Morphological Operator Distributions Based On Monotonicity And The Problem Posed By Digital Disk-Shaped Structuring Elements
Morphological Operator Distributions Based On Monotonicity And The Problem Posed By Digital Disk-Shaped Structuring Elements
复制标题
基于单调性的形态算子分布及数字盘形结构元提出的问题
作者：
R. Vogt
结构元素S = {S1，…，SN}是递增的，如果它具有这样的性质：对于每一个i， Si+i, Si。一般来说，这种序列是由形状相似但大小不同的元素组成的；例如，线条、正方形、八边形和圆盘。对于任意集合X，如果（X′F Si+1） 2 (X′F Si), Vi（单调递增）或（X AIF Si） 3 (X III Si+1), Vi（单调递减）膨胀是单调递增而侵蚀是单调递减，则形态运算ψ是单调递增的。这些性质使得通过将每个像素与序列s中的一个元素相关联，可以明确地对二值图像中的每个像素进行分类。形态学的开口和闭合也是单调的，但前提是序列5具有一个附加性质，即对于每个i，存在一个结构元素T使得Si+1 = (Si⊕T)，或者换句话说，Si和Si+1必须在形状上相似直到膨胀。在数字世界中，正方形、六边形和八边形是相似的，但磁盘的数字近似不是。这给试图基于非常精确的数字磁盘生成形态形状和大小分布带来了问题。本文证明了侵蚀、膨胀、开口和闭合的单调性，并展示了如何从这些属性中生成基于形状的像素分布或分类。它还讨论了数字磁盘带来的问题，并描述了一种绕过它的方法。
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.