In binary morphology, dilation is a shift-invariant operator, equivalent to Minkowski addition. A binary image is viewed in mathematical morphology as a subset of a Euclidean spaceRd or the integer gridZd, 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 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, where Bs denotes the symmetric of B, that is,.
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.
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 and the structuring function by b, 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 in the form where. In this case, the dilation is greatly simplified, and given by , z = , then x − z = In the bounded, discrete case, 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.
Dilation on complete lattices
s are partially ordered sets, where every subset has an infimum and a supremum. In particular, it contains a least element and a greatest element. 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. That is, the following are true:
