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Mapping polygons to the grid with small Hausdorff and Fréchet distance [article]

Quirijn W. Bouts, Irina Kostitsyna, Marc van Kreveld, Wouter Meulemans, Willem Sonke, Kevin Verbeek
2016 arXiv   pre-print
We show how to represent a simple polygon P by a grid (pixel-based) polygon Q that is simple and whose Hausdorff or Fréchet distance to P is small.  ...  For any simple polygon P, a grid polygon exists with constant Hausdorff distance between their boundaries and their interiors.  ...  Hausdorff distance We consider the problem of constructing a grid polygon Q with small Hausdorff distance to P . In Appendix A we prove the theorem below.  ... 
arXiv:1606.06660v1 fatcat:muobhxyvfjf3femmg53rxlm7s4

Mapping Polygons to the Grid with Small Hausdorff and Fréchet Distance *

Quirijn Bouts, Irina Kostitsyna, Marc Van Kreveld, Wouter Meulemans, Willem Sonke, Kevin Verbeek
unpublished
We show how to represent a simple polygon P by a (pixel-based) grid polygon Q that is simple and whose Hausdorff or Fréchet distance to P is small.  ...  For any simple polygon P , a grid polygon exists with constant Hausdorff distance between their boundaries and their interiors.  ...  ( 1 ) 1 Q is simply connected, and (2) the Hausdorff distance between P (∂P ) and Q (∂Q) is small.  ... 
fatcat:m6ctdya4yvhqrpoet7zchpwfdy

Optimizing the Geometrical Accuracy of 2D Curvilinear Meshes

Jean-François Remacle, Jonathan Lambrechts, Christophe Geuzaine, Thomas Toulorge
2014 Procedia Engineering  
This area based distance is shown to be strongly related to standard distance measures such as the Hausdorff distance, while being largely faster to compute.  ...  An area based distance is proposed to compute the geometrical discrepancy between the mesh and its underlying geometry.  ...  Discrete distances between polygonal curves Discrete Hausdorff and Fréchet distances δ H (M, N) and δ F (M, N) compute respectively the Hausdorff and the Fréchet distance restricted to discrete points.  ... 
doi:10.1016/j.proeng.2014.10.386 fatcat:2dhzzznytfculmtgh3edz4yis4

Walking your dog in the woods in polynomial time

Erin Wolf Chambers, Éric Colin de Verdière, Jeff Erickson, Sylvain Lazard, Francis Lazarus, Shripad Thite
2008 Proceedings of the twenty-fourth annual symposium on Computational geometry - SCG '08  
The Fréchet distance between two curves in the plane is the minimum length of a leash that allows a dog and its owner to walk along their respective curves, from one end to the other, without backtracking  ...  We describe a polynomial-time algorithm to compute the homotopic Fréchet distance between two given polygonal curves in the plane minus a given set of obstacles, which are either points or polygons.  ...  It is therefore possible for two curves to have small Hausdorff distance but large Fréchet distance.  ... 
doi:10.1145/1377676.1377694 dblp:conf/compgeom/ChambersVELLT08 fatcat:b7kuo5qnrrbwrn2n5iz7gtsndu

Homotopic Fréchet distance between curves or, walking your dog in the woods in polynomial time

Erin Wolf Chambers, Éric Colin de Verdière, Jeff Erickson, Sylvain Lazard, Francis Lazarus, Shripad Thite
2010 Computational geometry  
The Fréchet distance between two curves in the plane is the minimum length of a leash that allows a dog and its owner to walk along their respective curves, from one end to the other, without backtracking  ...  We describe a polynomial-time algorithm to compute the homotopic Fréchet distance between two given polygonal curves in the plane minus a given set of polygonal obstacles.  ...  We would like to thank the anonymous referees for their careful reading of the paper and their numerous suggestions for improvement.  ... 
doi:10.1016/j.comgeo.2009.02.008 fatcat:vqtemc2z6nffxd7si62be32s7u

Review and Perspective for Distance Based Trajectory Clustering [article]

Philippe Besse , Jean-Michel Loubes, Royer François
2015 arXiv   pre-print
We finally compare this new distance to the others according to their corresponding clustering results obtained using both hierarchical clustering and affinity propagation methods.  ...  Using clustering technics based on the choice of a distance between the observations, we first provide a comprehensive review of the different distances used in the literature to compare trajectories.  ...  With python, DTW, LCSS and Discrete Fréchet distances are the fastest methods, while Hausdorff and SSPD are the fastest with cython because of its ability to declare static variables and to use the C math  ... 
arXiv:1508.04904v1 fatcat:ktxxkclj3je43lz3otuy6qep2a

Review and Perspective for Distance-Based Clustering of Vehicle Trajectories

Philippe C. Besse, Brendan Guillouet, Jean-Michel Loubes, Francois Royer
2016 IEEE transactions on intelligent transportation systems (Print)  
We finally compare this new distance to the others according to their corresponding clustering results obtained using both hierarchical clustering and affinity propagation methods.  ...  Using clustering technics based on the choice of a distance between the observations, we first provide a comprehensive review of the different distances used in the literature to compare trajectories.  ...  With python, DTW, LCSS and Discrete Fréchet distances are the fastest methods, while Hausdorff and SSPD are the fastest with cython because of its ability to declare static variables and to use the C math  ... 
doi:10.1109/tits.2016.2547641 fatcat:7hvo5s2kwnht3otwugpbv4jb4i

