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Generative model for a generic signal
Basic concepts in tracking/filtering
• State variables x; observation y: both are vectors
• Discrete time: x(t), y(t), x(t+1), y(t+1)
• Probability P
• pdf [density] p(v) of vector variable v :
p(v*) = lim P(v* < v < v*+dv) / dv dv->0 .
Basic concepts:Gaussian pdf
A Gaussian pdf is completely characterized by 2 parameters:
• its mean vector
• its covariance matrix
Basic concepts: prior and likelihood
• Prior pdf of variable v: in tracking, this is usually the probability conditional on the previous estimate: p[ v(t) | v(t-1) ]
• Likelihood: pdf of the observation, given the state variables: p[ y(t) | x(t) ]
Basic concepts:Bayes’ theorem
• Posterior pdf is proportional to prior pdf times likelihood:
p[ x(t) | x(t-1), y(t) ] =
p[ x(t) | x(t-1) ] p[ y(t) | x(t) ] / Zwhere
Z = p[ y(t) ]
Basic concepts:recursive Bayesian estimation
Posterior pdf given the set y(1:t) of all observations up to time t:
p[ x(t) | y(1:t) ] =
p[ y(t) | x(t) ] . p[ x(t) | x(t-1) ] .
p[ x(t-1) | y(1:t-1) ] / Z1
Basic concepts:recursive Bayesian estimation
p[ x(t) | y(1:t) ] =
p[ y(t) | x(t) ] . p[ x(t) | x(t-1) ] .
p[ y(t-1) | x(t-1) ] . p[ x(t-1) | x(t-2) ] .
p[ x(t-2) | y(1:t-2) ] / Z2
Basic concepts:recursive Bayesian estimation
p[ x(t) | y(1:t) ] =
p[ y(t) | x(t) ] . p[ x(t) | x(t-1) ] .
p[ y(t-1) | x(t-1) ] . p[ x(t-1) | x(t-2) ] .
p[ y(t-2) | x(t-2) ] . p[ x(t-2) | x(t-3) ] .
… / Z*
Kalman model
Kalman model in words
• Dynamical model: the current state x(t) is a linear (vector) function of the previous state x(t-1) plus additive Gaussian noise
• Observation model: the observation y(t) is a linear (vector) function of the state x(t)
plus additive Gaussian noise
Problems in visual tracking
• Dynamics is nonlinear, non-Gaussian
• Pose and shape are nonlinear, non-Gaussian functions of the system state
• Most important: what is observed is not image coordinates, but pixel grey-level values: a nonlinear function of object shape and pose, with non-additive, non-Gaussian noise
More detailed model
Back to Kalman
• A Gaussian pdf, propagated through a linear system, remains Gaussian
• If Gaussian noise is added to a variable with Gaussian pdf, the resulting pdf is still Gaussian (sum of covariances)
---> The predicted state pdf is Gaussian if the previous state pdf was Gaussian
---> The observation pdf is Gaussian if the state pdf is Gaussian
Kalman dynamics
Kalman observation
Kalman posterior pdf
• The product of 2 Gaussian densities is still Gaussian (sum of inverse covariances)
---> the posterior pdf of the state is Gaussian if prior pdf and likelihood are Gaussian
Kalman filter
• Operates in two steps: prediction and update
• Prediction: propagate mean and covariance of the state through the dynamical model
• Update: combine prediction and innovation (defined below) to obtain the state estimate with maximum posterior pdf
Note on the symbols
• From now on, the symbol x represents no longer the ”real” state (which we cannot know) but the mean of the posterior Gaussian pdf
• The symbol A represents the covariance of the posterior Gaussian pdf
• x and A represent mean and covariance of the prior Gaussian pdf
Kalman prediction
• Prior mean: previous mean vector times dynamical matrix:
x(t) = D x(t-1)
• Prior covariance matrix: previous covariance matrix pre- AND post-multiplied by dynamical matrix PLUS noise covariance:
A(t) = DT A(t-1) D + N
Kalman update
In the update step, we must reason backwards, from effect (observation) to cause (state): we must ”invert” the generative process.
Hence the update is more complicated than the prediction.
Kalman update (continued)
Basic scheme:
• Predict the observation from the current state estimate
• Take the difference between predicted and actual observation (innovation)
• Project the innovation back to update the state
Kalman innovation
Observation matrix F
The innovation v is given by:
v = y - F x
Observation-noise covariance R
The innovation has covariance W:
W = F T A F + R
Kalman update: state mean vector
• Posterior mean vector: add weighted innovation to predicted mean vector
• weigh the innovation by the relative covariances of state and innovation:
larger covariance of the innovation
--> larger uncertainty of the innovation
--> smaller weight of the innovation
Kalman gain
• Predicted state covariance A
• Innovation covariance W
• Observation matrix F
• Kalman gain K = A F T W-1
• Posterior state mean:
s = s + K v
Kalman update: state covariance matrix
• Posterior covariance matrix: subtract weighted covariance of the innovation
• weigh the covariance of the innovation by the Kalman gain:
A = A - K T W K• Why subtract? Look carefully at the equation:
larger innovation covariance
--> smaller Kalman gain K
--> smaller amount subtracted!
Kalman update: state covariance matrix (continued)
• Another equivalent formulation requires matrix inversion (sum of inverse covariances)
Advanced note:• The equations given here are for the usual
covariance form of the Kalman filter • It is possible to work with inverse covariance
matrices all the time (in prediction and update): this is called the information form of the Kalman filter
Summary of Kalman equations• Prediction :
x(t) = D x(t-1)
A(t) = DT A(t-1) D + N• Update:innovation: v = y - F x
innov. cov: W = F T A F + R
Kalman gain: K = A F T W-1
posterior mean: s = s + K v
posterior cov: A = A - K T W K
Kalman equationswith control input u
• Prediction :x(t) = D x(t-1) + C u(t-1)
A(t) = DT A(t-1) D + N• Update:innovation: v = y - F x
innov. cov: W = F T A F + R
Kalman gain: K = A F T W-1
posterior mean: s = s + K v
posterior cov: A = A - K T W K
