Model-Driven Matching and Segmentation of Trajectories

Transcription

1 Model-Driven Matching and Segmentation of Trajectories Jiangwei Pan (Duke University) joint work with Swaminathan Sankararaman (Akamai Technologies) Pankaj K. Agarwal (Duke University) Thomas Mølhave (Scalable Algorithmics USA) Arnold P. Boedihardjo (U.S. Army Corps of Engineers)

2 Trajectories Definition: function from time domain to R d

3 Trajectories Definition: function from time domain to R d Observed: a sequence of sample points

4 Trajectories Definition: function from time domain to R d Observed: a sequence of sample points Examples: GPS traces of vehicles Object movement in videos Animal migration trajectories

5 Outline Matching A new model/algorithm for matching similar portions of two trajectories distinguish deviation (gaps) from noise handle non-uniform sampling Deviation/Gap Noise Segmentation Discover common patterns from a collection of trajectories Experiments

6 Trajectory Matching

7 Existing Approaches for Matching Two trajectories (sequences): P = p 1,, p m, Q = q 1,, q n Dynamic Time Warping (DTW) person walk on P, dog on Q, no backtrack minimize average leash length For trajectory matching: not meaningful at significant deviations

8 Existing Approaches for Matching Two trajectories (sequences): P = p 1,, p m, Q = q 1,, q n Dynamic Time Warping (DTW) person walk on P, dog on Q, no backtrack minimize average leash length For trajectory matching: not meaningful at significant deviations Biological Sequence Alignment (Seq-Align) allow gaps maximize score function one-to-one matching For trajectory matching: cannot handle non-uniform sampling

9 Existing Approaches for Matching Two trajectories (sequences): P = p 1,, p m, Q = q 1,, q n Dynamic Time Warping (DTW) person walk on P, dog on Q, no backtrack minimize average leash length Biological Sequence Alignment (Seq-Align) allow gaps maximize score function one-to-one matching For trajectory matching: not meaningful at significant deviations For trajectory matching: cannot handle non-uniform sampling Both can be computed in quadratic time.

10 Our Matching Model Combine the advantages of DTW and Seq-Align Handle non-uniform sampling: allow multiple-to-one matching (as DTW) Distinguish deviation from noise: allow gaps (as Seq-Align) Deviation/Gap Noise

11 Trajectory Matching - Assignment (Use to denote a gap) An assignment for P and Q is a pair of functions α : P Q { } β : Q P { }

12 Trajectory Matching - Assignment (Use to denote a gap) An assignment for P and Q is a pair of functions α : P Q { } β : Q P { } View as directed graph: α, β decide outgoing edges. Each point at most one outgoing edge could have multiple incoming edges

13 Trajectory Matching - Assignment (Use to denote a gap) An assignment for P and Q is a pair of functions α : P Q { } β : Q P { } View as directed graph: α, β decide outgoing edges. Each point at most one outgoing edge could have multiple incoming edges

14 Trajectory Matching - Assignment (Use to denote a gap) An assignment for P and Q is a pair of functions α : P Q { } β : Q P { } View as directed graph: α, β decide outgoing edges. Each point at most one outgoing edge could have multiple incoming edges

15 Trajectory Matching - Assignment (Use to denote a gap) An assignment for P and Q is a pair of functions α : P Q { } β : Q P { } View as directed graph: α, β decide outgoing edges. Each point at most one outgoing edge could have multiple incoming edges

16 Trajectory Matching - Score of Assignment Score of assignment α, β (λ > 0, θ < 0, > 0 are parameters) Score for matched edges Score for gaps E: set of matching edges gap: maximal contiguous sequence of points assigned to Γ: set of gaps Objective: find assignment α, β with maximum score

17 Trajectory Matching - Score of Assignment Score of assignment α, β (λ > 0, θ < 0, > 0 are parameters) Score for matched edges Score for gaps E: set of matching edges gap: maximal contiguous sequence of points assigned to Γ: set of gaps Objective: find assignment α, β with maximum score We propose a dynamic programming algoirthm that computes optimal assignment in O(mn) time

18 Segmentation of Trajectories

19 Related Work [Lee, Han, Whang, 2007]: partition and cluster subtrajectories [Buchin et al., 2011]: use Fréchet distance to discover popular subtrajectories [Chen, Su, Huang, Zhang, Guibas, 2013, this conference]: formulate the segmentation problem as an integer linear program

