Model-Driven Matching and Segmentation of Trajectories Jiangwei Pan - - PowerPoint PPT Presentation

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)

Trajectories

Definition: function from time domain to Rd

Trajectories

Definition: function from time domain to Rd Observed: a sequence of sample points

Trajectories

Definition: function from time domain to Rd Observed: a sequence of sample points Examples:

◮ GPS traces of vehicles ◮ Object movement in videos ◮ Animal migration

trajectories

Outline

Matching A new model/algorithm for matching similar portions of two trajectories

◮ distinguish deviation (gaps)

from noise

◮ handle non-uniform sampling

Segmentation Discover common patterns from a collection of trajectories Experiments

Deviation/Gap Noise

Trajectory Matching

Existing Approaches for Matching

Two trajectories (sequences): P = p1, · · · , pm, Q = q1, · · · , qn 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

Existing Approaches for Matching

Two trajectories (sequences): P = p1, · · · , pm, Q = q1, · · · , qn 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

Existing Approaches for Matching

Two trajectories (sequences): P = p1, · · · , pm, Q = q1, · · · , qn 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.

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

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 ∪ {⊥}

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

• utgoing edges. Each point

◮ at most one outgoing edge ◮ could have multiple incoming

edges

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

• utgoing edges. Each point

◮ at most one outgoing edge ◮ could have multiple incoming

edges

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

• utgoing edges. Each point

◮ at most one outgoing edge ◮ could have multiple incoming

edges

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

• utgoing edges. Each point

◮ at most one outgoing edge ◮ could have multiple incoming

edges

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

• f points assigned to ⊥

◮ Γ: set of gaps

Objective: find assignment α, β with maximum score

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

• f 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

Segmentation of Trajectories

Related Work

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

Trajectory Segmentation

Given a set of k trajectories T = {T1, · · · , Tk} Goal:

◮ segment trajectories into fragments ◮ represent trajectories compactly

Segmentation Algorithm

set of k trajectories T = {T1, · · · , Tk} 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

Segmentation Algorithm - Labeling

(Assign a label L(p) ⊆ {1, · · · , k} to each point p) Intuition: j ∈ L(p) means trajectory Tj “pass” p

Segmentation Algorithm - Labeling

(Assign a label L(p) ⊆ {1, · · · , k} to each point p) Intuition: j ∈ L(p) means trajectory Tj “pass” p Run matching algorithm between every pair of trajectories Labeling: L(p) contains

◮ trajectory that contains p ◮ trajectories that p is

matched to

Segmentation Algorithm - Labeling

(Assign a label L(p) ⊆ {1, · · · , k} to each point p) Intuition: j ∈ L(p) means trajectory Tj “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).

Segmentation Algorithm - Clustering

Fragment: maximal contiguous subsequences of points with same label

Experiments

Datasets

WorkOut: 330 trajectories from road cycling and running

◮ ∼ 1M points ◮ low noise, uniform (1 second) sampling rate

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

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

Matching Results

On a pair of trajectories from the Bus dataset.

(a) Dynamic Time Warping (DTW)

Matching Results

On a pair of trajectories from the Bus dataset.

(a) Dynamic Time Warping (DTW) (b) DTW-Pruned (prune long edges)

Matching Results

On a pair of trajectories from the Bus dataset.

(a) Dynamic Time Warping (DTW) (c) Sequence Alignment (Seq-Align) (b) DTW-Pruned (prune long edges)

Matching Results

On a pair of trajectories from the Bus dataset.

(a) Dynamic Time Warping (DTW) (c) Sequence Alignment (Seq-Align) (b) DTW-Pruned (prune long edges) (d) Our matching algorithm (Assignment)

DTW-Pruned/Seq-Align produce many spurious gaps

Matching Results

Assignment vs. DTW-Pruned Each point in figure: result on

• ne pair of trajectories

(a) GeoLife dataset

Assignment has fewer gaps, DTW-Pruned has many unnecessary gaps

Matching Results

Assignment vs. DTW-Pruned Each point in figure: result on

• ne pair of trajectories

(b) WorkOut dataset

Assignment has fewer gaps, DTW-Pruned has many unnecessary gaps

Segmentation Results

WorkOut, Assignment GeoLife, Assignment

Segmentation Results - Quantitative Comparison

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

Segmentation Results - Quantitative Comparison

Assignment covers more points with the same number of fragments.

DTW-Pruned,Workout DTW-Pruned,GeoLife Assignment,Workout Assignment,GeoLife

.7 .7 5 .8 .8 5 .9 .9 5 1 .0

Fra ction of Poin ts

2 4 6 8 1 1 2 1 4

N u m b e r

• f

F r a g m e n t s

vary ρ: minimum number of points in a fragment

Segmentation Results - Quantitative Comparison

Assignment covers more points with the same number of fragments.

DTW-Pruned,Workout DTW-Pruned,GeoLife Assignment,Workout Assignment,GeoLife

.2 .3 .4 .5 .6 .7 .8 .9

Fra ction of Poin ts

1 2 3 4 5 6 7 8

Nu m be r of Fra g m e n ts

vary η: minimum size of fragment label

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

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

Thank you.