Barbara Langfeld’s research while affiliated with Kiel University and other places

The paper deals with an inverse problem of reconstructing matrices from their marginal sums. More precisely, we are interested in the existence of r×s matrices for which only the following information is available: The entries belong to known subsets of c distinguishable abelian groups, and the row and column sums of all entries from each group are given. This generalizes Ryser’s classical problem of characterizing the set of all 0–1-matrices with given row and column sums and is a basic problem in (polyatomic) discrete tomography. We show that the problem is closely related to packings of trees in bipartite graphs, prove consistency results, give algorithms and determine its complexity. In particular, we find a somewhat unusual complexity behavior: the problem is hard for “small” but easy for “large” matrices.

Two sets in $\mathbb{R}^d$ are called homometric if they have the same covariogram, where the covariogram of a finite subset K of $\mathbb{R}^d$ is the function associating to each $u \in \mathbb{R}^d$ the cardinality of $K \cap (K+u)$. Understanding the structure of homometric sets is important for a number of areas of mathematics and applications. If two sets are homometric but do not coincide up to translations and point reflections, we call them nontrivially homometric. We study nontrivially homometric pairs of lattice-convex sets, where a set K is called lattice-convex with respect to a lattice $\mathbb{M} \subseteq \mathbb{R}^d$ if K is the intersection of $\mathbb{M}$ and a convex subset of $\mathbb{R}^d$. This line of research was initiated in 2005 by Daurat, G\'erard and Nivat and, independently, by Gardner, Gronchi and Zong. All pairs of nontrivially homometric lattice-convex sets that have been known so far can be written as direct sums $S \oplus T$ and $S \oplus (-T)$, where T is lattice-convex and the underlying lattice is the direct sum of T and some sublattice. We study pairs of nontrivially homometric lattice-convex sets assuming this particular form and establish a necessary and a sufficient condition for the lattice-convexity of $S \oplus T$. This allows us to explicitly describe all nontrivially homometric pairs in dimension two, under the above assumption, and to construct examples of nontrivially homometric pairs of lattice-convex sets for each $d \ge 3$.

Discrete tomography is concerned with the retrieval of finite point sets in some ℝ d from their X-rays in a given number m of directions u 1 ,...,u m . In the present paper we focus on uniqueness issues. The first remark gives a uniform treatment and extension of known uniqueness results. In particular, we introduce the concept of J-additivity and give conditions when a subset J of possible positions is already determined by the given data. As a by-product, we settle a conjecture of Brunetti and Daurat on planar lattice convex sets. Remark 2 resolves a problem of Kuba posed in 1997 on the uniqueness in the case d=m=3 with u 1 ,u 2 ,u 3 being the standard unit vectors. Remark 3 determines the computational complexity of finding a smallest set J of positions whose disclosure yields uniqueness. As a corollary, we obtain a hardness result for 0-1-polytopes.

A finite subset K of ℤd is said to be lattice-convex if K is the intersection of ℤd with a convex set. The covariogram g K of K⊆ℤd is the function associating to each u∈ℤd the cardinality of K∩(K+u). Daurat, Gérard, and Nivat and independently Gardner, Gronchi, and Zong raised the problem of the reconstruction of lattice-convex sets K from g K . We provide a partial positive answer to this problem by showing that for d=2 and under mild extra assumptions, g K determines K up to translations and reflections. As a complement to the theorem on reconstruction we also extend the known counterexamples (i.e., planar lattice-convex sets which are not reconstructible, up to translations and reflections) to an infinite family of counterexamples.

We introduce the notion of cyclic convolutional codes and briefly survey some recent results that were derived with the aid of Gröbner-type theory.

We give a characterization of when the index of Siegel grids is finite. As a main application, we solve a basic decomposition problem for the discrete tomography of quasicrystals that live on finitely generated Z-modules in some Rs.

Die Arbeit untersucht drei grundlegende Fragen der Diskreten Tomographie auf Moduln. Im ersten Teil wird charakterisiert, unter welchen Bedingungen das vollständige tomographische Grid in endlich viele Translate des zu Grunde liegenden Moduls zerfällt. Im zweiten Teil wird ein geometrisches Separationsproblem studiert, das in natürlicher Weise bei der Rekonstruktion quasikristalliner Punktmengen aus X-Ray-Daten auftritt. Wir zeigen, wie sich das Separationsproblem in einem semialgebraischen Kontext algorithmisch effizient lösen lässt. Im dritten Teil untersuchen wir das Problem, eine minimale Anzahl an Gridpunkten zu finden, sodass die Fixierung dieser Punkte als (Nicht-)Positionen die eindeutige Rekonstruktion eines gegebenen Musters aus den X-Ray-Daten garantiert. Wir beweisen die NP-Schwere dieses Problems und leiten verwandte Eindeutigkeitsresultate für Polytope ab.

Discrete tomography is a well-established method to investigate finite point sets, in particular finite subsets of periodic systems. Here, we start to develop an efficient approach for the treatment of finite subsets of mathematical quasicrystals. To this end, the class of cyclotomic model sets is introduced, and the corresponding consistency, reconstruction and uniqueness problems of the discrete tomography of these sets are discussed.

Discrete tomography is a well-established method to investigate finite point sets, in particular finite subsets of periodic systems. Here, we start to develop an efficient approach for the treatment of finite subsets of mathematical quasicrystals. To this end, the class of cyclotomic model sets is introduced, and the corresponding consistency, reconstruction and uniqueness problems of the discrete tomography of these sets are discussed.

This text is a report on work progress. We introduce the class of cyclotomic model sets (mathematical quasicrystals) Λ ⊂ Z [ξn], where Z [ξn] is the ring of integers in the nth cyclotomic field Q (ξn), and discuss the corresponding decomposition, consistency and reconstruction problems of the discrete tomography of these sets. Our solution of the so-called decomposition problem also applies to the case of the square lattice Z2 = Z [ξ4], which corresponds to the classical setting of discrete tomography.