Dilation (morphology)

Dilation (usually represented by ⊕) 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.

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 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.

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.

Each pixel of every superimposed B is included in the dilation of A by B.

Here are some properties of the binary dilation operator In grayscale morphology, images are functions mapping a Euclidean space or grid E into

is an element greater than any real number, and

It is common to use flat structuring elements in morphological applications.

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).)

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.

be a complete lattice, with infimum and supremum symbolized by

be a collection of elements from L. A dilation is any operator

that distributes over the supremum, and preserves the least element.

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