Mathematical Properties of the Pseudomedian Filter

Mark A. Schulze

M.S. Thesis

The University of Texas at Austin, 1990


The pseudomedian filter was designed to be a computationally efficient alternative to the median filter. However, a thorough analysis of the pseudomedian filter reveals some important differences between its response and that of the median filter. Several theorems describe the set of signals that are invariant to pseudomedian filtering, and show that this set is a subset of the set of signals invariant to median filtering, with the difference between the sets consisting only of fast-fluctuating signals. The pseudomedian filter does not completely remove impulses, as does the median filter, but both filters preserve edges. The responses of these filters to edges and impulses contrasts with those of the average and midrange filters, which neither preserve edges nor remove impulses. A generalization of the filters to continuous time reveals characteristics of the filter responses to periodic signals, particularly the ability of the pseudomedian filter to block high frequency signals that the median filter cannot. The response of the median filter to high-frequency periodic signals resembles that of the average filter, whereas the response of the pseudomedian filter resembles that of the midrange filter. A square-shaped two-dimensional definition for the pseudomedian filter preserves sharp corners and fine details better than the square-shaped two-dimensional median filter. As is true for the one-dimensional filters, the two-dimensional median filter is susceptible to high-frequency periodic noise and the two-dimensional pseudomedian filter is not. Pseudomedian- and midrange-filtered images often have a "blocky" appearance, while similar median- and average-filtered images do not. These properties of the pseudomedian filter distinguish it from the median, average, and midrange filters and show its superior performance on signals and images without highly impulsive noise and with fine details, sharp corners, or high frequency periodic noise.

© Copyright by Mark A. Schulze, 1990.

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Mark A. Schulze

Last Updated: 16 July 2003