1. Tracking
We are given a contour G1 with coordinates G1={x1 , x2 , … , xN} at the initial frame t=1, were the image is It=1 . We are interested in tracking this contour through several image frames, say through T image frames given by
We will denote each of these contours by Gt and so G1 = Gt=1 . We will track each of the coordinates of the initial contour, i.e., we will focus on the tracking of the N initial coordinates. Thus, at any time the new contour will be characterized by Gt = {x1 , x2 , … , xN }, and these coordinates need not be connected. In order to create a connected contour at each frame one may fit a spline or another form of linking of this sequence of N coordinates. In a Bayesian framework we attempt to obtain the probability
We can make measurements at each image frame, accumulate them over time denoting by and we ask “Which state (coordinates) the contour will be at that time given all the measurements ?”
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2. Propagating Probability Density
An interesting expansion of (1) is given by:
Correct or make
Measurements
Predict
Assuming that such probabilities can be obtained, namely
than we have a recursive method to estimate
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3. Assumptions
Independence assumption on data generation, i.e., the current state completely defines the probability of the measurements/data (often the previous data is also neglected)
First order Markov process, i.e., the current state of the system is completely described by the immediate previous state, and possibly the data (often the data is also neglected).
This is a normalization constant that can always be computed by normalizing the recurrence equation (2)
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4. Linear Dynamics Examples Random Walk
Points moving with constant velocity
Points moving with constant acceleration
Periodic motion
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5. Random Walk of a Point
Drift: New position is the previous one plus noise (sd)
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6. Point moving with constant velocity
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7. Point moving with constant acceleration
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8. Point moving with periodic motion
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9. Tracking one point (dynamic programming like)
Equation (2) is written as S S t
t t
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10. Kalman Filter
Simplify calculation of the probabilities due to good Gaussian properties (applied to )
Reminder: From equation (2) we have
1D case: For one dimension point and Gaussian distributions
Kalman observed the recurrence: if
then
where
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11. Kalman Filter and Gaussian Properties
Proof: Assume
Then
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12. Kalman Filter …(continuing the proof)
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13. Using Kalman Filter
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14. Kalman Filter (Generalization to N-D)
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