Non-parametric Estimation of Probability Distributions from Sampled Signals

This paper is concerned with the non-parametric estimation of probability distributions from band-limited and at least critically sampled signals such as images. Conventional approaches employing histograms or Parzen windows often perform unsatisfactorily since they ignore two important properties of such signals, namely that they are ordered and contain sufficient information to reconstruct the original continuous band-limited signal exactly. Based on these observations we propose a method to estimate the density and distribution of such signals. The technique has a number of interesting and useful properties. First, that domain resolution of the estimate is independent of the number of sample points. Second, the estimate is continuous and consequently no arbitrary bin widths or smoothing kernel parameters have to be set. Third, if a suitable analytic form of the band-limited signal or equivalently the sampling pre-filter is available then the resulting density is exact (aside from signal quantisation).