Dilation (morphology)

The dilation of a dark-blue square by a disk, resulting in the light-blue square with rounded corners.

In binary morphology, dilation is a shift-invariant (translation invariant) operator, equivalent to Minkowski addition.

A binary image is viewed in mathematical morphology as a subset of a Euclidean space R^d or the integer grid Z^d, for some dimension d. Let E be a Euclidean space or an integer grid, A a binary image in E, and B a structuring element regarded as a subset of R^d.

The dilation of A by B is defined by

${\displaystyle A\oplus B=\bigcup _{b\in B}A_{b},}$

where A_b is the translation of A by b.

Dilation is commutative, also given by ${\displaystyle A\oplus B=B\oplus A=\bigcup _{a\in A}B_{a}}$.

If B has a center on the origin, then the dilation of A by B can be understood as the locus of the points covered by B when the center of B moves inside A. The dilation of a square of size 10, centered at the origin, by a disk of radius 2, also centered at the origin, is a square of side 14, with rounded corners, centered at the origin. The radius of the rounded corners is 2.

The dilation can also be obtained by ${\displaystyle A\oplus B=\{z\in E\mid (B^{s})_{z}\cap A\neq \varnothing \}}$, where B^s denotes the symmetric of B, that is, ${\displaystyle B^{s}=\{x\in E\mid -x\in B\}}$.

    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 1 1 1 1 1 0              1 1 1
    0 1 1 0 0 0 1 1 1 1 0              1 1 1
    0 1 1 0 0 0 1 1 1 1 0              1 1 1
    0 1 1 0 0 0 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

    1 1 1 1 1 1 1 1 1 1 1
    1 1 1 1 1 1 1 1 1 1 1   
    1 1 1 1 1 1 1 1 1 1 1  
    1 1 1 1 1 1 1 1 1 1 1
    1 1 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 0 0   
    1 1 1 1 1 1 1 1 1 0 0

In grayscale morphology, images are functions mapping a Euclidean space or grid E into ${\displaystyle \mathbb {R} \cup \{\infty ,-\infty \}}$, where ${\displaystyle \mathbb {R} }$ is the set of reals, ${\displaystyle \infty }$ is an element greater than any real number, and ${\displaystyle -\infty }$ is an element less than any real number.

Grayscale structuring elements are also functions of the same format, called "structuring functions".

Denoting an image by f(x) and the structuring function by b(x), the grayscale dilation of f by b is given by

${\displaystyle (f\oplus b)(x)=\sup _{y\in E}[f(y)+b(x-y)],}$

where "sup" denotes the supremum.

Flat structuring functions

Example of dilation on a grayscale image using a 5x5 flat structuring element. The top figure demonstrates the application of the structuring element window to the individual pixels of the original image. The bottom figure shows the resulting dilated image.

It is common to use flat structuring elements in morphological applications. Flat structuring functions are functions b(x) in the form

${\displaystyle b(x)=\left\{{\begin{array}{ll}0,&x\in B,\\-\infty ,&{\text{otherwise}},\end{array}}\right.}$

where ${\displaystyle B\subseteq E}$.

In this case, the dilation is greatly simplified, and given by

${\displaystyle (f\oplus b)(x)=\sup _{y\in E}[f(y)+b(x-y)]=\sup _{z\in E}[f(x-z)+b(z)]=\sup _{z\in B}[f(x-z)].}$

(Suppose x = (px, qx), z = (pz, qz), then x − z = (px − pz, qx − qz).)

In the bounded, discrete case (E is a grid and B is bounded), the supremum operator can be replaced by the maximum. Thus, dilation is a particular case of order statistics filters, returning the maximum value within a moving window (the symmetric of the structuring function support B).

Complete lattices are partially ordered sets, where every subset has an infimum and a supremum. In particular, it contains a least element and a greatest element (also denoted "universe").

Let ${\displaystyle (L,\leq )}$ be a complete lattice, with infimum and supremum symbolized by ${\displaystyle \wedge }$ and ${\displaystyle \vee }$, respectively. Its universe and least element are symbolized by U and ${\displaystyle \varnothing }$, respectively. Moreover, let ${\displaystyle \{X_{i}\}}$ be a collection of elements from L.

A dilation is any operator ${\displaystyle \delta :L\rightarrow L}$ that distributes over the supremum, and preserves the least element. That is, the following are true:

• ${\displaystyle \bigvee _{i}\delta (X_{i})=\delta \left(\bigvee _{i}X_{i}\right),}$
• ${\displaystyle \delta (\varnothing )=\varnothing .}$

• Buffer (GIS)
• Closing (morphology)
• Erosion (morphology)
• Mathematical morphology
• Opening (morphology)
• Minkowski addition

• 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)