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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. (x 3 ,y 3 ), t 3.
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Interpolation for Trajectories Marc van Kreveld (UU) reporting on master thesis research ofBart 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 measurementsData 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 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 11:10, velocity 0 m/s 11:00, velocity 12 m/s What velocity at 11:05?
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 consistency scale invariant - - - - yes - yes - continuity C0 C1 C1 C0 C0 C1 C-1 C0 speed - yes - yes yes yes yes - heading - yes - - yes yes yes - linear cubic Bezier measured cubic Bezier inferred speed constancy, linear speed constancy, PL speed constancy, heading extrapolation Brownian bridges
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