1. Trajectory Simplification
Marc van Kreveld
Dept. of Information and Computing Sciences
Utrecht University
Joint work with Kevin Buchin and Maike Buchin

2. What is a trajectory?
A trajectory is a model for a motion path of a moving object (animal, car, human, hurricane, …)
A common representation is a polygonal line in the plane where the vertices have a time stamp
(xn,yn,tn)
(x2,y2,t2)
(xi,yi,ti)
(x1,y1,t1)

4. Assumptions
Due to lack of further information: constant speed on each segment

5. Assumptions
Due to lack of further information: constant speed on each segment
For this presentation: equal time intervals
t=14
t=16
t=12
t=18
t=20

6. Assumptions
Speed is a vector  constant speed on each segment implies that speed is discontinuous both in magnitude and in direction at vertices

7. Why trajectory simplification?
Algorithms on trajectories are expensive
A trajectory consists of 100 – 1,000,000 vertices
We may analyze 1, 2, or many 1,000s of trajectories
Trajectory similarity
Sub-trajectory similarity
Trajectory self-similarity
Trajectory flocking

8. Algorithms on trajectories
Trajectory (shape) similarity based on the Hausdorff distance: O(n log n) time [Alt et al. 1991]
Trajectory (shape) similarity based on the Frechét distance: O(n log n) time [Alt et al. 1995]

9. Algorithms on trajectories
Sub-trajectory similarity based on average distance: O(n) time [Buchin et al. 2009]
Same start
Duration = T
Duration ≥ T
Different start
O(n)
O(n3 / 2)
O(n4 / 2)

10. trajectory simplification = line simplification?
Line simplification is only about shape preservation 1 2 3 4 5 6 1 6 1 6 5

11. Previous research on trajectory simplification
Put trajectories in time-space (3D) and apply 3D line simplification [Cao et al. 2006] [Gudmundsson et al. 2007]
Where-at (t) gives the location at time t ; the locations in the original and simplified trajectories should be close

12. Where-at (t) t

13. Where-at (t) t

14. When-at (x,y) When-at (x,y) gives the time when the object is at (x,y)
We cannot easily require closeness in time for When-at (x,y) of the original and simplified trajectory t
(x,y)

15. Close in time and space ...
Use a Douglas-Peucker type of approach in 3-space, where a vertex is close enough if a ball with radius  centered at the vertex intersects the approximating line segment
If the approximation is not close enough, add the furthest vertex in 3-space t

16. Alternatives
Use an Imai-Iri type of approach: needs a test to establish whether a shortcut is allowed
Euclidean distance vertices – line segment  

17. Alternatives
Use an Imai-Iri type of approach: needs a test to establish whether a shortcut is allowed
Euclidean distance vertices – line segment  

18. Alternatives
Use an Imai-Iri type of approach: needs a test to establish whether a shortcut is allowed
Euclidean distance vertices – line segment  

19. Alternatives
Use an Imai-Iri type of approach: needs a test to establish whether a shortcut is allowed
Euclidean distance vertices – line segment  

20. Efficiency
The graph can have ~n2 edges; testing one shortcut takes time linear in the number of vertices in between
The Imai-Iri algorithm avoids spending ~n3 time by testing all shortcuts from a single vertex in linear time

21. Efficiency
The graph can have ~n2 edges; testing one shortcut takes time linear in the number of vertices in between
The Imai-Iri algorithm avoids spending ~n3 time by testing all shortcuts from a single vertex in linear time

22. Imai-Iri for trajectory simplification
Replace Euclidean distance by time-corresponding distance to test whether a shortcut is allowed

23. Imai-Iri for trajectory simplification
Replace Euclidean distance by time-corresponding distance to test whether a shortcut is allowed
All good shortcuts can be computed in O(n2 log2 n) time, or approximately in O(n2) time as in the Euclidean case

24. Not (yet) speed-preserving
Both the Douglas-Peucker approach [Cao et al.] and the Imai-Iri approach do not necessarily preserve speed well
Worse: a short stop can be simplified away, whereas it may be significant

25. Speed-preserving adaptation
For any shortcut, test if the implied speed is sufficiently similar to the speeds on the edges it replaces
Still O(n2 log2 n) time, or approximately in O(n2) time

26. Preserve features
Preserving speed as a vector will maintain winding of the simplified trajectory automatically

27. Conclusions
Line simplifications are not directly suitable for trajectory simplification
Variations can be made that are suitable
Various properties can be preserved
Efficient implementations for optimal simplification exist