On the reverse Loomis-Whitney inequality

The present paper deals with the problem of computing (or at least estimating) the LW-number $\lambda(n)$, i.e., the supremum of all $\gamma$ such that for each convex body $K$ in $\mathbb{R}^n$ there exists an orthonormal basis $\{u_1,\ldots,u_n\}$ such that $$vol_n(K)^{n-1} \geq \gamma \prod_{i=1}^n vol_{n-1} (K|u_i^{\perp}) ,$$ where $K|u_i^{\perp}$ denotes the orthogonal projection of $K$ onto the hyperplane $u_i^{\perp}$ perpendicular to $u_i$. Any such inequality can be regarded as a reverse to the well-known classical Loomis--Whitney inequality. We present various results on such reverse Loomis--Whitney inequalities. In particular, we prove some structural results, give bounds on $\lambda(n)$ and deal with the problem of actually computing the LW-constant of a rational polytope

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

Measurements of 3D slip velocities and plasma column lengths of a gliding arc discharge

A non-thermal gliding arc discharge was generated at atmospheric pressure in an air flow. The dynamics of the plasma column and tracer particles were recorded using two synchronized highspeed cameras. Whereas the data analysis for such systems has previously been performed in 2D (analyzing the single camera image), we provide here a 3D data analysis that includes 3D reconstructions of the plasma column and 3D particle tracking velocimetry based on discrete tomography methods. The 3D analysis, in particular, the determination of the 3D slip velocity between the plasma column and the gas flow, gives more realistic insight into the convection cooling process. Additionally, with the determination of the 3D slip velocity and the 3D length of the plasma column, we give more accurate estimates for the drag force, the electric field strength, the power per unit length, and the radius of the conducting zone of the plasma column. (C) 2015 AIP Publishing LLC

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

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 $\mathbb{N}\mathbb{P}$-hardness

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 $\mathbb{N}\mathbb{P}$-hard if the gray levels of some pixels come with an error of $\pm1$ or larger. The $\mathbb{N}\mathbb{P}$-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 $\mathbb{R}^d$). 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