Dilation is one of the basic operations in mathematical morphology. Originally developed for binary images, it has been expanded first to grayscale images, and then to complete lattices. The dilation operation usually uses a structuring element for probing and expanding the shapes contained in the input image.
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 regarded as a subset of Rd.
The dilation of A by B is defined by:
where Ab is the translation of A by b.
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, .
Properties of binary dilation
Here are some properties of the binary dilation operator
- It is translation invariant.
- It is increasing, that is, if , then .
- It is commutative.
- If the origin of E belongs to the structuring element B, then it is extensive, i.e., .
- It is associative, i.e., .
- It is distributive over set union
In grayscale morphology, images are functions mapping a Euclidean space or grid E into , where is the set of reals, is an element greater than any real number, and 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
where "sup" denotes the supremum.
Flat structuring functions
It is common to use flat structuring elements in morphological applications. Flat structuring functions are functions b(x) in the form
In this case, the dilation is greatly simplified, and given by
(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).
Dilation on complete lattices
Let be a complete lattice, with infimum and supremum symbolized by and , respectively. Its universe and least element are symbolized by U and , respectively. Moreover, let be a collection of elements from L.
A dilation is any operator that distributes over the supremum, and preserves the least element. I.e.: