Opening (morphology)

In mathematical morphology, opening is the dilation of the erosion of a set A by a structuring element B:

where ${\displaystyle \ominus }$ and ${\displaystyle \oplus }$ denote erosion and dilation, respectively.

Together with closing, the opening serves in computer vision and image processing as a basic workhorse of morphological noise removal. Opening removes small objects from the foreground (usually taken as the bright pixels) of an image, placing them in the background, while closing removes small holes in the foreground, changing small islands of background into foreground. These techniques can also be used to find specific shapes in an image. Opening can be used to find things into which a specific structuring element can fit (edges, corners, ...).

One can think of B sweeping around the inside of the boundary of A, so that it does not extend beyond the boundary, and shaping the A boundary around the boundary of the element.

In morphological opening ${\displaystyle (A\ominus B)\oplus B}$ , erosion operation removes objects that are smaller than structuring element B and dilation operation restores the shape of remaining objects. However, restoring accuracy in dilation operation highly depends on the type of structuring element and the shape of restoring objects. The opening by reconstruction method is able to restore the objects completely after erosion applied. It is defined as the reconstruction by geodesic dilation of ${\displaystyle n}$ erosions of ${\displaystyle F}$ by ${\displaystyle B}$ with respect to ${\displaystyle F}$ :

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