Bayesian Filtering for Location Estimation Authors Dieter Fox

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

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

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

Bayes filters • Bel(xt) = P(xt | z 1, z 2, …, 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

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

Bayes filters example

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

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

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

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

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

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

Particle filter example

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

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

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

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