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In [[mathematical morphology]] and [[digital image processing]], '''top-hat transform''' is an operation that extracts small elements and details from given [[image]]s. There exist two types of top-hat transform: The ''white top-hat transform'' is defined as the difference between the input image and its [[opening (morphology)|opening]] by some [[structuring element]]; The ''black top-hat transform'' is defined dually as the difference between the [[closing (morphology)|closing]] and the input image. Top-hat transforms are used for various image processing tasks, such as [[feature extraction]], [[background equalization]], [[image enhancement]], and others.
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==Mathematical definitions==
 
Let <math>f:E\mapsto R</math> be a [[grayscale]] image, mapping points from an [[Euclidean space]] or discrete [[Lattice graph|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.  
 
Then, the white top-hat transform of ''f'' is given by:
 
:<math>T_w(f)=f-f \circ b</math>,
 
where <math>\circ</math> denotes the opening operation.
 
The black top-hat transform of ''f'' (sometimes called the ''bottom-hat'' transform<ref>{{Cite web|last=Tcheslavski|first=Gleb V.|url=http://www.ee.lamar.edu/gleb/dip/10-3%20-%20Morphological%20Image%20Processing.pdf|title=Morphological Image Processing: Gray-scale morphology|year=2010|accessdate=4 November 2013}}</ref>
) is given by:
 
:<math>T_b(f)=f\bullet b-f</math>,
 
where <math>\bullet</math> is the closing operation.
 
==Properties==
 
The white top-hat transform returns an image, containing those "objects" or "elements" of an input image that:
* Are "smaller" than the structuring element (i.e., places where the structuring element does not fit in), and
* are '''brighter''' than their surroundings.
 
The black top-hat returns an image, containing the "objects" or "elements" that:
* Are "smaller" than the structuring element, and
* are '''darker''' than their surroundings.  
 
The size, or width, of the elements that are extracted by the top-hat transforms can be controlled by the choice of the structuring element <math>b</math>. The bigger the latter, the larger the elements extracted.
 
Both top-hat transforms are images that contain only 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)
* ''Hands-on Morphological Image Processing'' by Edward R. Dougherty and R. Lotufo, ISBN 0-8194-4720-X (2003)
 
{{reflist}}
 
[[Category:Mathematical morphology]]
[[Category:Digital geometry]]
 
{{compu-graphics-stub}}
{{comp-sci-stub}}

Revision as of 23:26, 4 March 2014

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