Dilation (morphology)

Binary dilation

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 ${\displaystyle A\oplus B=\{z\in E\mid (B^s{)}_z\cap A\ne \varnothing \}}$, where B^s denotes the symmetric of B, that is, ${\displaystyle B^s=\{x\in E\mid -x\in B\}}$.

Example

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 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

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.

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

Properties of binary dilation

Here are some properties of the binary dilation operator

• It is translation invariant.
• It is increasing, that is, if ${\displaystyle A\subseteq C}$, then ${\displaystyle A\oplus B\subseteq C\oplus B}$.
• It is commutative.
• If the origin of E belongs to the structuring element B, then it is extensive, i.e., ${\displaystyle A\subseteq A\oplus B}$.
• It is associative, i.e., ${\displaystyle (A\oplus B)\oplus C=A\oplus (B\oplus C)}$.
• It is distributive over set union