2009. 11. 17. 15:48
Computer Vision
Kalman Filtering
http://academic.csuohio.edu/simond/courses/eec644/kalman.pdf
72-79p, Embedded Systems Programming f e a tur e, JUNE 2001
http://www.embedded.com/9900168?_requestid=49635
The Kalman filter update equations in C
http://www.embedded.com/9900168?pgno=2
matrix algebra reference
ftp://ftp.embedded.com/pub/2001/simon06
Dan Simon
http://academic.csuohio.edu/simond/
Kalman filter
: estimates system states that can only be observed indirectly or inaccurately by the system itself.
: estimates the variables of a wide range of processes.
: estimates the states of a linear system.
: minimizes the variance of the estimation error
Linear system
x: state of the system
u: known input to the system
y: measured output
w: process noise
z: measurement noise
Criteria of an Estimator
1) The expected value of the estimate should be equal to the expected value of the state.
2) The estimator should be with the smallest possible error variance.
Requirement of Kalman filter
1) The average value of w is zero and average value of z is zero.
2) No correlation exists between w and z. w_k and z_k are independent random variables.
Kalman filter equations
K matrix: Kalman gain
P matrix: estimation error covariance
"steady state Kalman filter"
- K matrix & P matrix are constant
"extended Kalman filter"
: an extension of linear Kalman filter theory to nonlinear systems
"Kalman smoother"
: to estimate the state as a function of time so to reconstruct the trajectory after the fact
H infinity filter
=> correlated noise problem
=> unknown noise covariances problem
http://academic.csuohio.edu/simond/estimation/
Rudolph Kalman
Peter Swerling, 1958
Karl Gauss's method of least squares, 1795
spacecraft navigation for the Apollo space program
> applications
all forms of navigation (aerospace, land, and marine)
nuclear power plant instrumentation
demographic modeling
manufacturing
the detection of underground radioactivity
fuzzy logic and neural network training
Gelb, A. Applied Optimal Estimation. Cambridge, MA: MIT Press, 1974.
Anderson, B. and J. Moore. Optimal Filtering. Englewood Cliffs, NJ: Prentice-Hall, 1979.
Grewal, M. and A. Andrews. Kalman Filtering Theory and Practice. Englewood Cliffs, NJ: Prentice-Hall, 1993.
Sorenson, H. Kalman Filtering: Theory and Application. Los Alamitos, CA: IEEE Press, 1985.
Peter Joseph’s Web site @http://ourworld.compuserve.com/homepages/PDJoseph/
http://academic.csuohio.edu/simond/courses/eec644/kalman.pdf
72-79p, Embedded Systems Programming f e a tur e, JUNE 2001
http://www.embedded.com/9900168?_requestid=49635
The Kalman filter update equations in C
http://www.embedded.com/9900168?pgno=2
matrix algebra reference
ftp://ftp.embedded.com/pub/2001/simon06
Dan Simon
http://academic.csuohio.edu/simond/
Kalman filter
: estimates system states that can only be observed indirectly or inaccurately by the system itself.
: estimates the variables of a wide range of processes.
: estimates the states of a linear system.
: minimizes the variance of the estimation error
Linear system
u: known input to the system
y: measured output
w: process noise
z: measurement noise
http://wiki.answers.com/Q/What_is_a_feedback_system
A feedback system, in general engineering terms, is a system whose output if fed back to the input, and depending on the output, your input is adjusted so as to reach a steady-state. In colloquial language, you adjust your input based on the output of your system so as to achieve a certain end, like minimizing disturbance, cancelling echo (in a speech system) and so on.
A feedback system, in general engineering terms, is a system whose output if fed back to the input, and depending on the output, your input is adjusted so as to reach a steady-state. In colloquial language, you adjust your input based on the output of your system so as to achieve a certain end, like minimizing disturbance, cancelling echo (in a speech system) and so on.
Criteria of an Estimator
1) The expected value of the estimate should be equal to the expected value of the state.
2) The estimator should be with the smallest possible error variance.
Requirement of Kalman filter
1) The average value of w is zero and average value of z is zero.
2) No correlation exists between w and z. w_k and z_k are independent random variables.
Kalman filter equations
K matrix: Kalman gain
P matrix: estimation error covariance
http://en.wikipedia.org/wiki/Three_sigma_rule
In statistics, the 68-95-99.7 rule, or three-sigma rule, or empirical rule, states that for a normal distribution, nearly all values lie within 3 standard deviations of the mean.
In statistics, the 68-95-99.7 rule, or three-sigma rule, or empirical rule, states that for a normal distribution, nearly all values lie within 3 standard deviations of the mean.
"steady state Kalman filter"
- K matrix & P matrix are constant
"extended Kalman filter"
: an extension of linear Kalman filter theory to nonlinear systems
"Kalman smoother"
: to estimate the state as a function of time so to reconstruct the trajectory after the fact
H infinity filter
=> correlated noise problem
=> unknown noise covariances problem
http://academic.csuohio.edu/simond/estimation/
Rudolph Kalman
Peter Swerling, 1958
Karl Gauss's method of least squares, 1795
spacecraft navigation for the Apollo space program
> applications
all forms of navigation (aerospace, land, and marine)
nuclear power plant instrumentation
demographic modeling
manufacturing
the detection of underground radioactivity
fuzzy logic and neural network training
Gelb, A. Applied Optimal Estimation. Cambridge, MA: MIT Press, 1974.
Anderson, B. and J. Moore. Optimal Filtering. Englewood Cliffs, NJ: Prentice-Hall, 1979.
Grewal, M. and A. Andrews. Kalman Filtering Theory and Practice. Englewood Cliffs, NJ: Prentice-Hall, 1993.
Sorenson, H. Kalman Filtering: Theory and Application. Los Alamitos, CA: IEEE Press, 1985.
Peter Joseph’s Web site @http://ourworld.compuserve.com/homepages/PDJoseph/