Tracking
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Tracking. We are given a contour G 1 with coordinates G 1 ={ x 1 , x 2 , … , x N } at the initial frame t=1, were the image is I t=1 . We are interested in tracking this contour through several image frames, say through T image frames given by

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Tracking

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Tracking

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 ?”

Computer Vision


Tracking

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

Computer Vision


Tracking

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)

Computer Vision


Tracking

Linear Dynamics

Examples

Random Walk

Points moving with constant velocity

Points moving with constant acceleration

Periodic motion

Computer Vision


Tracking

Random Walk of a Point

Drift: New position is the previous one plus noise (sd)

Computer Vision


Tracking

Point moving with constant velocity

Computer Vision


Tracking

Point moving with constant acceleration

Computer Vision


Tracking

Point moving with periodic motion

Computer Vision


Tracking

Tracking one point (dynamic programming like)

Equation (2) is written as

S

S

t

t-1 t

Computer Vision


Tracking

Kalman Filter (Special Case of a Linear System)

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

Computer Vision


Tracking

Kalman Filter and Gaussian Properties

Proof: Assume

Then

Computer Vision


Tracking

Kalman Filter …(continuing the proof)

Computer Vision


Tracking

Using Kalman Filter

Computer Vision


Tracking

Kalman Filter (Generalization to N-D)

Computer Vision


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