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Jamal Saboune - CRV10 Tutorial Day 1
Bayesian state estimation and application to
tracking Jamal Saboune
VIVA Lab - SITE - University of Ottawa
Jamal Saboune - CRV10 Tutorial Day 2
Dynamic state estimationA dynamic process described using a number of random variables (state variables) The evolution of the variables follows a modelIndication on all or some of the variables (observation)
Evaluate at time t in a recursive manner (using t-1) the process represented by its state vector Xt , given the history of observations Yt
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Dynamic state estimationThe Markov process is defined by its transition model and the initial state vector:
The observation is defined by the observation model :
Xt= ft (Xt-1 ,Πt )X0
Yt= ht (Xt ,vt )
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Dynamic state estimation- Bayesian approachEstimate the a posteriori probability density function P(Xt / Yt ) using the transition model, the observation model and the probability density function P(Xt-1 / Yt-1 )
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Kalman filterProbability density propagation = Theoretical solution not an analytical oneParticular case : The observation and process noises distributions are Gaussian + The transition and observation functions are linear The probability density functions are Gaussian mono-modal
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Kalman filterA number of equations using the transition/observation functions and covariance matrices Optimal estimation of the state vector
Minimizes the mean square error between the estimated state vector X’t and the ‘real’ state vector Xt E[(X’t - Xt )2] given the history of observations Yt
Extended Kalman Filter (EKF) is the non linear version of the KF = The transition and observation function can be non-linear 6
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Kalman filter example
Jamal Saboune - CRV10 Tutorial Day
Condensation algorithm (Isard, Blake 98)Multimodal and non Gaussian probability
densities
Model the uncertainty
Each possible configuration of the state vector is represented by a ‘particle'
The likelihood of a certain configuration is called ‘weight’
The posterior (a posteriori) density is represented using N ‘weighted’ particles 8
Jamal Saboune - CRV10 Tutorial Day
Selection
Prediction
Measure
Particles at t-1
Likelihood function
Chosen particle
CO
ND
EN
SA
TIO
N – tim
e t
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Tracking of a hand movement using an edge detector
Condensation algorithm (Isard, Blake 98)
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Hands and head movement tracking using color models and optical flow (Tung et al. 2008)
Condensation algorithm for tracking
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Head tracking with contour models (Zhihong et al. 2002)
Condensation algorithm for tracking
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Interval Particle Filtering for 3D motion capture (Saboune et al. 05,07,08)3D humanoid model
adapted to the height of the person 32 degrees of freedom to simulate the human movement
Find the best
fitting 3D model
configuration
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Interval Particle Filtering for 3D motion capture (Saboune et al. 05,07,08)Modify the Condensation algorithm and
adapt it to the human motion tracking Good estimation using a reduced number of particles
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Particle Filtering for multi-targets trackingJoint state vector for all targets and joint
likelihood function (Isard, MacCormick 2001, Zhao, Nevatia 2004)
Multiple particle filters (one/target) and combined global likelihood function (Koller-Maier 2001)
The Explorative particle filtering for 3D people tracking (Saboune, Laganiere 09)
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Q & A
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