Density Propagation

The most obvious difficulties of the visual tracking lie in modeling the uncertainty of the shape - or the probability density of the "state", which is commonly used in the control community. Most of the algorithms that use the state-space description are based on Kalman filtering. Kalman filtering has a beautiful mechanism that fuse the current noisy measurement information and the estimation based on the history - the previous states. Kalman filtering deals with the uncertainty of the state by carrying the covariance matrices of the states and the measurements, and it works optimally when the noise is Gaussian. This indicates that Kalman filtering algorithm can fail when the uncertainty in the system has multi-modal distribution. This is exactly what happens in the noisy measurement data from an image with severe visual clutter, so the Kalman filtering can not be used in this case.

The obvious answer to the above question is to propagate the whole probability density of the state over time. However, the multi-modal probability density does not have a closed form representation, and this can be done only by numerical or simulation method. CONDENSATION algorithm is one simulation method , which works very well even in the severe visual clutter in the image. The algorithm represents the probability density as a sampled set of states with the corresponding weights,

where  is the nth sample from the density and  is the corresponding weight.

To propagate the probability density over time, CONDENSATION uses the "factored sampling"[3] iteratively. First, N samples are generated from the effective prior 

The samples then undergo deterministic drift due to the given dynamic model  to form the current state samples. The weights associated with these states are computed using

and then normalized

The conditional density  is the observation model, which explains how likely the current observation is given the current shape-space vector. This will be explained in the section2.4.

A set of pairs  form an approximate representation of the posterior probability density  and they will act as a prior density for the next iteration step. By recursively propagating the sample states with their weights over time, CONDENSATION algorithm effectively propagate the probability density over time.

Tue Mar 14 02:05:32 PST 2000