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. 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. The dilation of A by B is defined by where Ab is the translation of A by b. Dilation is commutative, also given by A ⊕ B = B ⊕ A = ⋃ a ∈ A B a {displaystyle Aoplus B=Boplus A=igcup _{ain A}B_{a}} . 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 A ⊕ B = { z ∈ E ∣ ( B s ) z ∩ A ≠ ∅ } {displaystyle Aoplus B={zin Emid (B^{s})_{z}cap A eq varnothing }} , where Bs denotes the symmetric of B, that is, B s = { x ∈ E ∣ − x ∈ B } {displaystyle B^{s}={xin Emid -xin B}} .