Some properties of order statistic filters

Herbert A. David

doi:10.1117/12.19598

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1 July 1990 Some properties of order statistic filters

Herbert A. David

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Proceedings Volume 1247, Nonlinear Image Processing; (1990) https://doi.org/10.1117/12.19598
Event: Electronic Imaging: Advanced Devices and Systems, 1990, Santa Clara, CA, United States

Abstract

Let X₁, X₂, ... be a stationary sequence of random variables with Pr(X_t less than or equal to x) = F(x), t = 1, 2, ... . Also let X_i:n^(t), i = 1, ..., n, denote the i-th order statistic (OS) in the moving sample (X_t-N, ..., X_t, ..., X_t+N) of odd size n = 2N + 1. Then Y_t = (summation)a_iX_i:n^(t) with (summation)ai = 1 is an order statistics filter. In practice a_i greater than or equal to 0, i = 1, ..., n. For t > N, the sequence (Y_t) is also stationary. If X₁, X₂, ... are independent, the autocorrelation function ρ(r) = corr(Y_t, T_t+r) is zero for r > n-1 and for r less than or equal to n-l can be evaluated directly in terms of the means, variances, and covariances of the OS in random samples of size n + r from F(x). In special cases several authors have observed that (Y_t) produces a low-pass filter. It will be shown that this result holds generally under white noise. The effect of outliers (impulses) is also discussed.

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Herbert A. David "Some properties of order statistic filters", Proc. SPIE 1247, Nonlinear Image Processing, (1 July 1990); https://doi.org/10.1117/12.19598

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