7 research outputs found
On the reconstruction of binary and permutation matrices under (binary) tomographic constraints
AbstractThe paper studies the problem of reconstructing binary matrices constrained by binary tomographic information. We prove new NP-hardness results that sharpen previous complexity results in the realm of discrete tomography but also allow applications to related problems for permutation matrices. Hence our results can be interpreted in terms of other combinatorial problems including the queens’ problem
Solution of a uniqueness problem in the discrete tomography of algebraic Delone sets
We consider algebraic Delone sets in the Euclidean plane and
address the problem of distinguishing convex subsets of by X-rays
in prescribed -directions, i.e., directions parallel to nonzero
interpoint vectors of . Here, an X-ray in direction of a finite
set gives the number of points in the set on each line parallel to . It is
shown that for any algebraic Delone set there are four prescribed
-directions such that any two convex subsets of can be
distinguished by the corresponding X-rays. We further prove the existence of a
natural number such that any two convex subsets of
can be distinguished by their X-rays in any set of
prescribed -directions. In particular, this
extends a well-known result of Gardner and Gritzmann on the corresponding
problem for planar lattices to nonperiodic cases that are relevant in
quasicrystallography.Comment: 21 pages, 1 figur
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 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
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