Tracking using the Kalman Filter

Point Tracking Estimate the location of a given point along a sequence of images. (x 0,y 0 ) (x n,y n )

Object Tracking – Generate some conclusions about the motion of the scene, objects, or the camera, given a sequence of images. – Knowing this motion, predict where things are going to project in the next image, so that we don’t have so much work looking for them. – For example- unstable camera + Walking man: a. Stabilize the camera using the dominant motion ( find motion parameters ! ) b. Assume that the man translates horizontally.

Modeling “noise” or “uncertainty” rotation

The General Model Dynamics Process noise ~N(0,Q) Projection Measurement noise ~N(0,R)

Prediction Estimated state Estimated uncertainty / noise

Update Updated state Updated uncertainty / noise The weighting factor

Prediction Update Summery

Gaussian: “Normal” distribution 1D Gaussian: General Gaussian:

Adding two information sources We are given to information sources: Z 1 and Z 2 Both are normally distributed (v 1 > v 2 ) We would like to believe more to Z 2, but still use the information from Z 1 ! Mathematically:

The solution

The solution (cont’)

The merging of two Gaussians A “noisy” measure, be don’t believe it very much A more reliable measure

The merging of two Gaussians (cont’) The result is a new Gaussian with a smaller variance than the original ones !

Why to use the normal distribution? Simple to manipulate Minimize the squared error. The “big numbers” low. The distribution of many “natural” things.

What happens when we have a “wrong” estimation of the measurements variance ? The correct variance (The same variance that was used to simulate the points) The variance is too small: The estimation doesn’t converge The variance is too large: The convergence is very slow

Tracking using the Kalman Filter Two more examples.

The General Model Dynamics Process noise Projection Measurement noise

Example 1: Estimating a constant Measurement noise

Prediction: Update

We can combine the prediction and update

Claim1: Claim2: Conclusion: The Kalman filter gives a weighted mean !

Example 2 : Shihab4 In X: constant velocity In Y: constant acceleration

Example2 -dynamics

Example2 -measurements For each possible location, give a score Normalize the sum of the scores to 1. The result is a matrix of “probabilities” for each location. Fit a 2D Gaussian to this matrix, whose center is given by: Given an image of the missile (or other source of information):