Morphological Operator Distributions Based On Monotonicity And The Problem Posed By Digital Disk-Shaped Structuring Elements

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Abstract
A sequence of structuring elements S = {S1,...,SN} is said to be increasing if it has the property that for each i, Si+i ⊃ Si. In general, such sequences are made up of elements with similar shapes but different sizes; e. g., lines, squares, octagons, and disks. A morphological operation ψ is said to be monotonic with respect to an Increasing structuring element sequence S, if, for any set X, either: (X 'F Si+1) 2 (X 'F Si), Vi (Monotonic increasing) Or (X AIF Si) 3 (X III Si+1), Vi (Monotonic decreasing) Dilation is monotonic increasing while erosion is monotonic decreasing. These properties make it possible to unambiguously classify every pixel in a binary image by associating each with one of the elements in the sequence S. Morphological openings and closings are also monotonic, but only if an additional property holds for the sequence 5, namely, that for each i, there exists a structuring element T such that Si+1 = (Si ⊕ T), or in other words, Si and Si+1 must be similar in shape up to a dilation. In the digital world, squares, hexagons, and octagons are similar but digital approximations to disks are not. This poses problems for trying to generate morphological shape and size distributions based on very accurate digital disks. This paper proves the monotonicity properties for erosions, dilations, openings, and closings, and shows how pixel distributions or classifications based on shape can be generated from these properties. It also discusses the problem posed by digital disks, and describes one method of circumventing it.

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