Snap Rounding with Restore: an Algorithm for Producing Robust Geometric Datasets

This paper presents a new algorithm called Snap Rounding with Restore (SRR), which aims to make ge- ometric datasets robust and to increase the quality of geometric approximation and the preservation of topological structure. It is based on the well-known Snap Rounding algorithm, but improves it by eliminat- ing from the snap rounded arrangement the configurations in which the distance between a vertex and a non-incident edge is smaller than half-the-width of a pixel of the rounding grid. Therefore, the goal of SRR is exactly the same as the goal of another algorithm, Iterated Snap Rounding (ISR), and of its evolution, Iterated Snap Rounding with Bounded Drift (ISRBD). However, SRR produces an output with a quality of approximation that is on average better than ISRBD, both under the viewpoint of the distance from the original segments and of the conservation of their topological structure. The paper also reports some cases where ISRBD, notwithstanding the bounded drift, produces strong topological modifications while SRR does not. A statistical analysis on a large collection of input datasets confirms these differences. It follows that the proposed Snap Rounding with Restore algorithm is suitable for applications that require both robustness, a guaranteed geometric approximation and a good topological approximation

Robustness of Spatial Relation Evaluation

In the last few years the amount of spatial data available through the network has increased both in volume and in heterogeneity, so that dealing with this huge amount of information has become an interesting new research challenge. In particular, spatial data is usually represented through a vector model upon which several spatial relations have been defined. Such relations represent the basic tools for querying spatial data and their robust evaluation in a distributed heterogeneous environment is an important issue to consider, in order to allow an effective usage of this kind of data. Among all possible spatial relations, this report considers the topological ones, since they are the most widely available in existing systems and represent the building blocks for the implementation of other spatial relations. The conditions and the operations needed to make a dataset robust w.r.t. topological interpretations strictly depends on the adopted evaluation model. In particular, this report considers an environment where two different eval- uation models for topological relations exist, one in which equality is based on identity of geometric primitives, and the other one where a tolerance in equality evaluation is introduced. Given such premises, the report proposes a set of rules for guaranteeing the robustness in both models, and discusses the applicability of available algorithms of the Snap Rounding family, in order to preserve robustness in case of perturbations

3D Snap Rounding

Let P be a set of n polygons in R^3, each of constant complexity and with pairwise disjoint interiors. We propose a rounding algorithm that maps P to a simplicial complex Q whose vertices have integer coordinates. Every face of P is mapped to a set of faces (or edges or vertices) of Q and the mapping from P to Q can be done through a continuous motion of the faces such that (i) the L_infty Hausdorff distance between a face and its image during the motion is at most 3/2 and (ii) if two points become equal during the motion, they remain equal through the rest of the motion. In the worst case, the size of Q is O(n^{15}) and the time complexity of the algorithm is O(n^{19}) but, under reasonable hypotheses, these complexities decrease to O(n^{5}) and O(n^{6}sqrt{n})

Establishing Robustness of a Spatial Dataset in a Tolerance-Based Vector Model

Spatial data are usually described through a vector model in which geometries are rep- resented by a set of coordinates embedded into an Euclidean space. The use of a finite representation, instead of the real numbers theoretically required, causes many robustness problems which are well-known in literature. Such problems are made even worst in a distributed context, where data is exchanged between different systems and several perturbations can be introduced in the data representation. In this context, a spatial dataset is said to be robust if the evaluation of the spatial relations existing among its objects can be performed in different systems, producing always the same result.In order to discuss the robustness of a spatial dataset, two implementation models have to be distinguished, since they determine different ways to evaluate the relations existing among geometric objects: the identity and the tolerance model. The robustness of a dataset in the identity model has been widely discussed in [Belussi et al., 2012, Belussi et al., 2013, Belussi et al., 2015a] and some algorithms of the Snap Rounding (SR) family [Hobby, 1999, Halperin and Packer, 2002, Packer, 2008, Belussi et al., 2015b] can be successfully applied in such context. Conversely, this problem has been less explored in the tolerance model. The aim of this paper is to propose an algorithm inspired by the ones of SR family for establishing or restoring the robustness of a vector dataset in the tolerance model. The main ideas are to introduce an additional operation which spreads instead of snapping geometries, in order to preserve the original relation between them, and to use a tolerance region for such operation instead of a single snapping location. Finally, some experiments on real-world datasets are presented, which confirms how the proposed algorithm can establish the robustness of a dataset

Part of the Computer Sciences Commons Comments Victor Milenkovic & Elisha Sacks

We present two approximate Minkowski sum algorithms for planar regions bounded by line and circle segments. Both algorithms form a convolution curve, construct its arrangement, and use winding numbers to identify sum cells. The first uses the kinetic convolution and the second uses our monotonic convolution. The asymptotic running times of the exact algorithms are increased by km log m with m the number of segments in the convolution and with k the number of segment triples that are in cyclic vertical order due to approximate segment intersection. The approximate Minkowski sum is close to the exact sum of perturbation regions that are close to the input regions. We validate both algorithms on part packing tasks with industrial part shapes. The accuracy is near the floating point accuracy even after multiple iterated sums. The programs are 2% slower than direct floating point implementations of the exact algorithms. The monotonic algorithm is 42% faster than the kinetic algorithm

