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Bayesian state estimation and application to tracking. Jamal Saboune [email protected] VIVA Lab - SITE - University of Ottawa. Dynamic state estimation. A dynamic process described using a number of random variables (state variables) The evolution of the variables follows a model

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### Bayesian state estimation and application to tracking

Jamal Saboune

VIVA Lab - SITE - University of Ottawa

Jamal Saboune - CRV10 Tutorial Day

### Dynamic state estimation

A dynamic process described using a number of random variables (state variables)

The evolution of the variables follows a model

Indication 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

Jamal Saboune - CRV10 Tutorial Day

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### Dynamic state estimation

Xt= ft (Xt-1 ,Πt )

X0

The Markov process is defined by its transition model and the initial state vector:

The observation is defined by the observation model :

Yt= ht (Xt ,vt )

Jamal Saboune - CRV10 Tutorial Day

### Dynamic state estimation- Bayesian approach

Estimate 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 )

Jamal Saboune - CRV10 Tutorial Day

### Kalman filter

Probability density propagation = Theoretical solution not an analytical one

Particular case : The observation and process noises distributions are Gaussian + The transition and observation functions are linear  The probability density functions are Gaussian mono-modal

Jamal Saboune - CRV10 Tutorial Day

### Kalman filter

A 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 andthe ‘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

Jamal Saboune - CRV10 Tutorial Day

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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

Jamal Saboune - CRV10 Tutorial Day

CONDENSATION – time t

Selection

Prediction

Likelihood function

Measure

Jamal Saboune - CRV10 Tutorial Day

Chosen particle

Tracking of a hand movement using an edge detector

Jamal Saboune - CRV10 Tutorial Day

Hands and head movement tracking using color models and optical flow (Tung et al. 2008)

Jamal Saboune - CRV10 Tutorial Day

Head tracking with contour models (Zhihong et al. 2002)

Jamal Saboune - CRV10 Tutorial Day

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

Jamal Saboune - CRV10 Tutorial Day

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Particle Filtering for multi-targets tracking et al. 05,07,08)

• Joint 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 et al. 05,07,08)

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