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I get confuse with the basic theory of particle filter for tracking single object.. What I want to ask is:
Xn=f(Xn-1)+Wn where Wn is noise.. Why need noise in particle filter?The following papers are good resources:
Gordon, N. J.; Salmond, D. J. and Smith, A. F. M. (1993). "Novel approach to nonlinear/non-Gaussian Bayesian state estimation". IEE Proceedings F on Radar and Signal Processing 140 (2): 107–113.
Arulampalam, M.S.; Maskell, S.; Gordon, N.; Clapp, T.; (2002). "A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking". IEEE Transactions on Signal Processing 50 (2): 174–188.
Doucet, A.; Johansen, A.M.; (December 2008). "A tutorial on particle filtering and smoothing: fifteen years later". Technical report (Department of Statistics, University of British Columbia). (Available here)
I'd recommend you read at least the first one.
A particle filter works on a model specified by 2 equations (a hidden Markov model):
1) The state equation $x_{k+1} = f_k (x_k,w_k)$ which tells you how the state (minimal set of variables to describe a system) evolves as being driven by noise.
2) The measurement equation $y_k = h_k (x_k, v_k)$ which maps the state to what you observe.
Given the observations $y_0, \ldots, y_n$, estimate the distribution of $x_n$ is the problem the particle filter tries to solve. Once you have the approximated distribution of $x_n$ (which is the distribution of the particles), you can estimate $x_n$ (say, by averaging the particles).
The particle filter essentially starts with a bunch of samples (called particles), evolves the state by running each particle through the state equation and re-samples the particles based on the observation you see in order to make the distribution of particles consistent with the observations.
