chapter 9, Automatic Design of Morphological Operators

Excerpt

The key to successful morphological image processing is the selection of structuring elements. There are a myriad of algorithms for a multitude of imaging applications, but in each and every instance, algorithm performance depends on the structuring elements. The classical approach to morphological processing is to have a human being, or a group of human beings, use intuition and an understanding of the goals to design algorithms based on erosions, openings, hit-or-miss transforms and other basic morphological operators. This approach can work well if the task can be described in elementary geometric terms and the images under consideration are not too complex. It breaks down in situations where satisfactory filtering might require hundreds, or even thousands, of structuring elements. The present chapter introduces automatic algorithm design, where morphological operators are designed based on sample data, structural decomposition, and criteria set by the imaging scientist.

9.1 Boolean Functions

In this chapter, we will exploit the relationship between binary mathematical morphology and Boolean functions. This section is devoted to that relationship.

A binary-valued function ψ(x₁,x₂,…,x_n) of binary variables x₁,x₂,…,x_n is called a Boolean function. As a logical function, ψ possesses a logical sum-of-products disjunctive-normal-form representation in terms of the n variables x₁,x₂,…,x_n:

The truth table formulation of ψ corresponds directly to the disjunctive normal form of Eq. (9.1). ψ is defined by a 2ⁿ-row truth table of n variables in which each string t₁t₂…t_n of 0s and 1s is assigned a binary value ψ(t₁t₂…t_n).