# Mathematical Properties of the Pseudomedian Filter

## Mark A. Schulze

### M.S. Thesis

### The University of Texas at Austin, 1990

## Abstract

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

http://www.markschulze.net/

mark@markschulze.net

Last Updated: 16 July 2003