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Recently, Kadir and Brady proposed a method for estimating probability density functions (PDFs) for digital signals which they call the Nonparametric (NP) Windows method. The method involves constructing a continuous space representation of the discrete space and sampled signal by using a suitable interpolation method. NP Windows requires only a small number of observed signal samples to estimate the PDF and is completely data driven. In this short paper, we first develop analytical formulae to obtain the NP Windows PDF estimates for 1D, 2D, and 3D signals, for different interpolation methods. We then show that the original procedure to calculate the PDF estimate can be significantly simplified and made computationally more efficient by a judicious choice of the frame of reference. We have also outlined specific algorithmic details of the procedures enabling quick implementation. Our reformulation of the original concept has directly demonstrated a close link between the NP Windows method and the Kernel Density Estimator.

Original publication




Journal article


IEEE Trans Pattern Anal Mach Intell

Publication Date





1673 - 1680


Algorithms, Image Processing, Computer-Assisted, Models, Statistical, Signal Processing, Computer-Assisted, Statistics, Nonparametric