Quadratic Structuring Functions in Mathematical Morphology

Abstract

In this contribution we look at the quadratic structuring functions as alternatives to the often used “flat” structuring functions. The quadratic structuring functions (henceforth abbreviated as QSF’s) are the morphological counterpart of the Gaussian function in linear image processing in the sense that: the class of QSF’s is closed under dilation (i.e. dilating two QSF’s results in a third QSF), the QSF’s are easily dimensionally decomposed, (i.e. any n-dimensional QSF can be obtained through dilation of n one-dimensional QSF’s in n independent directions) and the class of QSF’s contains the unique rotational symmetric structuring function that can be dimensionally decomposed with respect to dilation.

The above three claims are most easily proven in the slope domain. The slope transform represents a function in the slope domain instead of the familiar spatial domain. Conceptually the slope transform is the morphological equivalent of the Fourier transform. Just as the Gaussian functions are the eigenfunctions of the Fourier transform, the QSF’s are the eigenfunctions of the slope transform.

Because the QSF’s can be dimensionally decomposed efficient algorithms to perform erosions and dilations using these structuring functions are possible. In this paper three algorithms are discussed. Two of these algorithms prove to be nearly independent of the size of the structuring function.