Bayesian Filtering for Location Estimation

1. Bayesian Filtering for Location Estimation Authors: Dieter Fox, Jeffrey Hightower, Lin Liao, Dirk Schulz, Gaetano Borriello -- PERVASIVE computing 2003

2. Outline • Motivation • Bayes filters • Implement Bayes filters • Kalman filter • Multi-hypothesis tracking • Grid-based approaches • Topological approaches • Particle filter • Experiment

3. Motivation • No location sensor takes perfect measurements or works well in all situations • Representing and operating on uncertainty with a statistical tool will benefit the measurements • Estimating location information is the most fundamental in many pervasive computing scenarios • Representing locations statistically enables a unified interface for location information, independent of the sensors used

4. Bayes filters • Bel(xt) = P(xt | z1, z2,…, zt) = α P(zt | xt) * ∫ P(xt | xt-1)* Bel(xt-1)dxt-1 = f(zt, Bel(xt-1)) • Bel(xt) : state distribution at time t • α : a normalizing constant • P(zt | xt) : measurement model • P(xt | xt-1) : state transition model

5. Bayes filters • P(xt | z1:t) = P(z1:t-1, zt | xt) * P(xt) / P(z1:t) = P(zt | xt, z1:t-1) * P(z1:t-1 | xt) * P(xt) / P(z1:t) = P(zt | xt) * P(z1:t-1 | xt) * P(xt) / P(z1:t) = P(zt | xt) * P(xt | z1:t-1) * P(z1:t-1) / P(z1:t) = P(zt | xt) * P(xt | z1:t-1) * P(z1:t-1) / (P(zt | z1:t-1) * P(z1:t-1)) = P(zt | xt) * P(xt | z1:t-1) / (P(zt | z1:t-1) = α * P(zt | xt) * P(xt | z1:t-1) = α * P(zt | xt) * ∫P(xt | xt-1) * P(xt-1 | z1:t-1) dxt-1

6. Bayes filters example

7. Implement Bayes filters • Require specifying • the measurement model P(zt | xt), • the state transition model P(xt | xt-1) , • and the representation of the belief Bel(xt) • Implementation examples • Kalman filters • Multi-hypothesis tracking • Grid-based approaches • Topological approaches • Particle filters

8. Kalman filters • Represents the belief as Gaussian distribution • Bel(xt) is Gaussian • Measurement model is linear function • State transition model is linear function • Advantage : • Computational efficiency, using efficient matrix operations on the mean and covariance • Disadvantage : • Representational power, can represent only Gaussian distribution

9. Multi-hypothesis tracking • Represents the belief as mixtures of Gaussian • wt(i) is proportional to the sensor measurements • Each hypothesis using a Kalman filter • Advantage : • more widely applicable than the Kalman filter • Disadvantage : • computationally more expensive • Require sophisticated techniques or heuristics to determine when to add or delete hypotheses

10. Grid-based approaches • Represents the belief on discrete, the integration in equations will replace to summation • For indoor location estimation, grid-based filters tessellate the environment into small patches • Advantage : • Can represent distributions over the discrete state space • Disadvantage : • Computational and space complexity are high

11. Topological approaches • Using a graph to represent the environment, each node is a location and the edges is the environment’s connectivity • Advantage : • Efficiency, because they represent distributions over small, discrete state spaces • Disadvantage : • The representation is coarseness • Require sensors related to the environment’s layout

12. Particle filters • Represent belief by sets of particles • xt(i) is a state, wt(i) is weight • Resampling by state transition model • Weighted by measurement model • Advantage : • Can represents probability densities • Can work on non-Gaussian, non-linear dynamic systems • Very efficient than grid-based approach, focus only on state space with high probability • Low implementation overhead • Disadvantage : • Worst case complexity grows exponentially, must careful when applying to high dimension estimation problems

13. Particle filter example

14. Experiment • Sensors • Ultrasound sensors & tags • 4.5-meter, Gaussian distribution of measured distance • Infrared sensors & badges • A no distance detection, in a specific area • Laser range finders • Several short beams form a shadow region indicating a person’s presence

15. Experiment • Implement approach • Particle filters • The state contains person’s location, orientation, and motion velocity

16. Experiment • Constrain the state space to locations on a Voronoi graph, which is a structure similar to a skeleton of an environment’s free space

17. Expriment • Tracking multiple people • Problem • Requires maintaining the hypotheses for possible track continuations • Proposed solution • Track individual people using Kalman filters • A particle filter maintains multi hypotheses regarding the ID of people