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) ] / Z where Z = p[ y(t) ]
Basic concepts:recursive Bayesian estimation Posterior pdf given the set y(1:t) ofall 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 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
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 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 • xand A represent mean and covariance of the prior Gaussian pdf
Kalman prediction • Prior mean: previous mean vector times dynamical matrix: x(t) = Dx(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 TA 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 TW-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) = Dx(t-1) A(t) = DT A(t-1) D + N • Update: innovation: v = y - F x innov. cov: W = F TA F + R Kalman gain: K = A F TW-1 posterior mean:s = s + K v posterior cov: A = A - K T W K
Kalman equationswith control inputu • Prediction : x(t) = Dx(t-1) + Cu(t-1) A(t) = DT A(t-1) D + N • Update: innovation: v = y - F x innov. cov: W = F TA F + R Kalman gain: K = A F TW-1 posterior mean:s = s + K v posterior cov: A = A - K T W K