interpolation for trajectories
Post on 03-Jan-2016
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Interpolation for Trajectories
Marc van Kreveld (UU)
reporting on master thesis research of Bart Liefers (UU)
also in collaboration with
Emiel van Loon (UvA)
Sampling and trajectories
• Usually we get movement data from a set of measured locations at known times
(x1,y1), t1 (x2,y2), t2
(x3,y3), t3
(x3,y3), t3
Sampling and trajectories
• Usually we get movement data from a set of measured locations at known times
(x1,y1), t1 (x2,y2), t2
(x3,y3), t3
(x3,y3), t3
(piecewise) linear interpolation
at time (t3+t2) /2
Sampling and trajectories
• Sometimes linear interpolation is not called for
Sampling and trajectories
• Sometimes linear interpolation is not called for
Sampling and trajectories
• Interpolated location implies velocity (speed and heading)
(x1,y1), t1 (x2,y2), t2
(x3,y3), t3
(x3,y3), t3
linear interpolation constant speed
at time (t3+t2) /2
The case of gulls
• Lesser black-backed gull
• Five gulls, colony on Texel• Sampling intervals irregular;
3 sec – 30 min• Also velocity measurements
Data from the UvA, computational geo-ecology,with Emiel van Loon and Judy Shamoun-Baranes
30 measurements5:22 hours
Interpolation affects basic properties
• Location at any time• Speed at any time• Trajectory length• Average speed
piecewise linear interpolation
spline interpolation
Interpolation affects basic properties
• Location at any time• Speed at any time• Trajectory length• Average speed
• Availability of velocityallows new interpolations
piecewise linear interpolation
spline interpolation
Interpolation affects basic properties
• Location at any time• Speed at any time• Trajectory length• Average speed
• Availability of velocityallows new interpolations
piecewise linear interpolation
spline interpolation
consistent spline interpolation
Interpolation issues
• Consistency location and velocity:– Position correct at the measured locations– Velocity correct at the measured locations– Integral of velocity = path between any two
measured locations
• Scale-invariance for trajectory length:– fewer sampled locations should not result in a
smaller trajectory length
Interpolation issues: gull specific
11:00, velocity 12 m/s 11:10, velocity 0 m/s
What velocity at 11:05?
What probably happened between 11:00 and 11:10 …
Interpolation issues: gull specific
• Speed constancy between measurements
12 m/s 1 m/s
t0 t1
t0 t1
speed profile
speed
time
Interpolation issues: gull specific
• Speed constancy between measurements
12 m/s 1 m/s
t0 t1
t0 t1
speed profile
speed
time
Interpolation models
• Linear model (basic, ignores velocity)• Cubic Bezier models
– Use measured velocity– Infer velocity from adjacent samples
• Speed constancy models– Linear interpolation for path (ignores heading)– Piecewise linear interpolation for path– Path from interpolation of heading
Interpolation models
• Extrapolation model (use velocity of nearest sample, location is not continuous)
• Brownian bridges model
half-time
Properties of the models
linearcubic Bezier measured
cubic Bezier inferredspeed constancy, linear
speed constancy, PLspeed constancy, heading
extrapolationBrownian bridges
continuity
C0
C1
C1
C0
C0
C1
C-1
C0
speed
-yes
-yesyesyesyes
-
heading
-yes
--
yesyesyes
-
scale invariant
----
yes-
yes-
consistency
Analysis using densely sampled trajectories
• Triples with 3-second intervals (exclude stationary birds)• Predict location & speed at middle sample from the
outer samples• Analyze coarser and coarser sampled trajectories
Analysis using densely sampled trajectories
• Triples with 3-second intervals (exclude stationary birds)• Predict location & speed at middle sample from the
outer samples• Analyze coarser and coarser sampled trajectories
Linear model
Analysis using densely sampled trajectories
• Triples with 3-second intervals (exclude stationary birds)• Predict location & speed at middle sample from the
outer samples• Analyze coarser and coarser sampled trajectories
Speed constancy model, linear
Analysis using densely sampled trajectories
• Triples with 3-second intervals (exclude stationary birds)• Predict location & speed at middle sample from the
outer samples• Analyze coarser and coarser sampled trajectories
Cubic Bezier model using velocity
Analysis of location
• At high resolution several models are best, about 20% better than linear interpolation
• At lower resolution the speed constancy model, linear, is best, about 30% better than linear interpolation
at lower resolutions, speed helps but heading doesn’t
Analysis of speed and heading
• At high resolutions, several models are best for speed, including linear interpolation
• At lower resolutions the speed constancy model, linear, is best for speed
• For heading, linear interpolation is best, especially for lower resolutions
Analysis of trajectory length
• The extrapolation model and piecewise linear speed constancy model are not biased by sampling rate, unlike all other models
• Simple integration of speed works best
Conclusions
• For interpolation, scale matters• The particular application matters
• At lower resolutions, speed helps but heading doesn’t• Speed consistency models appear to work well for
location and velocity
The end
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