Constructing Street-maps from GPS Trajectories

Transcription

1 Constructing Street-maps from GPS Trajectories Mahmuda Ahmed, Carola Wenk The University of Antonio Department of Computer Science Presented by Mahmuda Ahmed

2 Problem Statement Given a set of trajectories in the plane, compute a street map/graph that represents all trajectories in the set. Optional information may include parameters such as road width ω and precision ɛ. 2

3 Outline Problem Statement Data Models Related Works Machine Learning Approaches Computational Geometry Approach Preliminaries & Our Approach 3

4 How to Represent a Road-network? Geometric Data Model G(V,E) V: set of vertices. E: set of links between vertices which represents a road fragment. Region Based Model ωis the width of road fragment Each road fragment(γ) is represented as a polygonal region which is the MinkowskiSum of a polygonal curve and a disk B(s, ω/2), γ ω/2 ω/2 s Vertex(V) Road Width(ω) Edge(E) Road Fragment(γ) 4

5 Related Works Machine Learning (ML) or Artificial Intelligence (AI) Approaches Sub-trajectory Clustering [8] Force Simulation[4] Image Processing[9] Computational Geometry Approach D. Chen, L. Guibas, J. Hershberger, and J. Sun, Road Network Reconstruction for Organizing Paths, ACM-SIAM Symposium on Discrete Algorithms (SODA), Road Network Model Road Network is modeled as Geometric Graph G = (V,E) V represents the Sampling Points E represents linking line between two vertices Each road fragment is associated with a road width ω 5

6 Computational Geometry Approach 1. Compute bɛ-net 2. Compute Voronoi diagram 3. Compute Delaunay Triangulation 4. Compute Restricted Delaunay Triangulation 5. Identifying clean & not clean graph 6. Compute Structure Graph 7. Compute Reconstruction Graph D. Chen, L. Guibas, J. Hershberger, and J. Sun, Road Network Reconstruction for Organizing Paths, ACM-SIAM Symposium on Discrete Algorithms (SODA),

7 Outline Problem Statement Data Models Different Approaches Machine Learning Approaches Computational Geometry Approach Preliminaries & Our Approach 7

8 Incremental Map Construction works with Geometric Graph Input: Given graph(map) and a set of curves Task: Add one curve to the map in each iteration. Graph (G) Curve(l) Find paths in graph (G) that represents the sub-curves New Graph (G) 8

9 Preliminaries Fréchet Distance Between Two Curves δ ( f, g) : = inf max f( α( t)) g( β( t)) F α, β:[0,1] [0,1] t [0,1] where α and β range over continuous non-decreasing reparametrizations only. ɛ f >ɛ g ɛ g Decision variant solves the following problem: Findamonotonepathinfreespacefromlowerlefttoupper right endpoint. f 9

10 Preliminaries Map Matching Problem(uses weak Fréchet) Curve(l) Graph (G) Find a path in graph (G) that represents the curve Curve(l) Find an l-monotone path from any left end point to any right end point. H. Alt, A. Efrat, G. Rote, and C. Wenk. Matching planar maps. Journal of Algorithms, pages ,

11 Preliminaries Partial Curve Matching Black Region Weight:1 White Region Weight:0 Red Curve Matched portion Unmatched portion Blue Curve Find a monotone shortest path on weighted surface from lower left endpointtoupperrightendpoint. Kevin Buchin, MaikeBuchin, and YusuWang. Exact algorithm for partial curve matching via the Fréchet distance. In Proc. ACM-SIAM Symposium on Discrete Algorithms (SODA09), pages ,

12 Incremental Map Construction Combining the idea of Map Matching and Partial Curve Matching, Map Construction problem can be redefined as finding an l-monotone shortest path on free-space surface from any left end point to any right end point. How our case is different from partial curve matching ε Graph(G) Curve(l) Graph(G) Curve(l) 12

13 Incremental Map Construction Our Observations Skipping a portion of graph is allowed. Which means, we are only interested to maximize the matching on curve. The path length from any left end to any right end point would be the length of unmatched portion of curve. Thus, the problem of computing shortest path on weighted nonmanifold surface transformed into a problem of computing the length of unmatched portion of curve. 13

14 The Algorithm Step 1: Identifying Matched and Unmatched Portion Compute the free-space surface Project white regions down on the curve (white intervals) Fill up the gap between non-overlapping white intervals with black intervals Step 2: Create Edge for Unmatched Portion Insert the sub curve corresponding to each black interval as an edge to the graph 14

