Opening (morphology)

• Opening is idempotent, that is, ${\displaystyle (A\circ B)\circ B=A\circ B}$.
• Opening is increasing, that is, if ${\displaystyle A\subseteq C}$, then ${\displaystyle A\circ B\subseteq C\circ B}$.
• Opening is anti-extensive, i.e., ${\displaystyle A\circ B\subseteq A}$.
• Opening is translation invariant.
• Opening and closing satisfy the duality ${\displaystyle A\bullet B=(A^{c}\circ B^{c})^{c}}$, where ${\displaystyle \bullet }$ denotes closing.

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

${\displaystyle O_{R}^{(n)}(F)=R_{F}^{D}[(F\ominus nB)],}$[1]

where ${\displaystyle (F\ominus nB)}$ denotes a marker image and ${\displaystyle F}$ is a mask image in morphological reconstruction by dilation. ${\displaystyle R_{F}^{D}[(F\ominus nB)]=D_{F}^{(k)}[(F\ominus nB)],}$[1] ${\displaystyle D}$ denotes geodesic dilation with ${\displaystyle k}$ iterations until stability, i.e., such that ${\displaystyle D_{F}^{(k)}[(F\ominus nB)]=D_{F}^{(k-1)}[(F\ominus nB)].}$[1] Since ${\displaystyle D_{F}^{(1)}[(F\ominus nB)]=([(F\ominus nB)]\oplus B)\cap F}$,[1] the marker image is limited in the growth region by the mask image, so the dilation operation on the marker image will not expand beyond the mask image. As a result, the marker image is a subset of the mask image ${\displaystyle (F\ominus nB)\subseteq F.}$[1] (Strictly, this holds for binary masks only. However, similar statements hold when the mask is not binary.)

The images below present a simple opening-by-reconstruction example which extracts the vertical strokes from an input text image. Since the original image is converted from grayscale to binary image, it has a few distortions in some characters so that same characters might have different vertical lengths. In this case, the structuring element is an 8-pixel vertical line which is applied in the erosion operation in order to find objects of interest. Moreover, morphological reconstruction by dilation, ${\displaystyle R_{F}^{D}[(F\ominus nB)]=D_{F}^{(k)}[(F\ominus nB)]}$[1] iterates ${\displaystyle k=9}$ times until the resulting image converges.

Original image for opening by reconstruction

Marker image

Result of opening by reconstruction

• Mathematical morphology
• Closing
• Dilation
• Erosion

• 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)

• http://homepages.inf.ed.ac.uk/rbf/HIPR2/open.htm - Morphological Opening

• Digital Image Processing (Third Edition) by Rafael C. Gonzalez and Richard E. Woods, ISBN 978-93-325-7032-0(2008)