Arrondi d’une soupe de triangles

Let $\mathcal{P}$ be a set of $n$ polygons in $\mathbb{R}^3$, each of constant complexity and with pairwise disjoint interiors. We propose a rounding algorithm that maps $\mathcal{P}$ to a simplicial complex $\mathcal{Q}$ whose vertices have integer coordinates. Every face of $\mathcal{P}$ is mapped to a set of faces (or edges or vertices) of $\mathcal{Q}$ and the mapping from $\mathcal{P}$ to $\mathcal{Q}$ can be done through a continuous motion of the faces such that (i) the $L_\infty$ Hausdorff distance between a face and its image during the motion is at most 3/2 and (ii) if two points become equal during the motion, they remain equal through the rest of the motion. In the worse the size of $\mathcal{Q}$ is $O(n^{15})$ and the time complexity of the algorithm is $O(n^{19})$ but, under reasonable hypotheses, these complexities decrease to $O(n^{5})$ and $O(n^{6}\sqrt{n})$.Soit $\mathcal{P}$ un ensemble de $n$ polygones dans $\mathbb{R}^3$, chacun de complexité constante, et d'intérieurs disjoints.Nous présentons un algorithme d'arrondi tel que l'image de $\mathcal{P}$ soit un complexe simplicial $\mathcal{Q}$ dont les sommets ont des coordonnées entières. Chaque face de$\mathcal{P}$ est envoyée sur un ensemble de faces (ou arêtes ou sommets) de $\mathcal{Q}$ et, de plus, $\mathcal {P}$ peut être tranformé en $\mathcal {Q}$ par un mouvement continu des faces de telle sorte que (i) la distance de Hausdorff $L_\infty$ entre une face et son image pendant le mouvement est au plus $3/2$ et (ii) si deux points deviennent égaux pendant le mouvement, ils restent égaux durant le reste de le mouvement. La taille de $\mathcal {Q}$ est au pire $O (n ^ {15})$ et la complexité temporelle de l'algorithme est $O (n ^ {19})$. Cependant, sous des hypothèses raisonnables, ces complexités peuvent être ramenées à $O (n ^ {5})$ et $O (n ^ {6} \sqrt {n})$

Rounding meshes in 3D

International audienceLet $\mathcal{P}$ be a set of $n$ polygons in $\mathbb{R}^3$, each of constant complexity and with pairwise disjoint interiors. We propose a rounding algorithm that maps $\mathcal{P}$ to a simplicial complex $\mathcal{Q}$ whose vertices have integer coordinates. Every face of $\mathcal{P}$ is mapped to a set of faces (or edges or vertices) of $\mathcal{Q}$ and the mapping from $\mathcal{P}$ to $\mathcal{Q}$ can be done through a continuous motion of the faces such that (i) the $L_\infty$ Hausdorff distance between a face and its image during the motion is at most 3/2 and (ii) if two points become equal during the motion, they remain equal through the rest of the motion. In the worst case the size of $\mathcal{Q}$ is $O(n^{13})$ and the time complexity of the algorithm is $O(n^{15})$ but, under reasonable assumptions, these complexities decrease to $O(n^{4}\sqrt{n})$ and $O(n^{5})$. Furthermore, these complexities are likely not tight and we expect, in practice on non-pathological data, $O(n\sqrt{n})$ space and time complexities

Establishing Robustness of a Spatial Dataset in a Tolerance-Based Vector Model

Spatial data are usually described through a vector model in which geometries are represented by a set of coordinates embedded into an Euclidean space. The use of a finite representation, instead of the real numbers theoretically required, causes many robustness problems which are well known in the literature. Such problems are made even worse in a distributed context, where data is exchanged between different systems and several perturbations can be introduced in the data representation. In order to discuss the robustness of a spatial dataset, two implementation models have to be distinguished: the identity and the tolerance model. The robustness of a dataset in the identity model has been widely discussed in the literature and some algorithms of the Snap Rounding (SR) family can be successfully applied in such contexts. Conversely, this problem has been less explored in the tolerance model. The aim of this article is to propose an algorithm inspired by those of the SR family for establishing or restoring the robustness of a vector dataset in the tolerance model. The main ideas are to introduce an additional operation which spreads instead of snapping geometries, in order to preserve the original relation between them, and to use a tolerance region for such an operation instead of a single snapping location. Finally, some experiments on real-world datasets are presented, confirming how the proposed algorithm can establish the robustness of a dataset

Iterated snap rounding with bounded drift

Snap Rounding and its variant, Iterated Snap Rounding, are methods for converting arbitrary-precision arrangements of segments into a fixed-precision representation (we call them SR and ISR for short). Both methods approximate each original segment by a polygonal chain, and both may lead, for certain inputs, to rounded arrangements with undesirable properties: in SR the distance between a vertex and a non-incident edge of the rounded arrangement can be extremely small, inducing potential degeneracies. In ISR, a vertex and a non-incident edge are well separated, but the approximating chain may drift far away from the original segment it approximates. We propose a new variant, Iterated Snap Rounding with Bounded Drift, which overcomes these two shortcomings of the earlier methods. The new solution augments ISR with simple and efficient procedures that guarantee the quality of the geometric approximation of the original segments, while still maintaining the property that a vertex and a non-incident edge in the rounded arrangement are well separated. We investigate the properties of the new method and compare it with the earlier variants. We have implemented the new scheme on top of CGAL, the Computational Geometry Algorithms Library, and report on experimental results.