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4/28/15CMPS 3130/6130 Computational Geometry1 CMPS 3130/6130 Computational Geometry Spring 2015 Shape Matching Carola Wenk A B   (B,A)

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1 4/28/15CMPS 3130/6130 Computational Geometry1 CMPS 3130/6130 Computational Geometry Spring 2015 Shape Matching Carola Wenk A B   (B,A)

2 Are these shapes similar? Translate When are two shapes similar?

3 Are these shapes similar? Translate Translate & Rotate When are two shapes similar?

4 Are these shapes similar? Translate Translate & Rotate Translate & Scale When are two shapes similar?

5 Are these shapes similar? Translate Translate & Rotate Translate & Scale Translate & Reflect When are two shapes similar?

6 Are these shapes similar? Translate Translate & Rotate Translate & Scale Translate & Reflect Partial Matching When are two shapes similar?

7 Are these shapes similar? Translate Translate & Rotate Translate & Scale Translate & Reflect Partial Matching When are two shapes similar?

8 Two geometric shapes, each composed of a number of basic objects such as Given: An application dependant distance measure , e.g., the Hausdorff distance A set of transformations T, e.g., translations, rigid motions, or none Matching Task: Compute min  (T(A),B)  pointsline segmentstriangles Geometric Shape Matching 4/28/15CMPS 3130/6130 Computational Geometry8

9 Hausdorff distance Directed Hausdorff distance   (A,B) = max min || a-b || Undirected Hausdorff-distance  (A,B) = max (   (A,B),   (B,A) ) If A, B are discrete point sets, |A|=m, |A|=n In d dimensions: Compute in O(mn) time brute-force In 2 dimensions: Compute in O((m+n) log (m+n) time Compute   (A,B) by computing VD(B) and performing nearest neighbor search for every point in A A B   (B,A)   (A,B) a  A b  B 4/28/15CMPS 3130/6130 Computational Geometry9

10 Shape Matching - Applications Character Recognition Fingerprint Identification Molecule Docking, Drug Design Image Interpretation and Segmentation Quality Control of Workpieces Robotics Pose Determination of Satellites Puzzling... 4/28/15CMPS 3130/6130 Computational Geometry10

11 Hausdorff distance  (m 2 n 2 ) lower bound construction for directed Hausdorff distance of line segments under translations 4/28/15CMPS 3130/6130 Computational Geometry11

12 Comparing Curves and Trajectories 4/28/15CMPS 3130/6130 Computational Geometry12

13 Polygonal Curves Let f,g:[0,1]  R d be two polygonal curves (i.e., piecewise linear curves) What are good distance measures for curves? –Hausdorff distance? –Fréchet distance? f g 4/28/15CMPS 3130/6130 Computational Geometry13

14 When Are Two Curves „Similar“? Directed Hausdorff distance    (A,B) = max min || a-b || Undirected Hausdorff-distance   (A,B) = max (    (A,B),    (B,A) ) But: Small Hausdorff distance When considered as curves the distance should be large The Fréchet distance takes the continuity of the curves into account A B    (B,A)    (A,B) a  A b  B 4/28/15CMPS 3130/6130 Computational Geometry14

15  F (f,g) = inf max ||f(  (t))-g(  (t))||  :[0,1] [0,1] t  [0,1] where  and  range over continuous monotone increasing reparameterizations only. Man and dog walk on one curve each They hold each other at a leash They are only allowed to go forward  F is the minimal possible leash length f g Fréchet Distance for Curves [F06] M. Fréchet, Sur quelques points de calcul fonctionel, Rendiconti del Circolo Mathematico di Palermo 22: 1-74, 1906. 4/28/15CMPS 3130/6130 Computational Geometry15

16 Free Space Diagram Let  > 0 fixed (eventually solve decision problem) F  (f,g) = { (s,t)  [0,1] 2 | || f(s) - g(t)||   } white points free space of f and g The free space in one cell is an ellipse. ε g f 123456 f 0 1 g  >> 4/28/15CMPS 3130/6130 Computational Geometry16

17 Free Space Diagram Let  > 0 fixed (eventually solve decision problem) F  (f,g) = { (s,t)  [0,1] 2 | || f(s) - g(t)||   } white points free space of f and g The free space in one cell is an ellipse. g f f g 4/28/15CMPS 3130/6130 Computational Geometry17

18 Free Space Diagram  Monotone path encodes reparametrizations of f  and g  F (f,g)   iff there is a monotone path in the free space from (0,0) to (1,1)   g f g f 4/28/15CMPS 3130/6130 Computational Geometry18

19 Compute the Fréchet Distance  Solve the optimization problem  In practice in O(mn log b) time with binary search and b-bit precision  In O(mn log mn) time [AG95] using parametric search (using Cole‘s sorting trick)  In O(mn log 2 mn) expected time [CW09] with randomized red/blue intersections [AG95] H. Alt, M. Godau, Computing the Fréchet distance between two polygonal curves, IJCGA 5: 75-91, 1995. [CW10] A.F. Cook IV, C. Wenk, Geodesic Fréchet Distance Inside a Simple Polygon, ACM TALG 7(1), 19 pages, 2010.    Solve the decision problem  F (f,g)   in O(mn) time:  Find monotone path using DP:  On each cell boundary compute the interval of all points that are reachable by a monotone path from (0,0)  Compute a monotone path by backtracking f g 1 1 0 0 194/28/15CMPS 3130/6130 Computational Geometry

