Dilation (image processing)

Dilatation (from Latin : dilatare = to expand, expand) is a basic morphological operation in digital image processing .

Dilation using a structuring element

Dilation of a binary image with a circle as a structuring element

In digital image processing, dilation is generally used by means of structuring elements. From a mathematical point of view, in the case of binary images, this is the formation of the Minkowski sum of the image and the structuring element.

The dilation of an image with a structuring element is called . In the case of binary image morphology, this clearly means that the complete element is inserted at each pixel , the pixel is expanded (dilated) to the shape of the structuring element. ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle A\oplus X}$ ${\displaystyle A}$ ${\displaystyle X}$

Gray value image processing

On a grayscale image, the dilation with a structuring element acts similar to a maximum filter. Light structures are enlarged, darker structures are reduced. It applies

${\displaystyle (A\oplus X)(x,y)={\textrm {max}}\{A(x+s,y+t)+X(s,t)|(s,t)\epsilon D_{X}\},}$

where denotes the domain of definition of the structuring element. The gray value dilation clearly means that the gray value mountain range - the values ​​of the pixels are interpreted as height information - are scanned from above with a reference shape (the structuring element). ${\displaystyle D_{X}}$

generalization

A complete association is given . An operator on is a dilation if it is distributive with respect to the supremum formation, i.e. if: ${\displaystyle V}$ ${\displaystyle \delta }$ ${\displaystyle V}$ ${\displaystyle \bigvee }$

${\displaystyle \delta \left(\bigvee _{x_{i}\in V}x_{i}\right)=\bigvee _{x_{i}\in V}\delta (x_{i})}$

Binary images represent the elements of a (Boolean) lattice; the formation of the supremum is then the oring of images (an image point is set when it is set in one of the original images). In the case of gray value images, the maximum value of all images is taken at each point.

Adjunction of dilation and erosion

In mathematical morphology , dilatations and erosions form two isomorphic associations on a complete association. At any dilation there is an erosion of ${\displaystyle V}$ ${\displaystyle \delta }$ ${\displaystyle \varepsilon }$

${\displaystyle \varepsilon \left(X\right)=\bigvee \left\{A\in V|\delta (A)\leq X\right\}}$

and at any erosion dilation with ${\displaystyle \varepsilon }$ ${\displaystyle \delta }$

${\displaystyle \delta \left(Y\right)=\bigwedge \left\{B\in V|\varepsilon (B)\geq Y\right\}.}$

Thus for ${\displaystyle X,Y\in V}$

${\displaystyle \delta \left(X\right)\leq Y\Leftrightarrow X\leq \varepsilon \left(Y\right).}$