# Closing (morphology)

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.

## Example

Perform Dilation ( ${\displaystyle A\oplus B}$ ):

Suppose A is the following 11 x 11 matrix and B is the following 3 x 3 matrix:

```    0 0 0 0 0 0 0 0 0 0 0
0 1 1 1 1 0 0 1 1 1 0
0 1 1 1 1 0 0 1 1 1 0
0 1 1 1 1 1 1 1 1 1 0
0 1 1 1 1 0 0 0 1 1 0              1 1 1
0 1 1 1 1 0 0 0 1 1 0              1 1 1
0 1 0 0 1 0 0 0 1 1 0              1 1 1
0 1 0 0 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 0 0 0
0 1 1 1 1 1 1 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0
```

For each pixel in A that has a value of 1, superimpose B, with the center of B aligned with the corresponding pixel in A.

Each pixel of every superimposed B is included in the dilation of A by B.

The dilation of A by B is given by this 11 x 11 matrix.

${\displaystyle A\oplus B}$ is given by :

```    1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 0 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
```

Now, Perform Erosion on the result: (${\displaystyle A\oplus B}$) ${\displaystyle \ominus B}$

${\displaystyle A\oplus B}$ is the following 11 x 11 matrix and B is the following 3 x 3 matrix:

```    1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1              1 1 1
1 1 1 1 1 1 0 1 1 1 1              1 1 1
1 1 1 1 1 1 1 1 1 1 1              1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
```

Assuming that the origin B is at its center, for each pixel in ${\displaystyle A\oplus B}$ superimpose the origin of B, if B is completely contained by A the pixel is retained, else deleted.

Therefore the Erosion of ${\displaystyle A\oplus B}$ by B is given by this 11 x 11 matrix.

(${\displaystyle A\oplus B}$) ${\displaystyle \ominus B}$ is given by:

```    0 0 0 0 0 0 0 0 0 0 0
0 1 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 1 0
0 1 1 1 1 0 0 0 1 1 0
0 1 1 1 1 0 0 0 1 1 0
0 1 1 1 1 0 0 0 1 1 0
0 1 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 1 0
0 0 0 0 0 0 0 0 0 0 0
```

Therefore Closing Operation fills small holes and smoothes the object by filling narrow gaps.

## 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.