Can Kalman Filter be used to track Randomly Moving Target?

Question

i want to track random moving object with a camera using kalman filter...i have the following questions...

1. Randomly moving target means $Corelation(t) = E[ x(T)x(T+t) ]$ is very low...where $x(T)$ is the position of the target at time = $T$ along an axis say $x$ axis..eg...moving hand...moving bird..moving ant..what will be my state transition matrix $F(k)$ in $x(k+1) = F(k)x(k) + w(k)$ because i cannot describe it randomness...

2. is there any standard model to represent any random moving target?

   any suggestion is welcome....i am aware that any object can be determined in a given image frame using its characteristics by image processing techniques and hence tracked....but is it possible to use Kalman Filter?

Answer 1

It depends on what you mean by a "randomly moving object". If you are trying to track something that truly moves around in a totally uncorrelated manner from sample to sample (like, say, a laser pointer that flickers on and off and randomly changes position in your camera images) then a linear tracker will not give you insight into the object's state.

The examples you cited (a moving hand, bird, etc.) might exhibit random behavior on "large" time scales, but on "small" time scales their state is highly correlated. For such a system the most important point is to have a high data sampling rate relative to the dynamics of the system. The higher the sampling and measurement update rate, the more linear the underlying system dynamics will be from one measurement time to the next. Also if you know the object of interest can be randomly changing direction or accelerating you can incorporate this prior knowledge by increasing the process noise during the prediction phase of your Kalman filter, giving less weight to the predicted position during the "update" stage.

Answer 2

You should define a linear $F$ to use Kalman filtering recursions. However, I think, random movements can not be described very well with linear models (an object which is tied to certain and linear physical rules can be tracked by the Kalman filter). Therefore, your state space model will not be linear.

Then, your question about $F$ is critical. Because, in many times, the art of Bayesian modeling is to write appropriate models for underlying physical structure. Therefore, if you write a certain motion model, you can use particle filtering (Also for nonlinear models it would not be $Fx$, it will become $f(x)$ where $f$ is a nonlinear function). And, I do not know, you may write a piecewise linear model for a bird or ant, in that case, you can use the Kalman filter but you have to estimate switching variable.

Therefore, I suggest you to search about motion models of random movements. Then think about the inference mechanism.