# Statistical Properties, Fixed Points, and Decomposition with WMMR Filters

## Abstract

WMMR^{m} filters weight the m ordered values in the window with minimum range. If *m* is not specified, it is assumed to be *N +* 1 for a window of length 2*N* + 1. Previous work has demonstrated a subclass of these filters that may be optimized for edge enhancement in that their output converges to the closest perfect edge. In this work it is shown that normalized WMMRm filters, whose weights sum to unity, are affine equivariant. The concept of the breakpoint of a filter is discussed, and the optimality of median and WMMR filters under the breakpoint concept is demonstrated. The optimality of a WMMR^{m} filter and of a similar generalized-order-statistic (GOS) filter is demonstrated for various non-L^{p} criterion, which we call closeness measures. Fixed-point results similar to those derived by Gallagher and Wise (see N.C. Gallagher and G.L. Wise, IEEE Trans. Acoust.,Speech, Signal Process., vol. ASSP-29, 1981, pp. 1136-1141) for the median filter are derived for order-statistic (OS) and WMMR filters with convex weights (weights that sum to unity and are nonnegative), i.e., we completely classify the fixed points under the assumption of a finite-length signal with constant boundaries. These fixed points are shown to be almost always the class of piecewise-constant (PICO) signals. The use of WMMR filters for signal decomposition and filtering based on the Haar basis is discussed. WMMR filters with window width 2*N* + 1 are shown to be linear over the PICO(*N* + 1) signals (minimum constant length *N +* 1). Concepts similar to lowpass, highpass, and bandpass for filtering PICO signals are introduced. Application of the filters to 1-dimensional biological data (non-PICO) and images of printed-circuit boards is then demonstrated, as is application to images in general.

## Key words

WMMR filters LMS regression Haar basis piecewise constant signals## Preview

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