Bayesian Filtering for Location Estimation

1. Bayesian Filtering for Location Estimation D. Fox, J. Hightower, L. Liao, D. Schulz, and G. Borriello Presented by: Honggang Zhang

2. Outline • Basic idea of Bayes filters • Several types of Bayes filters • Some applications

3. System state dynamics Observation dynamics We are interested in: Belief or posterior density Bayes Filters Estimating system state from noisy observations

4. Recall “law of total probability” and “Bayes’ rule” From above, constructing two steps of Bayes Filters Predict: Update:

5. Assumptions: Markov Process Predict: Update:

6. Bayes Filter How to use it? What else to know? Motion Model Perceptual Model Start from:

7. Step 0: initialization Step 1: updating Example 1

8. Step 3: updating Step 4: predicting Step 2: predicting Example 1 (continue)

9. Several types of Bayes filters • They differs in how to represent probability densities • Kalman filter • Multihypothesis filter • Grid-based approach • Topological approach • Particle filter

10. Recall general problem Assumptions of Kalman Filter: Belief of Kalman Filter is actually a unimodal Gaussian Advantage: computational efficiency Disadvantage: assumptions too restrictive Kalman Filter

11. Multi-hypothesis Tracking • Belief is a mixture of Gaussian • Tracking each Gaussian hypothesis using a Kalman filter • Deciding weights on the basis of how well the hypothesis predict the sensor measurements • Advantage: • can represent multimodal Gaussian • Disadvantage: • Computationally expensive • Difficult to decide on hypotheses

12. Grid-based Approaches • Using discrete, piecewise constant representations of the belief • Tessellate the environment into small patches, with each patch containing the belief of object in it • Advantage: • Able to represent arbitrary distributions over the discrete state space • Disadvantage • Computational and space complexity required to keep the position grid in memory and update it

13. Topological approaches • A graph representing the state space • node representing object’s location (e.g. a room) • edge representing the connectivity (e.g. hallway) • Advantage • Efficiency, because state space is small • Disadvantage • Coarseness of representation

14. Particle filters • Also known as Sequential Monte Carlo Methods • Representing belief by sets of samples or particles • are nonnegative weights called importance factors • Updating procedure is sequential importance sampling with re-sampling

15. Step 0: initialization Each particle has the same weight Step 1: updating weights. Weights are proportional to p(z|x) Example 2: Particle Filter

16. Step 3: updating weights. Weights are proportional to p(z|x) Step 4: predicting. Predict the new locations of particles. Step 2: predicting. Predict the new locations of particles. Example 2: Particle Filter Particles are more concentrated in the region where the person is more likely to be

17. Compare Particle Filter with Bayes Filter with Known Distribution Updating Example 1 Example 2 Predicting Example 1 Example 2

18. Comments on Particle Filters • Advantage: • Able to represent arbitrary density • Converging to true posterior even for non-Gaussian and nonlinear system • Efficient in the sense that particles tend to focus on regions with high probability • Disadvantage • Worst-case complexity grows exponentially in the dimensions

19. Comparison + : good; 0 : neutral; - : weak

20. Example Applications • Particle Filters (unconstrained) • Particle Filters (constrained) • Combination of Particle Filters and Kalman Filters

21. Sensors • Ultra sound and infrared Sensors: • Less accurate but certain with identification • Laser range finder • Accurate but anonymous

22. Example Indoor Environment Red circles: ultra-sound ID sensors Blue squares: infrared ID sensors

23. Using Particle Filters (unconstrained) • Due to high noise level of ultrasound and infrared sensors, we use particle filters • Whenever detect the person, updating particles

24. Using Particle Filters (unconstrained) Another Example

25. Using Particle Filters (unconstrained) Another Example

26. Using Particle Filters (constrained) • A more efficient way to use particle filters • constraining the state space to locations on a Voronoi graph (a structure similar to a skeleton of an environment’s free space)

27. Combine Particle and Kalman Filters To Solve Data Association Problem Laser range finder Data Association Problem In area 5 and 6, resolving ambiguity, but need additional hypotheses In area 3 and 4, identities of A and B are known Area covered by ID sensors

28. Combine Particle and Kalman Filters To Solve Data Association Problem • Track individual people using Kalman filters (using laser range data) • A particle filter maintains multiple hypothesis wrt identities of people

29. Conclusion • “The Location Stack”: a general framework with publicly available implementation • Probabilistic techniques have tremendous potential for inference problems Questions?