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,


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

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.

Example
Perform Dilation ( $$A\oplus B$$ ):

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.

$$A \oplus B$$ 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: ($$A\oplus B$$) $$ \ominus B$$

$$A\oplus B$$ 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 $$A\oplus B$$ superimpose the origin of B, if B is completely contained by A the pixel is retained, else deleted.

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

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