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| ''For use of "Erosion" in dermatopathology, see [[Erosion (dermatopathology)]]'' | | I'm a 42 years old, married and study at the university (Optometry).<br>In my spare time I'm trying to learn Spanish. I have been there and look forward to go there [http://www.party24-7.com/board_Qyic63/641178 4Inkjets Promotional] sometime in the future. I love [http://www.acsikorea.or.kr/video/1230752 4inkjets Coupon Code 20% Off] to read, preferably on my kindle. I like to watch Game of Thrones and The Vampire Diaries as well as docus about nature. I like Bus spotting.<br><br>Also visit my web-site - [http://mytoclub.com/board_DGPQ52/91441 Online Coupon Coupon Code 4Inkjets] |
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| [[Image:Erosion.png|thumb|right|The erosion of the dark-blue square by a disk, resulting in the light-blue square.]]
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| Erosion is one of two fundamental operations (the other being [[Dilation (morphology)|dilation]]) in [[morphological image processing]] from which all other morphological operations are based. It was originally defined for [[binary image]]s, later being extended to [[grayscale]] images, and subsequently to [[complete lattice]]s.
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| == Binary erosion ==
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| In binary morphology, an image is viewed as a [[subset]] of a [[Euclidean space]] <math>\mathbb{R}^d</math> or the [[integer]] [[lattice graph|grid]] <math>\mathbb{Z}^d</math>, for some dimension ''d''.
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| The basic idea in binary morphology is to probe an image with a simple, pre-defined shape, drawing conclusions on how this shape fits or misses the shapes in the image. This simple "probe" is called [[structuring element]], and is itself a binary image (i.e., a subset of the space or grid).
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| Let ''E'' be a Euclidean space or an integer grid, and ''A'' a binary image in ''E''.
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| The '''erosion''' of the binary image ''A'' by the structuring element ''B'' is defined by:
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| ::<math>A \ominus B = \{z\in E | B_{z} \subseteq A\}</math>,
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| where ''B''<sub>''z''</sub> is the translation of ''B'' by the vector z, i.e., <math>B_z = \{b+z|b\in B\}</math>, <math>\forall z\in E</math>.
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| When the structuring element ''B'' has a center (e.g., a disk or a square), and this center is located on the origin of ''E'', then the erosion of ''A'' by ''B'' can be understood as the locus of points reached by the center of ''B'' when ''B'' moves inside ''A''. For example, the erosion of a square of side 10, centered at the origin, by a disc of radius 2, also centered at the origin, is a square of side 6 centered at the origin.
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| The erosion of ''A'' by ''B'' is also given by the expression: <math>A \ominus B = \bigcap_{b\in B} A_{-b}</math>.
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| === Example ===
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| Suppose A is a 13 x 13 matrix and B is a 3 x 3 matrix:
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| 1 1 1 1 1 1 1 1 1 1 1 1 1
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| 1 1 1 1 1 1 0 1 1 1 1 1 1
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| 1 1 1 1 1 1 1 1 1 1 1 1 1
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| 1 1 1 1 1 1 1 1 1 1 1 1 1
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| 1 1 1 1 1 1 1 1 1 1 1 1 1
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| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
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| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
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| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
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| 1 1 1 1 1 1 1 1 1 1 1 1 1
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| 1 1 1 1 1 1 1 1 1 1 1 1 1
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| 1 1 1 1 1 1 1 1 1 1 1 1 1
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| 1 1 1 1 1 1 1 1 1 1 1 1 1
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| 1 1 1 1 1 1 1 1 1 1 1 1 1
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| Assuming that the origin B is at its center, for each pixel in A '''superimpose''' the origin of B, if B is completely contained by A the pixel is retained, else deleted.
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| Therefore the '''Erosion''' of A by B is given by this 11 x 11 matrix.
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| 1 1 1 1 0 0 0 1 1 1 1
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| 1 1 1 1 0 0 0 1 1 1 1
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| 1 1 1 1 1 1 1 1 1 1 1
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| 1 1 1 1 1 1 1 1 1 1 1
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| 1 1 1 1 1 1 1 1 1 1 1
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| 1 1 1 1 1 1 1 1 1 1 1
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| 1 1 1 1 1 1 1 1 1 1 1
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| 1 1 1 1 1 1 1 1 1 1 1
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| 1 1 1 1 1 1 1 1 1 1 1
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| 1 1 1 1 1 1 1 1 1 1 1
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| 1 1 1 1 1 1 1 1 1 1 1
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| This means that only when B is '''completely contained''' inside A that the pixels values are retained, otherwise it gets deleted or eroded.
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| ===Properties===
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| * The erosion is [[Translational invariance|translation invariant]].
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| * It is [[increasing]], that is, if <math>A\subseteq C</math>, then <math>A\ominus B \subseteq C\ominus B</math>.
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| * If the origin of ''E'' belongs to the structuring element ''B'', then the erosion is ''anti-extensive'', i.e., <math>A\ominus B\subseteq A</math>.
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| * The erosion satisfies <math>(A\ominus B)\ominus C = A\ominus (B\oplus C)</math>, where <math>\oplus</math> denotes the [[dilation (morphology)|morphological dilation]].
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| * The erosion is [[distributive]] over [[set intersection]]
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| ==Grayscale erosion==
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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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| Denoting an image by ''f(x)'' and the grayscale structuring element by ''b(x)'', where B is the space that b(x) is defined, the grayscale erosion of ''f'' by ''b'' is given by
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| ::<math>(f\ominus b)(x)=\inf_{y\in B}[f(y)-b(y-x)]</math>,
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| where "inf" denotes the [[infimum]].
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| ==Erosions 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>\emptyset</math>, respectively. Moreover, let <math>\{ X_{i} \}</math> be a collection of elements from ''L''.
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| An erosion in <math>(L,\leq)</math> is any operator <math>\varepsilon: L\rightarrow L</math> that distributes over the infimum, and preserves the universe. I.e.:
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| * <math>\bigwedge_{i}\varepsilon(X_i)=\varepsilon\left(\bigwedge_{i} X_i\right)</math>,
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| * <math>\varepsilon(U)=U</math>.
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| ==See also==
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| *[[Mathematical morphology]]
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| *[[Dilation (morphology)|Dilation]]
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| *[[Opening (morphology)|Opening]]
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| *[[Closing (morphology)|Closing]]
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| == References ==
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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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| * ''Morphological Image Analysis; Principles and Applications'' by Pierre Soille, ISBN 3-540-65671-5 (1999)
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| * R. C. Gonzalez and R. E. Woods, ''Digital image processing'', 2nd ed. Upper Saddle River, N.J.: Prentice Hall, 2002.
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| [[Category:Image processing]]
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| [[Category:Digital geometry]]
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| [[Category:Mathematical morphology]]
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