MEDIAN FILTERS: THEORY AND APPLICATIONS

GONZALO RAMIRO ARCE, Purdue University

Abstract

Median filtering is a simple digital technique used for signal smoothing and entropy reduction. Because this filter has demonstrated good performance in some applications where linear filters are not adequate, it has increasingly being used in recent years. Because of the nonlinearity of the median filter, theoretical analysis is difficult. Recently, some deterministic, as well as some statistical properties have been developed. One main characteristic of the filtering process is that it maps the input signal space into a root signal space, where signals invariant to median filters are called roots of the signal. In this thesis we develop the theory of these root signals in a stochastic and deterministic sense. First we show that root signals of median filters may be described as a complex Markov processe. Using this Markov property a tree structure is developed, which yields more information about the signals. Using the tree structure and the stochastic Markov matrix, the stationary distributions for any recursive median filtered source with finite memory is obtained. The output distribution for a two dimensional recursively filtered source is also considered. Spectral characteristics of the recursive median filter as well as the entropy loss due to the filter are also analyzed. The properties of the root signals can be used to encode the truncated block used in Block Truncation Coding (BTC) of images. Because of the low distortion introduced and because the Root Signal Space is much smaller than the binary space, we obtain good encoding performance. Using one dimensional filtering along with a trellis encoder, we reduce the original BTC rate of 1.63 bits/pel to 1.12 bits/pel. Finally, we use recursive filtering to reduce the entropy of two tone images and hence develop a non-exact extension of run length coding.

Degree

Ph.D.

Subject Area

Electrical engineering

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