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?
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 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
The end
