# Closing (morphology)

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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,

${\displaystyle A\bullet B=(A\oplus B)\ominus B,\,}$

where ${\displaystyle \oplus }$ and ${\displaystyle \ominus }$ 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, ${\displaystyle (A\bullet B)\bullet B=A\bullet B}$.
• It is increasing, that is, if ${\displaystyle A\subseteq C}$, then ${\displaystyle A\bullet B\subseteq C\bullet B}$.
• It is extensive, i.e., ${\displaystyle A\subseteq A\bullet B}$.
• It is translation invariant.

## Bibliography

• Image Analysis and Mathematical Morphology by Jean Serra, ISBN 0-12-637240-3 (1982)
• Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances by Jean Serra, ISBN 0-12-637241-1 (1988)
• An Introduction to Morphological Image Processing by Edward R. Dougherty, ISBN 0-8194-0845-X (1992)