arXiv subjects

Peter Gritzmann

Publications and source records attributed to Peter Gritzmann.

14 recordsLinked to original sources

arXiv2022

Turning grain maps into diagrams

The present paper studies mathematical models for representing, imaging, and analyzing polycrystalline materials. We introduce various techniques for converting grain maps into diagram or tessellation representations that rely on constrained clustering. In particular, we show how to significantly accelerate the generalized balanced power diagram method from [1] and how to extend it to allow for optimization over all relevant parameters. A comparison of the accuracies of the proposed approaches is given based on a 3D real-world data set of $339\times 339 \times 599$ voxels.

math.NA↗

arXiv2022

Dynamic grain models via fast heuristics for diagram representations

The present paper introduces a mathematical model for studying dynamic grain growth. In particular, we show how characteristic measurements, grain volumes, centroids, and central second-order moments at discrete moments in time can be turned quickly into a continuous description of the grain growth process in terms of geometric diagrams (which largely generalize the well-known Voronoi and Laguerre tessellations). We evaluate the computational behavior of our algorithm on real-world data.

physics.comp-ph↗

arXiv2022

On resolution coresets for constrained clustering

Specific data compression techniques, formalized by the concept of coresets, proved to be powerful for many optimization problems. In fact, while tightly controlling the approximation error, coresets may lead to significant speed up of the computations and hence allow to extend algorithms to much larger problem sizes. The present paper deals with a weight-balanced clustering problem from imaging in materials science. Here, the class of desired coresets is naturally confined to those which can be viewed as lowering the resolution of the data. Hence one would expect that such resolution coresets are inferior to unrestricted coreset. We show, however, that the restrictions are more than compensated by utilizing the underlying structure of the data. In particular, we prove bounds for resolution coresets which improve known bounds in the relevant dimensions and also lead to significantly faster algorithms practice.

math.OC↗

arXiv2022

Coresets for Weight-Constrained Anisotropic Assignment and Clustering

The present paper constructs coresets for weight-constrained anisotropic assignment and clustering. In contrast to the well-studied unconstrained least-squares clustering problem, approximating the centroids of the clusters no longer suffices in the weight-constrained anisotropic case, as even the assignment of the points to best sites is involved. This assignment step is often the limiting factor in materials science, a problem that partially motivates our work. We build on a paper by Har-Peled and Kushal, who constructed coresets of size $\mathcal{O}\bigl(\frac{k^3}{\epsilon^{d+1}}\bigr)$ for unconstrained least-squares clustering. We generalize and improve on their results in various ways, leading to even smaller coresets with a size of only $\mathcal{O}\bigl(\frac{k^2}{\epsilon^{d+1}}\bigr)$ for weight-constrained anisotropic clustering. Moreover, we answer an open question on coreset designs in the negative, by showing that the total sensitivity can become as large as the cardinality of the original data set in the constrained case. Consequently, many techniques based on importance sampling do not apply to weight-constrained clustering.

cs.DS↗

arXiv2018

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.

cs.DS↗

arXiv2018

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.

cs.DS↗

arXiv2017

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.

cs.DS↗

arXiv2017

On the reverse Loomis-Whitney inequality

The present paper deals with the problem of computing (or at least estimating) the LW-number $λ(n)$, i.e., the supremum of all $γ$ such that for each convex body $K$ in $\mathbb{R}^n$ there exists an orthonormal basis $\{u_1,\ldots,u_n\}$ such that $$ vol_n(K)^{n-1} \geq γ\prod_{i=1}^n vol_{n-1} (K|u_i^{\perp}) , $$ 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 $λ(n)$ and deal with the problem of actually computing the LW-constant of a rational polytope.

math.MG↗

arXiv2017

Constrained clustering via diagrams: A unified theory and its applications to electoral district design

The paper develops a general framework for constrained clustering which is based on the close connection of geometric clustering and diagrams. Various new structural and algorithmic results are proved (and known results generalized and unified) which show that the approach is computationally efficient and flexible enough to pursue various conflicting demands. The strength of the model is also demonstrated practically on real-world instances of the electoral district design problem where municipalities of a state have to be grouped into districts of nearly equal population while obeying certain politically motivated requirements.

cs.DS↗

arXiv2017

Reconstructing binary matrices under window 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.

cs.DS↗

arXiv2016

A balanced k-means algorithm for weighted point sets

The classical $k$-means algorithm for partitioning $n$ points in $\mathbb{R}^d$ into $k$ clusters is one of the most popular and widely spread clustering methods. The need to respect prescribed lower bounds on the cluster sizes has been observed in many scientific and business applications. In this paper, we present and analyze a generalization of $k$-means that is capable of handling weighted point sets and prescribed lower and upper bounds on the cluster sizes. We call it weight-balanced $k$-means. The key difference to existing models lies in the ability to handle the combination of weighted point sets with prescribed bounds on the cluster sizes. This imposes the need to perform partial membership clustering, and leads to significant differences. For example, while finite termination of all $k$-means variants for unweighted point sets is a simple consequence of the existence of only finitely many partitions of a given set of points, the situation is more involved for weighted point sets, as there are infinitely many partial membership clusterings. Using polyhedral theory, we show that the number of iterations of weight-balanced $k$-means is bounded above by $n^{O(dk)}$, so in particular it is polynomial for fixed $k$ and $d$. This is similar to the known worst-case upper bound for classical $k$-means for unweighted point sets and unrestricted cluster sizes, despite the much more general framework. We conclude with the discussion of some additional favorable properties of our method.

math.OC↗

arXiv2015

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.

cond-mat.mtrl-sci↗

arXiv2015

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.

physics.flu-dyn↗

arXiv2013

Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving

We derive efficient algorithms for coarse approximation of algebraic hypersurfaces, useful for estimating the distance between an input polynomial zero set and a given query point. Our methods work best on sparse polynomials of high degree (in any number of variables) but are nevertheless completely general. The underlying ideas, which we take the time to describe in an elementary way, come from tropical geometry. We thus reduce a hard algebraic problem to high-precision linear optimization, proving new upper and lower complexity estimates along the way.

math.AG↗