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