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 .

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 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. • A combinatorial perspective on the Neighborhood Grid spatial data structure • Introducing a polynomial time construction of stable states ( Theorem 3.1 ) • Analyzing the run time of our construction ( Theorem 4.3 and Corollary 4.4 ) • Disproving an implicit uniqueness assumption of previous work ( Proposition 5.5 ) • Sharing several enumerative open problems and conjectures

The need to understand the structure of hierarchical or high-dimensional data is present in a variety of fields. Hyperbolic spaces have proven to be an important tool for embedding computations and analysis tasks as their non-linear nature lends itself well to tree or graph data. Subsequently, they have also been used in the visualization of high-dimensional data, where they exhibit increased embedding performance. However, none of the existing dimensionality reduction methods for embedding into hyperbolic spaces scale well with the size of the input data. That is because the embeddings are computed via iterative optimization schemes and the computation cost of every iteration is quadratic in the size of the input. Furthermore, due to the non-linear nature of hyperbolic spaces, euclidean acceleration structures cannot directly be translated to the hyperbolic setting. This article introduces the first acceleration structure for hyperbolic embeddings, building upon a polar quadtree. We compare our approach with existing methods and demonstrate that it computes embeddings of similar quality in significantly less time. Implementation and scripts for the experiments can be found at https://graphics.tudelft.nl/accelerating-hyperbolic-tsne.

In this paper, we define and evaluate a weighting scheme for neighborhoods in point sets. Our weighting takes the shape of the geometry, i.e., the normal information, into account. This causes the obtained neighborhoods to be more reliable in the sense that connectivity also depends on the orientation of the point set. We utilize a sigmoid to define the weights based on the normal variation. For an evaluation of the weighting scheme, we turn to a Shannon entropy model for feature classification that can be proven to be non-degenerate for our family of weights. Based on this model, we evaluate our weighting terms on a large scale of both clean and real-world models. This evaluation provides results regarding the choice of optimal parameters within our weighting scheme. Furthermore, the large-scale evaluation also reveals that neighborhood sizes should not be fixed globally when processing models. Finally, we highlight the applicability of our weighting scheme within the application context of denoising.