en>R.e.b.: /* Stable vector bundles over curves */ Adding/removing wikilink(s)

2012-12-06T17:33:39Z

Stable vector bundles over curves: Adding/removing wikilink(s)

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	~~Hi there~~, ~~I am Caprice. To engage in golfing~~ is ~~the factor I adore most. Considering that I was 18 I've been performing as~~ a ~~dispatcher and the income has been genuinely satisfying. Montana has often been my living place and I will not approach on altering it. My spouse and I retain~~ a [~~http://browse~~.~~deviantart.com/?qh=&section=&global=1&q=web-site web-site~~]. ~~You might want to check it out here: http://wiki.acceed.de/index.php/Benutzer_Diskussion:JasminVESX<br><br>~~		In [[mathematical morphology]], '''granulometry''' is an approach to compute a size distribution of grains in binary images, using a series of [[opening (morphology)\|morphological opening]] operations. It was introduced by [[Georges Matheron]] in the 1960s, and is the basis for the characterization of the concept of ''size'' in mathematical morphology.

	~~Here~~ is ~~my web page~~ [~~http~~://~~wiki~~.~~acceed~~.de/~~index~~.~~php~~/~~Benutzer_Diskussion~~:~~JasminVESX Nike free run~~]		== Granulometry generated by a structuring element ==

			Let ''B'' be a [[structuring element]] in an [[Euclidean space]] or [[Lattice graph\|grid]] ''E'', and consider the family <math>\{B_k\}</math>, <math>k=0,1,\ldots</math>, given by:

			:<math>B_k=\underbrace{B\oplus\ldots\oplus B}_{k\mbox{ times}}</math>,

			where <math>\oplus</math> denotes [[dilation (morphology)\|morphological dilation]]. By convention, <math>B_0</math> is the set containing only the origin of ''E'', and <math>B_1=B</math>.

			Let ''X'' be a [[Set (mathematics)\|set]] (i.e., a [[binary image]] in mathematical morphology), and consider the series of sets <math>\{\gamma_k(X)\}</math>, <math>k=0,1,\ldots</math>, given by:

			:<math>\gamma_k(X)=X\circ B_k</math>,

			where <math>\circ</math> denotes the morphological opening.

			The ''granulometry function'' <math>G_k(X)</math> is the [[cardinality]] (i.e., [[area]] or [[volume]], in continuous Euclidean space, or number of elements, in grids) of the image <math>\gamma_k(X)</math>:

			:<math>G_k(X)=\|\gamma_k(X)\|</math>.

			The '''pattern spectrum''' or '''size distribution''' of ''X'' is the collection of sets <math>\{PS_k(X)\}</math>, <math>k=0,1,\ldots</math>, given by:

			:<math>PS_k(X) = G_{k}(X)-G_{k+1}(X)</math>.

			The parameter ''k'' is referred to as ''size'', and the component ''k'' of the pattern spectrum <math>PS_k(X)</math> provides a rough estimate for the amount of grains of size ''k'' in the image ''X''. Peaks of <math>PS_k(X)</math> indicate relatively large quantities of grains of the corresponding sizes.

			== Sieving axioms ==

			The above common method is a particular case of the more general approach derived by Matheron.

			The [[France\|French]] mathematician was inspired by [[sieving]] as a means of characterizing ''size''. In sieving, a [[granular]] sample is worked through a series of [[sieve]]s with decreasing hole sizes. As a consequence, the different [[grain]]s in the sample are separated according to their sizes.

			The operation of passing a sample through a sieve of certain hole size "''k''" can be mathematically described as an operator <math>\Psi_k(X)</math> that returns the subset of elements in ''X'' with sizes that are smaller or equal to ''k''. This family of operators satisfy the following properties:

			# ''Anti-extensivity'': Each sieve reduces the amount of grains, i.e., <math>\Psi_k(X)\subseteq X</math>,
			# ''Increasingness'': The result of sieving a subset of a sample is a subset of the sieving of that sample, i.e., <math>X\subseteq Y\Rightarrow\Psi_k(X)\subseteq\Psi_k(Y)</math>,
			# "''Stability''": The result of passing through two sieves is determined by the sieve with smallest hole size. I.e., <math>\Psi_k\Psi_m(X)=\Psi_m\Psi_k(X)=\Psi_{\max(k,m)}(X)</math>.

			A granulometry-generating family of operators should satisfy the above three axioms.

			In the above case (granulometry generated by a structuring element), <math>\Psi_k(X)=\gamma_k(X)=X\circ B_k</math>.

			Another example of granulometry-generating family is when <math>\Psi_k(X)=\bigcup_{i=1}^{N} X\circ (B^{(i)})_k</math>, where <math>\{B^{(i)}\}</math> is a set of linear structuring elements with different directions.

			== References ==

			* ''Random Sets and Integral Geometry'', by Georges Matheron, Wiley 1975, ISBN 0-471-57621-2.
			* ''Image Analysis and Mathematical Morphology'' by Jean Serra, ISBN 0-12-637240-3 (1982)
			* ''Image Segmentation By Local Morphological Granulometries,'' Dougherty, ER, Kraus, EJ, and Pelz, JB., Geoscience and Remote Sensing Symposium, 1989. IGARSS'89, DOI: 10.1109/IGARSS.1989.576052 (1989)
			* ''An Introduction to Morphological Image Processing'' by Edward R. Dougherty, ISBN 0-8194-0845-X (1992)
			* ''Morphological Image Analysis; Principles and Applications'' by Pierre Soille, ISBN 3-540-65671-5 (1999)

			[[Category:Mathematical morphology]]

en>Bender235: /* References */

2012-06-08T22:42:30Z

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Hi there, I am Caprice. To engage in golfing is the factor I adore most. Considering that I was 18 I've been performing as a dispatcher and the income has been genuinely satisfying. Montana has often been my living place and I will not approach on altering it. My spouse and I retain a [http://browse.deviantart.com/?qh=&section=&global=1&q=web-site web-site]. You might want to check it out here: http://wiki.acceed.de/index.php/Benutzer_Diskussion:JasminVESX<br><br>

Here is my web page [http://wiki.acceed.de/index.php/Benutzer_Diskussion:JasminVESX Nike free run]

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en>R.e.b.: /* Stable vector bundles over curves */ Adding/removing wikilink(s)

en>Bender235: /* References */