitative effect of taking one more X-ray?
Our main theoretical result gives the first nontrivial unconditioned
(and best possible) stability result. On the algorithmic side we show that
while there always is a certain inherent stability, the possibility of
making (worst-case) efficient use of it is rather limited.


The second part of the talk deals with the discrete tomography of
quasicrystals that live on finitely generated $ Z$-modules in some
$ R^s$.
Focussing on aspects in which the discrete tomography of quasicrystals
differs from that in the classical lattice case, we solve a basic
decomposition problem for the discrete tomography of quasicystals. More
generally, we study the problem of existence of pseudodiophantine
solutions to certain systems of linear equations over the reals and give a
complete characterization of when the index of Siegel grids is finite.


The results on stability are joint work with Andreas Alpers, that on
Siegel grids are joint work with Barbara Langfeld.