20 Trajectory Segmentation Given a set of k trajectories T = {T 1,, T k } Goal: segment trajectories into fragments represent trajectories compactly

21 Segmentation Algorithm set of k trajectories T = {T 1,, T k } V : set of all trajectory points The algorithm 1. Labeling: assign a label L(p) {1,, k} to each point p V 2. Clustering: cluster points into fragments based on their labels

22 Segmentation Algorithm - Labeling (Assign a label L(p) {1,, k} to each point p) Intuition: j L(p) means trajectory T j pass p

23 Segmentation Algorithm - Labeling (Assign a label L(p) {1,, k} to each point p) Intuition: j L(p) means trajectory T j pass p Run matching algorithm between every pair of trajectories Labeling: L(p) contains trajectory that contains p trajectories that p is matched to

24 Segmentation Algorithm - Labeling (Assign a label L(p) {1,, k} to each point p) Intuition: j L(p) means trajectory T j pass p Run matching algorithm between every pair of trajectories Labeling: L(p) contains trajectory that contains p trajectories that p is matched to Note: other matching algorithms can also be used in labeling (comparison in experiment).

25 Segmentation Algorithm - Clustering Fragment: maximal contiguous subsequences of points with same label

26 Experiments

27 Datasets WorkOut: 330 trajectories from road cycling and running 1M points low noise, uniform (1 second) sampling rate

28 Datasets WorkOut: 330 trajectories from road cycling and running 1M points low noise, uniform (1 second) sampling rate Bus: 143 trajectories of school buses in Athens, Greece 65K points high noise, uniform sampling rate

29 Datasets WorkOut: 330 trajectories from road cycling and running 1M points low noise, uniform (1 second) sampling rate Bus: 143 trajectories of school buses in Athens, Greece 65K points high noise, uniform sampling rate GeoLife (Microsoft Research Asia): 17,621 trajectories of 182 users in Beijing, China 5M points high noise, non-uniform sampling rate

30 Matching Results On a pair of trajectories from the Bus dataset. (a) Dynamic Time Warping (DTW)

31 Matching Results On a pair of trajectories from the Bus dataset. (a) Dynamic Time Warping (DTW) (b) DTW-Pruned (prune long edges)

32 Matching Results On a pair of trajectories from the Bus dataset. (a) Dynamic Time Warping (DTW) (b) DTW-Pruned (prune long edges) (c) Sequence Alignment (Seq-Align)

33 Matching Results On a pair of trajectories from the Bus dataset. (a) Dynamic Time Warping (DTW) (b) DTW-Pruned (prune long edges) (c) Sequence Alignment (Seq-Align) (d) Our matching algorithm (Assignment) DTW-Pruned/Seq-Align produce many spurious gaps

34 Matching Results Assignment vs. DTW-Pruned Each point in figure: result on one pair of trajectories (a) GeoLife dataset Assignment has fewer gaps, DTW-Pruned has many unnecessary gaps

35 Matching Results Assignment vs. DTW-Pruned Each point in figure: result on one pair of trajectories (b) WorkOut dataset Assignment has fewer gaps, DTW-Pruned has many unnecessary gaps

36 Segmentation Results WorkOut, Assignment GeoLife, Assignment

37 Segmentation Results - Quantitative Comparison Recall: other matching algorihtms can also be used in Labeling step Compare using Assignment and DTW-Pruned in labeling

38 Segmentation Results - Quantitative Comparison Assignment covers more points with the same number of fragments. DTW-Pruned,Workout Assignment,Workout DTW-Pruned,GeoLife Assignment,GeoLife 1.00 Fraction of Points Number of Fragments vary ρ: minimum number of points in a fragment

39 Segmentation Results - Quantitative Comparison Assignment covers more points with the same number of fragments. DTW-Pruned,Workout Assignment,Workout DTW-Pruned,GeoLife Assignment,GeoLife Fraction of Points Number of Fragments vary η: minimum size of fragment label

40 Conclusion In this paper, we proposed a new model for matching two trajectories that can handle non-uniform sampling and distinguish between deviation and noise. a segmentation algorithm that discovers a collection of fragments from a set of trajectories

41 Conclusion In this paper, we proposed a new model for matching two trajectories that can handle non-uniform sampling and distinguish between deviation and noise. a segmentation algorithm that discovers a collection of fragments from a set of trajectories Future work: we are working on better labeling scheme for the segmentation algorithm handle noisy and sparse trajectories

42 Thank you.