Human-Computer Interaction Kalman Filter Hanyang University Jong-Il Park.
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Transcript of Human-Computer Interaction Kalman Filter Hanyang University Jong-Il Park.
Human-Computer InteractionHuman-Computer Interaction Kalman FilterKalman Filter
Hanyang University
Jong-Il Park
Rudolf Emil KalmanRudolf Emil Kalman
Born 1930 in Hungary Born 1930 iHungary
BS and MS from MIT BS and MS from MIT
PhD 1957 from Columbia Filter developed in 1960-61 Filter Now retired Now retired
What is a Kalman filter?What is a Kalman filter?
Just some applied math. A linear system: f(a+b) = f(a) + f(b). Noisy data in hopefully less noisy
out. But delay is the price for filtering... Pure KF does not even adapt to the
data.
What is it used for?What is it used for?
Tracking missiles Tracking missiles Tracking heads/hands/drumsticks Extracting lip motion from video Fitting Bezier patches to point data Lots of computer vision
applications Economics Navigation Navigation
Typical Kalman filter Typical Kalman filter applicationapplication
system
measuring device
Kalmanfilter
measurement noise
system noise
systemstate(desired but not known )
controls
observedmeasurement
optimalestimate
(system state)
But suppose we are But suppose we are movingmoving……
Not all the difference is error Some may be motion Some may be
motion KF can include a motion model Estimate velocity and position
Process modelProcess model
Describes how the state changes over time
The state for the first example was scalar
The process model was nothing changes
A better model might be A State is a 2-vector [ position, velocity ] positionn+1= positionn + velocityn * time
velocityn+1= velocityn
Predict Predict Correct Correct
KF operates by KF operates by Predicting the new state and its
uncertainty Correcting with the new
measurement Correcting with the new measurementPredict
Correct
Kalmanfilter
Kalman filter assumes Kalman filter assumes linearitylinearity Only matrix operations allowed Measurement is a linear function of
state Next state is linear function of
previous state state
If nonlinear No way?projection
If nonlinear If nonlinear Extended Kalman Extended Kalman ilter(EKF)ilter(EKF)Nonlinear Process (Model)
Process dynamics: A becomes a(x)Measurement: H becomes h(x)
Filter ReformulationUse functions instead of matricesUse Jacobians to project forward, and
to relate measurement to state
JacobianJacobian
Partial derivative of measurement with respect to state
If measurement is a vector of length M and state has length N
Jacobian of measurement fucntion will be MxN matrix of numbers (not equations)
Often evaluating h(x) and Jacobian(h(x)) at the same time cost only a little extra
Concluding remarksConcluding remarks Kalman filter is a very useful and
powerful Optimal in many sense Linear system -> Kalman filter Nonlinear system -> Extended Kalman
filter Effect of initial value
Covariance matrix – less sensitive Measurement noise
Over-estimated -> slow convergence Under-estimated -> noisy output