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In [[mathematical morphology]] and [[digital image processing]], a '''morphological gradient''' is the difference between the [[dilation (morphology)|dilation]] and the [[erosion (morphology)|erosion]] of a given image. It is an image where each [[pixel]] value (typically non-negative) indicates the contrast intensity in the close neighborhood of that pixel. It is useful for [[edge detection]] and [[Segmentation (image processing)|segmentation]] applications.
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==Mathematical definition and types==
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Let <math>f:E\mapsto R</math> be a grayscale image, mapping points from a Euclidean space or discrete grid ''E'' (such as ''R''<sup>2</sup> or ''Z''<sup>2</sup>) into the real line. Let <math>b(x)</math> be a grayscale [[structuring element]]. Usually, ''b'' is [[symmetry|symmetric]] and has [[short-support]], e.g.,  
:<math>b(x)=\left\{\begin{array}{ll}0,&|x|\leq 1,\\-\infty,&\mbox{otherwise}\end{array}\right.</math>.
 
Then, the morphological gradient of ''f'' is given by:
 
:<math>G(f)=f\oplus b-f\ominus b</math>,
 
where <math>\oplus</math> and <math>\ominus</math> denote the dilation and the erosion, respectively.
 
An '''internal gradient''' is given by:
 
:<math>G_i(f)=f-f\ominus b</math>,
 
and an '''external gradient''' is given by:
 
:<math>G_e(f)=f\oplus b-f</math>.
 
The internal and external gradients are "thinner" than the gradient, but the gradient peaks are located ''on'' the edges, whereas the internal and external ones are located at each side of the edges. Notice that <math>G_i+G_e=G</math>.
 
If <math>b(0)\geq 0</math>, then all the three gradients have non-negative values at all pixels.
 
==References==
* <cite id=serra82>''Image Analysis and Mathematical Morphology'' by Jean Serra, ISBN 0-12-637240-3 (1982)</cite>
* ''Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances'' by Jean Serra, ISBN 0-12-637241-1 (1988)
* ''An Introduction to Morphological Image Processing'' by Edward R. Dougherty, ISBN 0-8194-0845-X (1992)
 
== External links ==
* [http://cmm.ensmp.fr/~beucher/publi/morph_grad.pdf Morphological gradients], Centre de Morphologie Mathématique, École_des_Mines_de_Paris
 
[[Category:Mathematical morphology]]
[[Category:Digital geometry]]
 
{{compu-graphics-stub}}
{{comp-sci-stub}}

Latest revision as of 11:00, 19 June 2014

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