Discretized Approaches to Schematization [article]

Wouter Meulemans
2016 arXiv   pre-print
With the former, S is a simple cycle C in G and we quantify resemblance via the Fréchet distance.  ...  We prove that it is NP-hard to compute a cycle that approximates the minimal Fréchet distance over all simple cycles in a plane graph G.  ...  The author would like to thank: Bettina Speckmann and her Applied Geometric Algorithms group, Kevin Buchin, Bart Jansen, Arthur van Goethem, Marc van Kreveld and Aidan Slingsby for inspiring discussions  ... 
arXiv:1606.06488v1 fatcat:h33mxq7mobdyxlaars6v33novm

Shortest Path Problems on a Polyhedral Surface

Atlas F. Cook, Carola Wenk
2012 Algorithmica  
We develop algorithms to compute edge sequences, Voronoi diagrams, shortest path maps, the Fréchet distance, and the diameter for a polyhedral surface.  ...  The link diameter of N can be computed in O(M 19 3 log 3.11 M ) time and O(M 3 ) space. Proof. Partition each of the O(M ) edges of N into O(M 3 ) edgelets with Lemma 3.  ...  Sections 3.3 and 3.4 contain algorithms to compute the Fréchet distance and shortest path maps.  ... 
doi:10.1007/s00453-012-9723-6 fatcat:ebu7n3elgbertijtsoh7mwoipa

Fréchet Distance for Curves, Revisited [chapter]

Boris Aronov, Sariel Har-Peled, Christian Knauer, Yusu Wang, Carola Wenk
2006 Lecture Notes in Computer Science  
Fréchet distance has also been extended to graphs (maps) [AERW03], to piecewise smooth curves [Rot05], to simple polygons [BBW06] and to surfaces [AB05].  ...  [AG95] present an algorithm to compute the Fréchet distance between two polygonal curves of n and m vertices, respectively, in time O(nm log 2 (nm)).  ...  Figure 1 : 1 (a) Light and dark curves are close under Hausdorff but far under Fréchet distance.  ... 
doi:10.1007/11841036_8 fatcat:ex4of3d7ajhhrisrx6qz5nxufq

Approximating the Fréchet distance for realistic curves in near linear time

Anne Driemel, Sariel Har-Peled, Carola Wenk
2010 Proceedings of the 2010 annual symposium on Computational geometry - SoCG '10  
We present a simple and practical (1 + ε)-approximation algorithm for the Fréchet distance between two polygonal curves in R d .  ...  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.  ...  The authors would also like to thank the anonymous referees for their insightful comments.  ... 
doi:10.1145/1810959.1811019 dblp:conf/compgeom/DriemelHW10 fatcat:o3fjc2hwefca3k7hfpkq7kd72e

Approximating the Fréchet Distance for Realistic Curves in Near Linear Time

Anne Driemel, Sariel Har-Peled, Carola Wenk
2012 Discrete & Computational Geometry  
We present a simple and practical (1 + ε)-approximation algorithm for the Fréchet distance between two polygonal curves in R d .  ...  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.  ...  The authors would also like to thank the anonymous referees for their insightful comments.  ... 
doi:10.1007/s00454-012-9402-z fatcat:jhkejni6ovewlp7qfwjftjxohm

A comparison and evaluation of map construction algorithms using vehicle tracking data

Mahmuda Ahmed, Sophia Karagiorgou, Dieter Pfoser, Carola Wenk
2014 Geoinformatica  
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  ...  to map construction algorithms.  ...  map and the output map, with a guaranteed Fréchet distance between those portions.  ... 
doi:10.1007/s10707-014-0222-6 fatcat:rkgn2tffqndq7l6rpvhazahfte

Geometric Optimization Revisited [chapter]

Pankaj K. Agarwal, Esther Ezra, Kyle Fox
2019 Lecture Notes in Computer Science  
We review the progress made in recent years on a number of such geometric optimization problems, with an emphasis on how geometry has been exploited to develop better algorithms.  ...  Instead of discussing many problems, we focus on a few problems, namely, set cover, hitting set, independent set, and computing maps between point sets.  ...  We thank Timothy Chan, Sariel Har-Peled, and Micha Sharir for their helpful comments.  ... 
doi:10.1007/978-3-319-91908-9_5 fatcat:3l3jwv5bevhkvoyi3gtb65g7ae

Computing the Fréchet Distance Between Polygons with Holes [chapter]

Amir Nayyeri, Anastasios Sidiropoulos
2015 Lecture Notes in Computer Science  
For general polygons in R 2 , it has been shown to be NP-hard, and the best known polynomial-time algorithm works only for polygons with at most a single puncture [Buchin et al., 2010] .  ...  We study the problem of computing the Fréchet distance between subsets of Euclidean space.  ...  They first show that a map f s between the boundaries of a convex polygon and a simple polygon can be extended to a map between the polygons with Fréchet length arbitrarily close to δ F (f s ).  ... 
doi:10.1007/978-3-662-47672-7_81 fatcat:qs3yrjfnijevhejygcyd6ctbvi
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