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Carola Wenk

Publications and source records attributed to Carola Wenk.

31 records · Page 2Linked to original sources
arXiv2020

Simplification of Indoor Space Footprints

Simplification is one of the fundamental operations used in geoinformation science (GIS) to reduce size or representation complexity of geometric objects. Although different simplification methods can be applied depending on one's purpose, a simplification that many applications employ is designed to preserve their spatial properties after simplification. This article addresses one of the 2D simplification methods, especially working well on human-made structures such as 2D footprints of buildings and indoor spaces. The method simplifies polygons in an iterative manner. The simplification is segment-wise and takes account of intrusion, extrusion, offset, and corner portions of 2D structures preserving its dominant frame.

cs.CG↗
arXiv2019

Threshold-Based Graph Reconstruction Using Discrete Morse Theory

Discrete Morse theory has recently been applied in metric graph reconstruction from a given density function concentrated around an (unknown) underlying embedded graph. We propose a new noise model for the density function to reconstruct a connected graph both topologically and geometrically.

cs.CG↗
arXiv2019

Distance Measures for Embedded Graphs

We introduce new distance measures for comparing straight-line embedded graphs based on the Fréchet distance and the weak Fréchet distance. These graph distances are defined using continuous mappings and thus take the combinatorial structure as well as the geometric embeddings of the graphs into account. We present a general algorithmic approach for computing these graph distances. Although we show that deciding the distances is NP-hard for general embedded graphs, we prove that our approach yields polynomial time algorithms if the graphs are trees, and for the distance based on the weak Fréchet distance if the graphs are planar embedded. Moreover, we prove that deciding the distances based on the Fréchet distance remains NP-hard for planar embedded graphs and show how our general algorithmic approach yields an exponential time algorithm and a polynomial time approximation algorithm for this case.

cs.CG↗
arXiv2019

Fast Frechet Distance Between Curves With Long Edges

Computing the Fréchet distance between two polygonal curves takes roughly quadratic time. In this paper, we show that for a special class of curves the Fréchet distance computations become easier. Let $P$ and $Q$ be two polygonal curves in $\mathbb{R}^d$ with $n$ and $m$ vertices, respectively. We prove four results for the case when all edges of both curves are long compared to the Fréchet distance between them: (1) a linear-time algorithm for deciding the Fréchet distance between two curves, (2) an algorithm that computes the Fréchet distance in $O((n+m)\log (n+m))$ time, (3) a linear-time $\sqrt{d}$-approximation algorithm, and (4) a data structure that supports $O(m\log^2 n)$-time decision queries, where $m$ is the number of vertices of the query curve and $n$ the number of vertices of the preprocessed curve.

cs.CG↗
arXiv2017

On Minimum Area Homotopies of Normal Curves in the Plane

In this paper, we study the problem of computing a homotopy from a planar curve $C$ to a point that minimizes the area swept. The existence of such a minimum homotopy is a direct result of the solution of Plateau's problem. Chambers and Wang studied the special case that $C$ is the concatenation of two simple curves, and they gave a polynomial-time algorithm for computing a minimum homotopy in this setting. We study the general case of a normal curve $C$ in the plane, and provide structural properties of minimum homotopies that lead to an algorithm. In particular, we prove that for any normal curve there exists a minimum homotopy that consists entirely of contractions of self-overlapping sub-curves (i.e., consists of contracting a collection of boundaries of immersed disks).

math.AT↗
arXiv2015

Fréchet Distance for Curves, Revisited

$\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}} \newcommand{\eps}{\varepsilon} \newcommand{\SetX}{\mathsf{X}} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)}$ We revisit the problem of computing Fréchet distance between polygonal curves under $L_1$, $L_2$, and $L_\infty$ norms, focusing on discrete Fréchet distance, where only distance between vertices is considered. We develop efficient algorithms for two natural classes of curves. In particular, given two polygonal curves of $n$ vertices each, a $\eps$-approximation of their discrete Fréchet distance can be computed in roughly $O(nκ^3\log n/\eps^3)$ time in three dimensions, if one of the curves is \emph{$κ$-bounded}. Previously, only a $κ$-approximation algorithm was known. If both curves are the so-called \emph{\backbone~curves}, which are widely used to model protein backbones in molecular biology, we can $\eps$-approximate their Fréchet distance in near linear time in two dimensions, and in roughly $O(n^{4/3}\log nm)$ time in three dimensions. In the second part, we propose a pseudo--output-sensitive algorithm for computing Fréchet distance exactly. The complexity of the algorithm is a function of a quantity we call the \emph{\bwnumber{}}, which is quadratic in the worst case, but tends to be much smaller in practice.

