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| '''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.
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| ==Binary operator==
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| [[Image:Dilation.png|thumb|right|The dilation of dark-blue square by a disk, resulting in the light-blue square with rounded corners.]]
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| In binary morphology, dilation is a shift-invariant ([[Translational invariance|translation invariant]]) operator, strongly related to the [[Minkowski addition]].
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| A binary image is viewed in mathematical morphology as a [[subset]] of a [[Euclidean space]] ''R''<sup>''d''</sup> or the integer grid ''Z''<sup>''d''</sup>, 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.
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| The dilation of ''A'' by ''B'' is defined by:
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| ::<math>A \oplus B = \bigcup_{b\in B} A_b</math>.
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| The dilation is commutative, also given by: <math>A \oplus B = B\oplus A = \bigcup_{a\in A} B_a</math>.
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| 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.
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| The dilation can also be obtained by: <math>A \oplus B = \{z \in E| (B^{s})_{z} \cap A\neq \varnothing\}</math>, where ''B''<sup>''s''</sup> denotes the [[rotational symmetry|symmetric]] of ''B'', that is, <math>B^{s}=\{x\in E | -x \in B\}</math>.
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| === Properties of binary dilation ===
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| Here are some properties of the binary dilation operator:
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| * It is [[Translational invariance|translation invariant]].
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| * It is [[increasing]], that is, if <math>A\subseteq C</math>, then <math>A\oplus B \subseteq C\oplus B</math>.
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| * It is [[commutative]].
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| * If the origin of ''E'' belongs to the structuring element ''B'', then it is [[extensive]], i.e., <math>A\subseteq A\oplus B</math>.
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| * It is [[associative]], i.e., <math>(A\oplus B)\oplus C = A\oplus (B\oplus C)</math>.
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| * It is [[distributive]] over [[set union]]
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| ==Grayscale dilation==
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| In [[grayscale]] morphology, images are [[Function (mathematics)|functions]] mapping a [[Euclidean space]] or [[lattice graph|grid]] ''E'' into <math>\mathbb{R}\cup\{\infty,-\infty\}</math>, where <math>\mathbb{R}</math> is the set of [[real numbers|reals]], <math>\infty</math> is an element larger than any real number, and <math>-\infty</math> is an element smaller than any real number.
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| Grayscale structuring elements are also functions of the same format, called "structuring functions".
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| Denoting an image by ''f(x)'' and the structuring function by ''b(x)'', the grayscale dilation of ''f'' by ''b'' is given by
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| ::<math>(f\oplus b)(x)=\sup_{y\in E}[f(y)+b(x-y)]</math>,
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| where "sup" denotes the [[supremum]].
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| ===Flat structuring functions===
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| It is common to use flat structuring elements in morphological applications. Flat structuring functions are functions ''b(x)'' in the form
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| ::<math>b(x)=\left\{\begin{array}{ll}0,&x\in B,\\-\infty,&\mbox{otherwise}\end{array}\right.</math>, | |
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| where <math>B\subseteq E</math>.
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| In this case, the dilation is greatly simplified, and given by
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| ::<math>(f\oplus b)(x)=\sup_{y\in E}[f(y)+b(x-y)]=\sup_{z\in E}[f(x-z)+b(z)]=\sup_{z\in B}[f(x-z)]</math>.
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| (Suppose x=(px,qx), z=(pz,qz), then x-z=(px-pz, qx-qz).)
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| 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'').
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| ==Dilation on complete lattices==
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| [[Complete lattice]]s are [[partially ordered set]]s, where every subset has an [[infimum]] and a [[supremum]]. In particular, it contains a [[least element]] and a [[greatest element]] (also denoted "universe").
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| Let <math>(L,\leq)</math> be a complete lattice, with infimum and supremum symbolized by <math>\wedge</math> and <math>\vee</math>, respectively. Its universe and least element are symbolized by ''U'' and <math>\varnothing</math>, respectively. Moreover, let <math>\{ X_{i} \}</math> be a collection of elements from ''L''.
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| A dilation is any operator <math>\delta: L\rightarrow L</math> that distributes over the supremum, and preserves the least element. I.e.:
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| * <math>\bigvee_{i}\delta(X_i)=\delta\left(\bigvee_{i} X_i\right)</math>,
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| * <math>\delta(\varnothing)=\varnothing</math>.
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| ==See also==
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| *[[Buffer (GIS)]]
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| *[[Closing (morphology)|Closing]]
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| *[[Erosion (morphology)|Erosion]]
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| *[[Mathematical morphology]]
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| *[[Opening (morphology)|Opening]]
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| ==Bibliography==
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| * ''Image Analysis and Mathematical Morphology'' by Jean Serra, ISBN 0-12-637240-3 (1982)
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| * ''Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances'' by Jean Serra, ISBN 0-12-637241-1 (1988)
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| * ''An Introduction to Morphological Image Processing'' by Edward R. Dougherty, ISBN 0-8194-0845-X (1992)
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| {{DEFAULTSORT:Dilation (Morphology)}}
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| [[Category:Mathematical morphology]]
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| [[Category:Digital geometry]]
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