Related papers: Uniqueness in Discrete Tomography of Planar Model …
We consider algebraic Delone sets $\varLambda$ in the Euclidean plane and address the problem of distinguishing convex subsets of $\varLambda$ by X-rays in prescribed $\varLambda$-directions, i.e., directions parallel to nonzero interpoint…
We address the problem of determining finite subsets of Delone sets $\varLambda\subset\R^d$ with long-range order by $X$-rays in prescribed $\varLambda$-directions, i.e., directions parallel to non-zero interpoint vectors of $\varLambda$.…
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…
We report recent progress in the problem of distinguishing convex subsets of cyclotomic model sets $\varLambda$ by (discrete parallel) X-rays in prescribed $\varLambda$-directions. It turns out that for any of these model sets $\varLambda$…
We consider the problem of distinguishing convex subsets of $n$-cyclotomic model sets $\varLambda$ by (discrete parallel) X-rays in prescribed $\varLambda$-directions. In this context, a `magic number' $m_{\varLambda}$ has the property that…
For a finite set $U$ of directions in the Euclidean plane, a convex non-degenerate polygon $P$ is called a $U$-polygon if every line parallel to a direction of $U$ that meets a vertex of $P$ also meets another vertex of $P$. We characterize…
Scheme-theoretic methods are used to classify ternary quadratic forms with values in line bundles over arbitrary schemes and to canonically determine the isomorphisms between them. The association of a quadratic bundle to its even Clifford…
The affinely regular polygons in certain planar sets are characterized. It is also shown that the obtained results apply to cyclotomic model sets and, additionally, have consequences in the discrete tomography of these sets.
We study tilings of polygons $R$ with arbitrary convex polygonal tiles. Such tilings come in continuous families obtained by moving tile edges parallel to themselves (keeping edge directions fixed). We study how the tile shapes and areas…
The aim of this paper is to present a new method of approximation of planar data set using only arcs or segments. The first problem we are trying to solve is the following: the CNC machines can work only with simple curves (arcs or…
We develop criteria for deciding the contravariant finiteness status of a subcategory $A \subseteq \Lambda\text{-mod}$, where $\Lambda$ is a finite dimensional algebra. In particular, given a finite dimensional $\Lambda$-module $X$, we…
We extend the classical LR characterization of chirotopes of finite planar families of points to chirotopes of finite planar families of pairwise disjoint convex bodies: a map \c{hi} on the set of 3-subsets of a finite set I is a chirotope…
In this work, we present an end-to-end deep learning framework for X-ray image diagnosis. As the first step, our system determines whether a submitted image is an X-ray or not. After it classifies the type of the X-ray, it runs the…
The relation of s-convexity and sets modeling physical quasicrystals is explained for quasicrystals related to quadratic unitary Pisot numbers. We show that 1-dimensional model sets may be characterized by s-convexity for finite set of…
Multivariate distributions are fundamental to modeling. Discrete copulas can be used to construct diverse multivariate joint distributions over random variables from estimated univariate marginals. The space of discrete copulas admits a…
Tilings of the plane resemble the simplicial and other complexes from algebraic topology, but have not been studied from this perspective. We construct finite categories corresponding to polygons with labeled directed edges, and introduce…
We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This provides a generalization of the enumeration of…
Segmentation of overlapping convex objects has various applications, for example, in nanoparticles and cell imaging. Often the segmentation method has to rely purely on edges between the background and foreground making the analyzed images…
The problem of classifying high-dimensional shapes in real-world data grows in complexity as the dimension of the space increases. For the case of identifying convex shapes of different geometries, a new classification framework has…
An increasing number of public datasets have shown a marked impact on automated organ segmentation and tumor detection. However, due to the small size and partially labeled problem of each dataset, as well as a limited investigation of…