cs.CG↗
arXiv2015

Path-Based Distance for Street Map Comparison

Comparing two geometric graphs embedded in space is important in the field of transportation network analysis. Given street maps of the same city collected from different sources, researchers often need to know how and where they differ. However, the majority of current graph comparison algorithms are based on structural properties of graphs, such as their degree distribution or their local connectivity properties, and do not consider their spatial embedding. This ignores a key property of road networks since similarity of travel over two road networks is intimately tied to the specific spatial embedding. Likewise, many current street map comparison algorithms focus on the spatial embeddings only and do not take structural properties into account, which makes these algorithms insensitive to local connectivity properties and shortest path similarities. We propose a new path-based distance measure to compare two planar geometric graphs embedded in the plane. Our distance measure takes structural as well as spatial properties into account by imposing a distance measure between two road networks based on the Hausdorff distance between the two sets of travel paths they represent. We show that this distance can be approximated in polynomial time and that it preserves structural and spatial properties of the graphs.

cs.CG↗
arXiv2014

A Comparison and Evaluation of Map Construction Algorithms

Map construction methods automatically produce and/or update road network datasets using vehicle tracking data. Enabled by the ubiquitous generation of georeferenced tracking data, there has been a recent surge in map construction algorithms coming from different computer science domains. A cross-comparison of the various algorithms is still very rare, since (i) algorithms and constructed maps are generally not publicly available and (ii) there is no standard approach to assess the result quality, given the lack of benchmark data and quantitative evaluation methods. This work represents a first comprehensive attempt to benchmark map construction algorithms. We provide an evaluation and comparison of seven algorithms using four datasets and four different evaluation measures. In addition to this comprehensive comparison, we make our datasets, source code of map construction algorithms and evaluation measures publicly available on mapconstruction.org. This site has been established as a repository for map con- struction data and algorithms and we invite other researchers to contribute by uploading code and benchmark data supporting their contributions to map construction algorithms.

cs.CG↗
arXiv2013

Simple Curve Embedding

Given a curve f and a surface S, how hard is it to find a simple curve f' in S that is the most similar to f? We introduce and study this simple curve embedding problem for piecewise linear curves and surfaces in R^2 and R^3, under Hausdorff distance, weak Frechet distance, and Frechet distance as similarity measures for curves. Surprisingly, while several variants of the problem turn out to have polynomial-time solutions, we show that in R^3 the simple curve embedding problem is NP-hard under Frechet distance even if S is a plane, as well as under weak Frechet distance if S is a terrain. Additionally, these results give insight into the difficulty of computing the Frechet distance between surfaces, and they imply that the partial Frechet distance between non-planar surfaces is NP-hard as well.

cs.CG↗
arXiv2012

Approximating the Fréchet Distance for Realistic Curves in Near Linear Time

We present simple and practical $(1+\eps)$-approximation algorithm for the Frechet distance between curves. To analyze this algorithm we introduce a new realistic family of curves, $c$-packed curves, that is closed under simplification. We believe the notion of $c$-packed curves to be of independent interest. We show that our algorithm has near linear running time for $c$-packed curves, and show similar results for other input models.

cs.CG↗
arXiv2011

Computing the Fréchet Distance Between Folded Polygons

Computing the Fréchet distance for surfaces is a surprisingly hard problem and the only known algorithm is limited to computing it between flat surfaces. We adapt this algorithm to create one for computing the Fréchet distance for a class of non-flat surfaces which we call folded polygons. Unfortunately, the original algorithm cannot be extended directly. We present three different methods to adapt it. The first of which is a fixed-parameter tractable algorithm. The second is a polynomial-time approximation algorithm. Finally, we present a restricted class of folded polygons for which we can compute the Fréchet distance in polynomial time.

cs.CG↗
arXiv2008

Geodesic Fréchet Distance Inside a Simple Polygon

We unveil an alluring alternative to parametric search that applies to both the non-geodesic and geodesic Fréchet optimization problems. This randomized approach is based on a variant of red-blue intersections and is appealing due to its elegance and practical efficiency when compared to parametric search. We present the first algorithm for the geodesic Fréchet distance between two polygonal curves $A$ and $B$ inside a simple bounding polygon $P$. The geodesic Fréchet decision problem is solved almost as fast as its non-geodesic sibling and requires $O(N^{2\log k)$ time and $O(k+N)$ space after $O(k)$ preprocessing, where $N$ is the larger of the complexities of $A$ and $B$ and $k$ is the complexity of $P$. The geodesic Fréchet optimization problem is solved by a randomized approach in $O(k+N^{2\log kN\log N)$ expected time and $O(k+N^{2)$ space. This runtime is only a logarithmic factor larger than the standard non-geodesic Fréchet algorithm (Alt and Godau 1995). Results are also presented for the geodesic Fréchet distance in a polygonal domain with obstacles and the geodesic Hausdorff distance for sets of points or sets of line segments inside a simple polygon $P$.

cs.DS↗