Tracking

Describes the fundamental mathematics of tracking with explanation of Kalman Filter and Particle Filter

Model for Tracking

State

State $x(t)$ is a vector of variable which denotes the system state at time $t$. For example state of system with a car on road along a axis can be defined by its position & velocity or by its position, velocity and acceleration.

Observation

Many times, you does not have any information about the state of system but can only have some observation about it. Observation is generally denoted by $y(t)$or $z(t)$. Sometimes you can completely or partially observe the state of the system. For ex, in above example if can directly observe position & velocity then its fully observable and if you can observe either position or velocity then its partially observable. In case of object trackin, we consider object bouding boxes as its states $x(t)$and as we can observe the image we call it the observation $z(t)$. But you will see that practically we consider x(t) as directly observable because we have using detectors we can directly get $x(t)$.

To understand this fully, we can draw parallel to control systems concept of state-space model:

$$x(t+1) = Ax(x) + Bu(t) \\ y(t) = Cx(t) + Du(t)$$

In this $u(t)$is the input to the system, which we can ignore in our computer tracking problem. So you can see that, our observation $y(t)$ directly depends upon the state $x(t)$.

Also, the above two equations are called transition model and observation model in our tracking literature. As they denotes, how our model is gonna transition and what is the relation between bservation and state.

State and observation Diagram

Tracking: Prediction and Filtering

Kalman Filter

Particle Filter

From the course course on Computer Vision

• Prediction:-

• Correction :-

• The Summary Video for 7B-L1 is very good for overall concept.

Kalman Filter:-

• Linear Model:- Linear Dynamics Model - Linear measurement model -

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• Tracking with Kalman Filter:-

• Extended Kalman Filter is used for NON-LINEAR Models.

Particle Filter

• Watch this for basic idea :- https://youtu.be/aUkBa1zMKv4

• Note that they assume that MAP OF ENVIRONMENT IS KNOWN.

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• Bayes Filter :- Here P(z|x) is probability that if given particle at x then what is probability of measurement z at there.

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• 7C - L2 - Very good for understanding robot motion using particle filters.

• 9th video 7C-L3 particle filter for practical consideration is important.