Mathematical morphology operators

Dilation and erosion are the basic operators of mathematical morphology [143,144]. The dilation of a set of points X by a structural element B is written $X \oplus B$ and is defined as

$$X \oplus B = \{ \; x + b \vert x \in X \mathrm{and} \; b \in B \}$$ (3.10)

The erosion, written $X \ominus B$, is the dual of dilation, i.e. the complement of a dilation performed on the complement set of X. Other morphological operators can be derived by combining dilation and erosion. For instance, the opening and closing are defined as

  

$$X \circ B (X \ominus B) \oplus B$$ = (3.11)

$$X \bullet B (X \oplus B) \ominus B$$ = (3.12)

Symmetrical and circular structural elements (SE) play a central role in mathematical morphology in the continuous plane, because they provide an isotropic treatment of the image. On the other hand, for digital images, circular SE are rarely used because other shapes are easier and faster to implement.