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

2016
*
arXiv
*
pre-print

We show how

arXiv:1606.06660v1
fatcat:muobhxyvfjf3femmg53rxlm7s4
*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. ...##
###
Mapping Polygons to the Grid with Small Hausdorff and Fréchet Distance *

unpublished

We show how

fatcat:m6ctdya4yvhqrpoet7zchpwfdy
*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*. ...##
###
Optimizing the Geometrical Accuracy of 2D Curvilinear Meshes

2014
*
Procedia Engineering
*

This area based

doi:10.1016/j.proeng.2014.10.386
fatcat:2dhzzznytfculmtgh3edz4yis4
*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. ...##
###
Walking your dog in the woods in polynomial time

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*. ...

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

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. ...

##
###
Review and Perspective for Distance Based Trajectory Clustering
[article]

2015
*
arXiv
*
pre-print

We finally compare this new

arXiv:1508.04904v1
fatcat:ktxxkclj3je43lz3otuy6qep2a
*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 ...##
###
Review and Perspective for Distance-Based Clustering of Vehicle Trajectories

2016
*
IEEE transactions on intelligent transportation systems (Print)
*

We finally compare this new

doi:10.1109/tits.2016.2547641
fatcat:7hvo5s2kwnht3otwugpbv4jb4i
*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 ...##
###
Discretized Approaches to Schematization
[article]

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 ...

##
###
Shortest Path Problems on a Polyhedral Surface

2012
*
Algorithmica
*

We develop algorithms

doi:10.1007/s00453-012-9723-6
fatcat:ebu7n3elgbertijtsoh7mwoipa
*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*. ...##
###
Fréchet Distance for Curves, Revisited
[chapter]

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*. ...

##
###
Approximating the Fréchet distance for realistic curves in near linear time

2010
*
Proceedings of the 2010 annual symposium on Computational geometry - SoCG '10
*

We present a simple

doi:10.1145/1810959.1811019
dblp:conf/compgeom/DriemelHW10
fatcat:o3fjc2hwefca3k7hfpkq7kd72e
*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. ...##
###
Approximating the Fréchet Distance for Realistic Curves in Near Linear Time

2012
*
Discrete & Computational Geometry
*

We present a simple

doi:10.1007/s00454-012-9402-z
fatcat:jhkejni6ovewlp7qfwjftjxohm
*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. ...##
###
A comparison and evaluation of map construction algorithms using vehicle tracking data

2014
*
Geoinformatica
*

A cross-comparison of

doi:10.1007/s10707-014-0222-6
fatcat:rkgn2tffqndq7l6rpvhazahfte
*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. ...##
###
Geometric Optimization Revisited
[chapter]

2019
*
Lecture Notes in Computer Science
*

We review

doi:10.1007/978-3-319-91908-9_5
fatcat:3l3jwv5bevhkvoyi3gtb65g7ae
*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. ...##
###
Computing the Fréchet Distance Between Polygons with Holes
[chapter]

2015
*
Lecture Notes in Computer Science
*

For general

doi:10.1007/978-3-662-47672-7_81
fatcat:qs3yrjfnijevhejygcyd6ctbvi
*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 ). ...
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