Dilation (morphology) - Binary Operator

Binary Operator

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

A binary image is viewed in mathematical morphology as a subset of a Euclidean space Rd or the integer grid Zd, 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.

The dilation of A by B is defined by:

.

The dilation is commutative, also given by: .

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 side 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:, where Bs denotes the symmetric of B, that is, .