The Histogram Filter

Table of Contents:

Need for Dimensionality Reduction
PCA

An alternative to continuous distributions are piecewise constant approximations, such as histograms. These are nonparametric filters, since they do not utilize parameters like a mean and covariance \((\mu, \Sigma)\) to define a distribution. Thus, nonparametric filters do not rely on a fixed functional form of the posterior, like the Gaussian does [1].

The Histogram Filter is a type of nonparametric filters that discretizes the state space into a finite number of regions. The histogram assigns to each region a single cumulative probability.

1-D Histogram Filter

for all \(k\) do:
- \( \hat{p}{k,t} = \sum\limits_i p(X_t = x_k \mid u_t, X{t-1} = x_i) p_{i,t-1} \)
- \( p_{k,t} = \eta p(z_t \mid X_t = x_k) \hat{p}_{k,t}) \)

2-D Histogram Filter and Grid Localization

References

[1] Sebastian Thrun, Wolfram Burgard, Dieter Fox. Probabilistic Robotics. The MIT Press, Cambridge, MA, 2005.