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Concept
Dilation (morphology)
Summary
Dilation (usually represented by ⊕) is one of the basic operations in mathematical morphology. Originally developed for , 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.
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, .
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
Here are some properties of the binary dilation operator
It is translation invariant.
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