15 The Algorithm(cntd.) Step 3: Updating the Matched Portion Compute simplified(low complexity) representative curve of two curves, one is the edge of graph and another one is the matched portion of input curves. We used adaptive version of min-link algorithm[6]. Min-Link Problem: Given an ordered set of convex objects compute a stabbing curve with minimum complexity that stabs all the objects in the given order and vertices of the stabbing curve lie inside the convex hull of consecutive two objects. 15

16 Quality of the Reconstruction Graph Assumptions degree 2 Good Vertex π/2 Intersections p 1 α < 3ε α 2 α 1 Bad Vertex degree>2 p 2 <π/2 Good Road Fragment: No other road fragment is within 3εdistance Assumptions on Input Data: Each input path is within Fréchet distance ε/2of a sub curve of a path on G. For each road fragment, there exists an input path l such that, γ ε/2 contains that. 16

17 Quality of the Reconstruction Graph(cntd) Lemmas Lemma 1: A polygonal curve with n vertices, which is within ε/2fréchet distance from a k-link road fragment, can be simplified to a curve having at most k links. This guarantees that the number of good vertices in our reconstructed graph is at most the number of good vertices in the original road network graph. Lemma 2: If two road fragments γ 1 and γ 2 come close to each other (distance < 3ε), then our algorithm will detect the vertex no farther than 3ε /sin α from its original position. This guarantees maximum displacement of a vertex in reconstruction graph. 17

18 Quality of the Reconstruction Graph(cntd) Lemmas Lemma 3: A good road fragment in the original road network will be reconstructed with length at least len(γ)-(3ε/sin α 1 + 3ε/sin α 2 ), where α 1 and α 2 are the minimum angles with the terminal vertices of γ. This guarantees the length of road fragment in reconstruction graph. 18

19 Experimental Setup We applied our algorithm on real data. The tracking data was obtained by sampling vehicle movements at a rate of 30 seconds. The dataset consists of 3237 vehicle trajectories consisting of a total of position samples. The tracking data was collected from taxi cabs in the municipal area of Berlin, Germany, in

20 Experimental Results Original Map of Berlin, Germany (Map data by TeleAtlas) Reconstructed map of Berlin generated by our algorithm 20

21 Future Work How to update or merge bad vertices How to bound the number of bad vertices How to handle self-intersecting curves. 21

22 Thank You 22

23 References [1] H. Alt, A. Efrat, G. Rote, and C. Wenk. Matching planar maps.journal of Algorithms, pages , [2] R. Bruntrup, S. Edelkamp, S. Jabbar, and B. Scholz. Incremental map generation with GPS traces. In Intelligent Transportation Systems, Proceedings IEEE, pages , Sept [3] Kevin Buchin, MaikeBuchin, and YusuWang. Exact algorithm for partial curve matching via the Fréchet distance. In Proc. ACM-SIAM Symposium on Discrete Algorithms (SODA09), pages , [4] LiliCao and John Krumm. From GPS traces to a routable road map. In Proc. of the 17th ACM SIGSPATIAL, GIS '09, pages 312, New York, NY, USA, ACM. [5] Daniel Chen, LeonidasGuibas, John Hershberger, and JianSun. Road network reconstruction for organizing paths. In Proceedings of the 21st ACM-SIAM Symposium on Discrete Algorithms, [6] L J Guibas, J E Hershberger, J S B Mitchell, and J S Snoeyink. Approximating polygons and subdivisions with minimum-link paths, int. J. of Computational Geometry and Applications, pages 34, [7] Tao Guo, K. Iwamura, and M. Koga. Towards high accuracy road maps generation from massive GPS traces data. In Geoscienceand Remote Sensing Symposium, IGARSS IEEE International, pages , July [8] ZhenhuiLi, Jae-Gil Lee, XiaoleiLi, and JiaweiHan. Incremental clustering for trajectories. In DASFAA (2), pages 3246, [9] HuijingZhao, Jun Kumagai, MasafumiNakagawa and RyosukeShibasaki. Semi-automatic road extraction from High Resolution Satellite image. In Proc. ISPRS Technical CommisionIII Symposium: Photometric Computer Vision, page A:406,

24 Runtime The worst case Runtime for adding the i-thcurve iso(m i-1 *n i + k u +(m i-1 + n i ) 2 log(m i-1 + n i ) + k c ) where, m i-1 is the complexity of graph after adding the (i-1)-thcurve, n i is the complexity of the i-thcurve and k u and k c are the number of edges updated or created respectively. 24