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Kalman Filter: for tracking # CSE 378 : Robotics

Background Subtraction: Subtract Background Image from Foreground Image, then threshold.
Computer Vision: Extracting information from images.
Causal Probabilty: P(E|H)
Diagnostic Probability: P(H|E)
Kinematics: Solve for points in space given system parameters.
Inverse Kinematics: Solve for System Parameters given points in space.
Markov Process: A Stochastic Process that has the Markov Property.
Markov Property: Given a system's current state, the system's future state is conditionally independent of the past.
Robotics: The study of robots.
Robot: A mechanical agent.

Rx = $\left[\begin{array}{ccc}1& 0& 0\\ 0& cos\theta & -sin\theta \\ 0& sin\theta & cos\theta \end{array}\right]$ Ry = $\left[\begin{array}{ccc}cos\theta & 0& sin\theta \\ 0& 1& 0\\ -sin\theta & 0& cos\theta \end{array}\right]$ Rz = $\left[\begin{array}{ccc}cos\theta & -sin\theta & 0\\ sin\theta & cos\theta & 0\\ 0& 0& 1\end{array}\right]$ RV = eV⨯θ

Jacobian = $\left[\begin{array}{ccc}dF1/dX1& dF1/dX2& dF1/dX3\\ dF2/dX1& dF2/dX2& dF2/dX3\\ dF3/dX1& dF3/dX2& dF3/dX3\end{array}\right]$

dF = Jacobian * dX
dX = Jacobian-1 * dF

Convolution Theorem: ℱ{f * g} = ℱ{f} · ℱ{g} and ℱ{f · g} = ℱ{f} * ℱ{g}
Example: N(1,V₁)·N(1,V₂) = ℱ{ℱN(1,V₁)*ℱN(1,V₂)} = ℱ{N(1,1/V₁)*N(1,1/V₂)} = ℱ{N(1,1/V₁+1/V₂)} = N(1,1/(1/V₁+1/V₂))