Date of this Version


Document Type

Discussion Paper

Publication Details

Dawei Huang and William T. M. Dunsmuir (1993) A Boolean Convolution Theorem and Its Application to Computing Distributions of Non-linear Filters

School of Business Discussion Paper ; No. 40, Feb. 1993

© Copyright Dawei Huang, William T. M. Dunsmuir and the School of Business, Bond University


This paper solves the statistical distributional problem for detecting a step change of known orientation in a surface by moving median filters based on pixel windows of varying shapes and sizes. This had previously been thought to be a difficult problem. Our formulation allows us to derive the probability distribution, needed to evaluate the probability of failing to detect an edge when present ("edge miss probability") and the probability of falsely detecting a non-existent edge. The derivation is based on a basic theorem about Boolean representations. The theoretical results are applied to calculating the probabilities of missing vertical edges in a Gaussian noise contaminated surface with two levels when medians based on a variety of pixel window shapes are considered. In particular the power for testing the null hypothesis that the edge is not present against a series of alternatives that its height is h are calculated for three basic window shapes.