Structuring Element - Mathematical Particulars and Examples

Mathematical Particulars and Examples

Structuring elements are particular cases of binary images, usually being small and simple. In mathematical morphology, binary images are subsets of an Euclidean space Rd or the integer grid Zd, for some dimension d. Here are some examples of widely used structuring elements (denoted by B):

• Let E=R2; B is an open disk of radius r, centered at the origin.
• Let E=Z2; B is a 3x3 square, that is, B={(-1,-1),(-1,0),(-1,1),(0,-1),(0,0),(0,1),(1,-1),(1,0),(1,1)}.
• Let E=Z2; B is the "cross" given by: B={(-1,0),(0,-1),(0,0),(0,1),(1,0)}.

In the discrete case, a structuring element can also be represented as a set of pixels on a grid, assuming the values 1 (if the pixel belongs to the structuring element) or 0 (otherwise).

When used by a hit-or-miss transform, usually the structuring element is a composite of two disjoint sets (two simple structuring elements), one associated to the foreground, and one associated to the background of the image to be probed. In this case, an alternative representation of the composite structuring element is as a set of pixels which are either set (1, associated to the foreground), not set (0, associated to the background) or "don't care".