20  Navigation systems answer shortest path queries based on travel times on road segments  Usually based on static travel-time weights derived from speed limits  Goal: Develop a navigation system which answers routing queries based on the current traffic situation A B  Current traffic situation:  Database of current travel times per road segment  Use GPS trajectory data from vehicle fleets GPS Trajectories for Dynamic Routing 4/28/15CMPS 3130/6130 Computational Geometry20

21 GPS Trajectories from Vehicles Road map of Athens GPS trajectoriy Corresponding path in the road map 1)Measurement error: GPS points generally do not lie on the road map 2)Sampling error: The GPS trajectory is a by-product and is usually sampled every 30s  The GPS trajectory does not lie on the road map Map matching: Find a path in the graph which corresponds to the GPS trajectory (curve). Find a path in the graph with minimal distance to the GPS curve (partial matching) 214/28/15CMPS 3130/6130 Computational Geometry

22 Free Space Surface Glue the free space diagrams FD i,j together according to adjacency information in G Free space surface of f and G G [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. 4/28/15CMPS 3130/6130 Computational Geometry22

23 Free Space Surface Task: Find a monotone path in the free space surface that –starts in a lower left corner –and ends in an upper right corner G [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. 4/28/15CMPS 3130/6130 Computational Geometry23

24 Sweep G Sweep all FD i,j with a sweep line from left to right [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. 4/28/15CMPS 3130/6130 Computational Geometry24

25 G Sweep [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. Sweep all FD i,j with a sweep line from left to right 4/28/15CMPS 3130/6130 Computational Geometry25

26 G Sweep [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. Sweep all FD i,j with a sweep line from left to right 4/28/15CMPS 3130/6130 Computational Geometry26

27 G Sweep [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. Sweep all FD i,j with a sweep line from left to right 4/28/15CMPS 3130/6130 Computational Geometry27

28 G Sweep [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. Sweep all FD i,j with a sweep line from left to right 4/28/15CMPS 3130/6130 Computational Geometry28

29 G Sweep [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. Sweep all FD i,j with a sweep line from left to right 4/28/15CMPS 3130/6130 Computational Geometry29

30 G Sweep [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. Sweep all FD i,j with a sweep line from left to right 4/28/15CMPS 3130/6130 Computational Geometry30

31 Compute Reachable Points G During the sweep: Compute points on the free space surface, to the left of the sweep line, which are reachable by a monotone path from a lower left corner. [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. 4/28/15CMPS 3130/6130 Computational Geometry31

32 G [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. Compute Reachable Points During the sweep: Compute points on the free space surface, to the left of the sweep line, which are reachable by a monotone path from a lower left corner. 4/28/15CMPS 3130/6130 Computational Geometry32

33 G [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. Compute Reachable Points During the sweep: Compute points on the free space surface, to the left of the sweep line, which are reachable by a monotone path from a lower left corner. 4/28/15CMPS 3130/6130 Computational Geometry33

34 G [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. Compute Reachable Points During the sweep: Compute points on the free space surface, to the left of the sweep line, which are reachable by a monotone path from a lower left corner. 4/28/15CMPS 3130/6130 Computational Geometry34

35 Update Reachable Points During the sweep: –Update reachable points Dijkstra-style –Use a data structure which supports reachability queries in the free space surface [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. G 4/28/15CMPS 3130/6130 Computational Geometry35

36 [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. G During the sweep: –Update reachable points Dijkstra-style –Use a data structure which supports reachability queries in the free space surface Update Reachable Points 4/28/15CMPS 3130/6130 Computational Geometry36

37 [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. G During the sweep: –Update reachable points Dijkstra-style –Use a data structure which supports reachability queries in the free space surface Update Reachable Points 4/28/15CMPS 3130/6130 Computational Geometry37

38 [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. G During the sweep: –Update reachable points Dijkstra-style –Use a data structure which supports reachability queries in the free space surface Update Reachable Points 4/28/15CMPS 3130/6130 Computational Geometry38

39 [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. G During the sweep: –Update reachable points Dijkstra-style –Use a data structure which supports reachability queries in the free space surface Update Reachable Points 4/28/15CMPS 3130/6130 Computational Geometry39

40 G [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. During the sweep: –Update reachable points Dijkstra-style –Use a data structure which supports reachability queries in the free space surface Update Reachable Points 4/28/15CMPS 3130/6130 Computational Geometry40

41 G [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. During the sweep: –Update reachable points Dijkstra-style –Use a data structure which supports reachability queries in the free space surface Update Reachable Points 4/28/15CMPS 3130/6130 Computational Geometry41

42 G [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. During the sweep: –Update reachable points Dijkstra-style –Use a data structure which supports reachability queries in the free space surface Update Reachable Points 4/28/15CMPS 3130/6130 Computational Geometry42

43 G [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. During the sweep: –Update reachable points Dijkstra-style –Use a data structure which supports reachability queries in the free space surface Update Reachable Points 4/28/15CMPS 3130/6130 Computational Geometry43

44 G Backtracking After the sweep: –Construct a monotone path via backtracking [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. 4/28/15CMPS 3130/6130 Computational Geometry44

45 Map-Matching  Algorithm for decision problem takes O(mn log(mn)) time and O(mn) space.  Optimization problem with parametric search: O(mn log 2 (mn)) time [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: 262-283, 2003. [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB 853-864, 2005. [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: 379-388, 2006. 4/28/15CMPS 3130/6130 Computational Geometry45

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