arXiv subjects

Ulrich Reitebuch

Publications and source records attributed to Ulrich Reitebuch.

6 recordsLinked to original sources

arXiv2023

Combinatorial and Asymptotical Results on the Neighborhood Grid

In various application fields, such as fluid-, cell-, or crowd-simulations, spatial data structures are very important. They answer nearest neighbor queries which are instrumental in performing necessary computations for, e.g., taking the next time step in the simulation. Correspondingly, various such data structures have been developed, one being the \emph{neighborhood grid}. In this paper, we consider combinatorial aspects of this data structure. Particularly, we show that an assumption on uniqueness, made in previous works, is not actually satisfied. We extend the notions of the neighborhood grid to arbitrary grid sizes and dimensions and provide two alternative, correct versions of the proof that was broken by the dissatisfied assumption. Furthermore, we explore both the uniqueness of certain states of the data structure as well as when the number of these states is maximized. We provide a partial classification by using the hook-length formula for rectangular Young tableaux. Finally, we conjecture how to extend this to all 2-dimensional cases.

math.CO↗

arXiv2023

Direct Construction of Aperiodic Tilings with the Hat Monotile

In 2023, the quest for an aperiodic monotile was answered by the hat monotile. In this article, structures in this aperiodic tiling are discovered, which allow for a direct computation of the tiling, similar to well-known methods for the Penrose tilings.

math.CO↗

arXiv2023

Feature-aware manifold meshing and remeshing of point clouds and polyhedral surfaces with guaranteed smallest edge length

Point clouds and polygonal meshes are widely used when modeling real-world scenarios. Here, point clouds arise, for instance, from acquisition processes applied in various surroundings, such as reverse engineering, rapid prototyping, or cultural preservation. Based on these raw data, polygonal meshes are created to, for example, run various simulations. For such applications, the utilized meshes must be of high quality. This paper presents an algorithm to derive triangle meshes from unstructured point clouds. The occurring edges have a close to uniform length and their lengths are bounded from below. Theoretical results guarantee the output to be manifold, provided suitable input and parameter choices. Further, the paper presents several experiments establishing that the algorithms can compete with widely used competitors in terms of quality of the output and timing and the output is stable under moderate levels of noise. Additionally, we expand the algorithm to detect and respect features on point clouds as well as to remesh polyhedral surfaces, possibly with features. Supplementary material, an extended preprint, a link to a previously published version of the article, utilized models, and implementation details are made available online: https://ms-math-computer.science/projects/guaranteed-smallest-edge-length-manifold-meshing.html

cs.CG↗

arXiv2022

Weaving patterns inspired by the pentagon snub subdivision scheme

Various computer simulations regarding, e.g., the weather or structural mechanics, solve complex problems on a two-dimensional domain. They mostly do so by splitting the input domain into a finite set of smaller and simpler elements on which the simulation can be run fast and efficiently. This process of splitting can be automatized by using subdivision schemes. Given the wide range of simulation problems to be tackled, an equally wide range of subdivision schemes is available. They create subdivisions that are (mainly) comprised of triangles, quadrilaterals, or hexagons. Furthermore, they ensure that (almost) all vertices have the same number of neighboring vertices. This paper illustrates a subdivision scheme that splits the input domain into pentagons. Repeated application of the scheme gives rise to fractal-like structures. Furthermore, the resulting subdivided domain admits to certain weaving patterns. These patterns are subsequently generalized to several other subdivision schemes. As a final contribution, we provide paper models illustrating the weaving patterns induced by the pentagonal subdivision scheme. Furthermore, we present a jigsaw puzzle illustrating both the subdivision process and the induced weaving pattern. These transform the visual and abstract mathematical algorithms into tactile objects that offer exploration possibilities aside from the visual.

cs.GR↗

arXiv2020

Experimental visually-guided investigation of sub-structures in three-dimensional Turing-like patterns

In his 1952 paper "The chemical basis of morphogenesis", Alan M. Turing presented a model for the formation of skin patterns. While it took several decades, the model has been validated by finding corresponding natural phenomena, e.g. in the skin pattern formation of zebrafish. More surprising, seemingly unrelated pattern formations can also be studied via the model, like e.g. the formation of plant patches around termite hills. In 1984, David A. Young proposed a discretization of Turing's model, reducing it to an activator/inhibitor process on a discrete domain. From this model, the concept of three-dimensional Turing-like patterns was derived. In this paper, we consider this generalization to pattern-formation in three-dimensional space. We are particularly interested in classifying the different arising sub-structures of the patterns. By providing examples for the different structures, we prove a conjecture regarding these structures within the setup of three-dimensional Turing-like pattern. Furthermore, we investigate - guided by visual experiments - how these sub-structures are distributed in the parameter space of the discrete model. We found two-fold versions of zero- and one-dimensional sub-structures as well as two-dimensional sub-structures and use our experimental findings to formulate several conjectures for three-dimensional Turing-like patterns and higher-dimensional cases.

q-bio.NC↗

arXiv2017

Robust and High Fidelity Mesh Denoising

This paper presents a simple and effective two-stage mesh denoising algorithm, where in the first stage, the face normal filtering is done by using the bilateral normal filtering in the robust statistics framework. Tukey's bi-weight function is used as similarity function in the bilateral weighting, which is a robust estimator and stops the diffusion at sharp edges to retain features and removes noise from flat regions effectively. In the second stage, an edge weighted Laplace operator is introduced to compute a differential coordinate. This differential coordinate helps the algorithm to produce a high-quality mesh without any face normal flips and makes the method robust against high-intensity noise.

cs.GR↗