149 research outputs found
The smallest sets of points not determined by their X-rays
Let be an -point set in with
and . A (discrete) X-ray of
in direction gives the number of points of on each line parallel to
. We define as the minimum number for which
there exist directions (pairwise linearly independent and
spanning ) such that two -point sets in exist
that have the same X-rays in these directions. The bound
has been observed many times in the
literature. In this note we show
for . For the
cases and , , this
represents the first upper bound on that is polynomial
in . As a corollary we derive bounds on the sizes of solutions to both the
classical and two-dimensional Prouhet-Tarry-Escott problem. Additionally, we
establish lower bounds on that enable us to prove a
strengthened version of R\'enyi's theorem for points in
3D particle tracking velocimetry using dynamic discrete tomography
Particle tracking velocimetry in 3D is becoming an increasingly important
imaging tool in the study of fluid dynamics, combustion as well as plasmas. We
introduce a dynamic discrete tomography algorithm for reconstructing particle
trajectories from projections. The algorithm is efficient for data from two
projection directions and exact in the sense that it finds a solution
consistent with the experimental data. Non-uniqueness of solutions can be
detected and solutions can be tracked individually
Generalized balanced power diagrams for 3D representations of polycrystals
Characterizing the grain structure of polycrystalline material is an
important task in material science. The present paper introduces the concept of
generalized balanced power diagrams as a concise alternative to voxelated
mappings. Here, each grain is represented by (measured approximations of) its
center-of-mass position, its volume and, if available, by its second-order
moments (in the non-equiaxed case). Such parameters may be obtained from 3D
x-ray diffraction. As the exact global optimum of our model results from the
solution of a suitable linear program it can be computed quite efficiently.
Based on verified real-world measurements we show that from the few parameters
per grain (3, respectively 6 in 2D and 4, respectively 10 in 3D) we obtain
excellent representations of both equiaxed and non-equiaxed structures. Hence
our approach seems to capture the physical principles governing the forming of
such polycrystals in the underlying process quite well
Citizen Empowerment by a Technical Approach for Privacy Enforcement
It is a fundamental right of every natural person to control which personal information is collected, stored and processed by whom, for what purposes and how long. In fact, many (cloud based) services can only be used if the user allows them broad data collection and analysis. Often, users can only decide to either give their data or not to participate in communities. The refusal to provide personal data results in significant drawbacks for social interaction. That is why we believe that there is a need for tools to control one\u27s own data in an easy and effective way as protection against economic interest of global companies and their cloud computing systems (as data collector from apps, mobiles and services). Especially, as nowadays everybody is permanently online using different services and devices, users are often lacking the means to effectively control the access to their private data. Therefore, we present an approach to manage and distribute privacy settings: PRIVACY-AVARE is intended to enable users to centrally determine their data protection preferences and to apply them on different devices. Thus, users gain control over their data when using cloud based services. In this paper, we present the main idea of PRIVACY-AVARE
Literature Survey on how to cluster and define Living Labs, Real World Laboratories and similar research infrastructures
In today\u27s world, where societal challenges in the areas of digitalization, demographic change and sustainability are becoming increasingly complex, new innovation structures are needed to meet these challenges. Living Labs or also Real World Laboratories prove to be such. Through their applied methods such as co-creation, they integrate users into research, making it more user-centric. Which other research infrastructures exist and how they can be differentiated is presented in this paper on the basis of a systematic literature research. Furthermore, methods for user integration are examined and provided in the form of an overview
ON DOUBLE-RESOLUTION IMAGING AND DISCRETE TOMOGRAPHY
Super-resolution imaging aims at improving the resolution of an image by
enhancing it with other images or data that might have been acquired using
different imaging techniques or modalities. In this paper we consider the task
of doubling, in each dimension, the resolution of grayscale images of binary
objects by fusion with double-resolution tomographic data that have been
acquired from two viewing angles. We show that this task is polynomial-time
solvable if the gray levels have been reliably determined. The problem becomes
-hard if the gray levels of some pixels come with an
error of or larger. The -hardness persists for any
larger resolution enhancement factor. This means that noise does not only
affect the quality of a reconstructed image but, less expectedly, also the
algorithmic tractability of the inverse problem itself.Comment: 26 pages, to appear in SIAM Journal on Discrete Mathematic
Reconstructing Binary Matrices underWindow Constraints from their Row and Column Sums
The present paper deals with the discrete inverse problem of reconstructing
binary matrices from their row and column sums under additional constraints on
the number and pattern of entries in specified minors. While the classical
consistency and reconstruction problems for two directions in discrete
tomography can be solved in polynomial time, it turns out that these window
constraints cause various unexpected complexity jumps back and forth from
polynomial-time solvability to -hardness
Dynamic discrete tomography
We consider the problem of reconstructing the paths of a set of points over
time, where, at each of a finite set of moments in time the current positions
of points in space are only accessible through some small number of their
X-rays. This particular particle tracking problem, with applications, e.g., in
plasma physics, is the basic problem in dynamic discrete tomography. We
introduce and analyze various different algorithmic models. In particular, we
determine the computational complexity of the problem (and various of its
relatives) and derive algorithms that can be used in practice. As a byproduct
we provide new results on constrained variants of min-cost flow and matching
problems.Comment: In Pres
On the Reconstruction of Static and Dynamic Discrete Structures
We study inverse problems of reconstructing static and dynamic discrete structures from tomographic data (with a special focus on the `classical' task of reconstructing finite point sets in ). The main emphasis is on recent mathematical developments and new applications, which emerge in scientific areas such as physics and materials science, but also in inner mathematical fields such as number theory, optimization, and imaging. Along with a concise introduction to the field of discrete tomography, we give pointers to related aspects of computerized tomography in order to contrast the worlds of continuous and discrete inverse problems
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