# Closing (morphology)

The closing of the dark-blue shape (union of two squares) by a disk, resulting in the union of the dark-blue shape and the light-blue areas.

In mathematical morphology, the closing of a set (binary image) A by a structuring element B is the erosion of the dilation of that set,

$A\bullet B = (A\oplus B)\ominus B, \,$

where $\oplus$ and $\ominus$ denote the dilation and erosion, respectively.

In image processing, closing is, together with opening, the basic workhorse of morphological noise removal. Opening removes small objects, while closing removes small holes.

## Properties

• It is idempotent, that is, $(A\bullet B)\bullet B=A\bullet B$.
• It is increasing, that is, if $A\subseteq C$, then $A\bullet B \subseteq C\bullet B$.
• It is extensive, i.e., $A\subseteq A\bullet B$.
• It is translation invariant.