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| [[Image:Opening.png|thumb|right|The opening of the dark-blue square by a disk, resulting in the light-blue square with round corners.]]
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| In [[mathematical morphology]], '''opening''' is the [[Dilation (morphology)|dilation]] of the [[Erosion (morphology)|erosion]] of a [[Set (mathematics)|set]] A by a [[structuring element]] B:
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| :<math>A\circ B = (A\ominus B)\oplus B, \, </math>
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| where <math>\ominus</math> and <math>\oplus</math> denote erosion and dilation, respectively.
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| Together with [[Closing (morphology)|closing]], the opening serves in [[computer vision]] and [[image processing]] as a basic workhorse of morphological noise removal. Opening removes small objects from the foreground (usually taken as the dark pixels) of an image, placing them in the background, while closing removes small holes in the foreground, changing small islands of background into foreground. These techniques can also be used to find specific shapes in an image. Opening can be used to find things into which a specific structuring element can fit (edges, corners, ...).
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| One can think of ''B'' sweeping around the inside of the boundary of ''A'', so that it does not extend beyond the boundary, and shaping the ''A'' boundary around the boundary of the element.
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| ==Properties==
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| * Opening is [[idempotent]], that is, <math>(A\circ B)\circ B = A\circ B</math>.
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| * Opening is [[increasing]], that is, if <math>A\subseteq C</math>, then <math>A\circ B \subseteq C\circ B</math>.
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| * Opening is [[anti-extensive]], i.e., <math>A\circ B\subseteq A</math>.
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| * Opening is [[Translational invariance|translation invariant]].
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| * Opening and closing satisfy the duality <math>A \bullet B = (A^{c} \circ B^{s})^{c}</math>, where <math>\bullet</math> denotes closing.
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| ==See also==
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| *[[Mathematical morphology]]
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| *[[Closing (morphology)|Closing]]
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| *[[Dilation (morphology)|Dilation]]
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| *[[Erosion (morphology)|Erosion]]
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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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| ==External links==
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| * http://homepages.inf.ed.ac.uk/rbf/HIPR2/open.htm - Morphological Opening
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| [[Category:Mathematical morphology]]
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| [[Category:Digital geometry]]
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| {{compu-graphics-stub}}
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| {{comp-sci-stub}}
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My hobby is mainly Card collecting. Seems boring? Not!
I also try to learn Arabic in my free time.
Also visit my blog bookbyte Discounts