Dynamical systems analysis of stack filters

TL;DR

This paper explores the dynamical properties of stack filters modeled as Boolean networks, revealing their chaotic nature when optimized for noise suppression and analyzing how continuous states influence their dynamics.

Contribution

It establishes an analytical link between stack filter coefficients, sensitivity, and dynamical behavior, and introduces a generalized model with continuous-valued states.

Findings

Optimal stack filters are dynamically chaotic.

Sensitivity relates to filter coefficients and RSPs.

Continuous-valued extensions alter the dynamical regime.

Abstract

We study classes of dynamical systems that can be obtained by constructing recursive networks with monotone Boolean functions. Stack filters in nonlinear signal processing are special cases of such systems. We show an analytical connection between coefficients used to optimize the statistical properties of stack filters and their sensitivity, a measure that can be used to characterize the dynamical properties of Boolean networks constructed from the corresponding monotone functions. A connection is made between the rank selection probabilities (RSPs) and the sensitivity. We also examine the dynamical behavior of monotone functions corresponding to filters that are optimal in terms of their noise suppression capability, and find that the optimal filters are dynamically chaotic. This contrasts with the optimal information preservation properties of critical networks in the case of small perturbations in Boolean networks, and highlights the difference between such perturbations and those corresponding to noise. We also consider a generalization of Boolean networks that is obtained by utilizing stack filters on continuous-valued states. It can be seen that for such networks, the dynamical regime can be changed when binary variables are made continuous.