Transcript Interpolation for Trajectories

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
(x3,y3), t3
(x3,y3), t3
(x1,y1), t1
(x2,y2), t2
Sampling and trajectories
• Usually we get movement data from a set of measured
locations at known times
(x3,y3), t3
at time (t3+t2) /2
(x3,y3), t3
(x1,y1), t1
(x2,y2), t2
(piecewise) linear interpolation
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)
(x3,y3), t3
at time (t3+t2) /2
(x3,y3), t3
(x1,y1), t1
(x2,y2), t2
linear interpolation  constant speed
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 measurements
5:22 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 velocity
allows 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 velocity
allows 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
11:10, velocity 0 m/s
11:00, velocity 12 m/s
What velocity at 11:05?
What probably happened between 11:00 and 11:10 …
Interpolation issues: gull specific
• Speed constancy between measurements
t0
t1
1 m/s
12 m/s
speed
speed profile
t0
t1
time
Interpolation issues: gull specific
• Speed constancy between measurements
t0
t1
1 m/s
12 m/s
speed
speed profile
t0
t1
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
continuity
linear
cubic Bezier measured
cubic Bezier inferred
speed constancy, linear
speed constancy, PL
speed constancy, heading
extrapolation
Brownian bridges
C0
C1
C1
C0
C0
C1
C-1
C0
consistency
speed heading
yes
yes
yes
yes
yes
-
yes
yes
yes
yes
-
scale
invariant
yes
yes
-
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