Arithmetical set - Revision history

en>Trovatore: /* Properties */ denumerable -> countable

2014-05-02T20:41:08Z

Properties: denumerable -> countable

← Older revision		Revision as of 22:41, 2 May 2014
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	~~{\| style="width:300px; float:right; border:1px solid #ccc; background:#f9f9f9; font-size:88%; line-height:1.5em; margin:5px;"~~		Hi there, I am Andrew Berryhill. The favorite hobby for him and his kids is to perform lacross and he would never give it up. My working day occupation is a travel agent. Some time in the past she chose to live in Alaska and her parents live close by.<br><br>My web-site [http://www.Prayerarmor.com/uncategorized/dont-know-which-kind-of-hobby-to-take-up-read-the-following-tips/ love psychic]
	~~\| [[File:RGB 24bits palette sample image.jpg]]~~
	~~\| [[File:RGB 24bits palette sample image - 3-bit RGB.png]]~~
	\|-
	~~\| colspan="2" width="300px" \| Example of a 24-bit RGB image dithered to [[List_of_monochrome_and_RGB_palettes#3-bit_RGB\|3-bit RGB]] using Floyd–Steinberg dithering~~
	\|}

	~~[[Image:Michelangelo's David - Floyd-Steinberg.png\|thumb\|right\|An image of the [[David (Michelangelo)\|Statue of David]]~~, ~~dithered with Floyd–Steinberg algorithm]]~~

	~~'''Floyd–Steinberg dithering''' is an image [[dithering]] algorithm first published in 1976 by [[Robert W~~. ~~Floyd]]~~ and ~~[[Louis Steinberg]]. It~~ is ~~commonly used by image manipulation software, for example when an image is converted into [[GIF]] format that is restricted~~ to ~~a maximum of 256 colors~~.

	The algorithm achieves dithering by pushing (adding) the residual [[quantization error]] of a [[pixel]] onto its neighboring pixels, to be dealt with later. It spreads the debt out according to the distribution (shown as a ~~map of the neighboring pixels):~~
	~~: <math>~~
	~~\begin{bmatrix}~~
	~~& & & & \\~~
	~~& & * & \frac{\displaystyle 7}{\displaystyle 16} & ... \\~~
	~~... & \frac{\displaystyle 3}{\displaystyle 16} & \frac{\displaystyle 5}{\displaystyle 16} & \frac{\displaystyle 1}{\displaystyle 16} & ... \\~~
	~~\end{bmatrix}~~
	~~</math>~~
	~~The pixel indicated with a star (*) indicates the pixel currently being scanned, and the blank pixels are the previously-scanned pixels.~~
	~~The algorithm scans the image from left to right, top to bottom, quantizing pixel values one by one~~. ~~Each~~ time the ~~quantization error is transferred~~ to the neighboring pixels, while not affecting the pixels that already have been quantized. Hence, if a number of pixels have been rounded downwards, it becomes more likely that the next pixel is rounded upwards, such that on average, the quantization error is close to zero.

	~~The diffusion coefficients have the property that if the original pixel values are exactly halfway~~ in ~~between the nearest available colors, the dithered result is a checkerboard pattern. For example 50% grey data could be dithered as a black-~~and~~-white checkerboard pattern. For optimal dithering, the counting of quantization errors should be in sufficient accuracy to prevent rounding errors from affecting the result~~.

	~~In some implementations, the horizontal direction of scan alternates between lines; this is called "serpentine scanning" or [[boustrophedon transform]] dithering.~~

	~~In [[pseudocode]]:~~
	~~'''for each''' y from top to bottom~~
	~~'''for each''' x from left to right~~
	~~oldpixel := pixel[x][y]~~
	~~newpixel := find_closest_palette_color(oldpixel)~~
	~~pixel[x][y] := newpixel~~
	~~quant_error := oldpixel - newpixel~~
	~~pixel[x+1][y ] := pixel[x+1][y ] + 7/16 * quant_error~~
	~~pixel[x-1][y+1] := pixel[x-1][y+1] + 3/16 * quant_error~~
	~~pixel[x ][y+1] := pixel[x ][y+1] + 5/16 * quant_error~~
	~~pixel[x+1][y+1] := pixel[x+1][y+1] + 1/16 * quant_error~~

	~~For example, to convert 16 bit to 8 bit~~ <~~code~~>~~find_closest_palette_color()~~<~~/code~~> ~~may perform just a simple rounding~~
	~~find_closest_palette_color(oldpixel) = oldpixel / 256~~

	~~==See also==~~
	*[[Error diffusion]]

	~~==References==~~
	*[http://www.~~visgraf~~.~~impa.br~~/~~Courses~~/~~ip00/proj/Dithering1/floyd_steinberg_dithering.html Floyd–Steinberg Dithering] (Graphics course project, Visgraf lab, Brazil)~~
	*R.W. Floyd, L. Steinberg, ''An adaptive algorithm for spatial grey scale''. Proceedings of ~~the Society of Information Display '''17''', 75–77 (1976).~~

	~~==External links==~~
	*[http://www.2038bug.com/ptrans/ptrans.html PTRANS] Stand-~~alone ANSI~~-~~C programming language implementation.~~
	*[http://michal.is/projects/image-~~dithering~~-in-~~matlab~~/ ~~Matlab implementation]~~

	~~{{DEFAULTSORT:Floyd-Steinberg dithering}}~~
	~~[[Category:Image processing]]~~
	~~[[Category:Articles with example pseudocode]]~~
	~~[[Category:Computer graphics algorithms]~~]

en>Hyacinth: /* References */ -->Further reading, uncited

2013-09-20T04:23:52Z

References: -->Further reading, uncited

← Older revision		Revision as of 06:23, 20 September 2013
Line 1:		Line 1:
	~~Hi there~~, ~~I am Andrew Berryhill~~. ~~She works~~ as a ~~journey agent but quickly she'll be on her personal~~. ~~Mississippi~~ is ~~where her house~~ is ~~but her spouse wants them~~ to ~~move~~. ~~To play lacross~~ is 1 of the ~~issues she enjoys most~~.<br><br>~~My weblog - tarot readings~~ ([http://~~gcjcteam~~.~~org~~/~~index~~.~~php?mid~~=~~etc_video&document_srl~~=~~696611&sort_index~~=~~regdate&order_type~~=~~desc click through the following website~~])		{\| style="width:300px; float:right; border:1px solid #ccc; background:#f9f9f9; font-size:88%; line-height:1.5em; margin:5px;"
			\| [[File:RGB 24bits palette sample image.jpg]]
			\| [[File:RGB 24bits palette sample image - 3-bit RGB.png]]
			\|-
			\| colspan="2" width="300px" \| Example of a 24-bit RGB image dithered to [[List_of_monochrome_and_RGB_palettes#3-bit_RGB\|3-bit RGB]] using Floyd–Steinberg dithering
			\|}

			[[Image:Michelangelo's David - Floyd-Steinberg.png\|thumb\|right\|An image of the [[David (Michelangelo)\|Statue of David]], dithered with Floyd–Steinberg algorithm]]

			'''Floyd–Steinberg dithering''' is an image [[dithering]] algorithm first published in 1976 by [[Robert W. Floyd]] and [[Louis Steinberg]]. It is commonly used by image manipulation software, for example when an image is converted into [[GIF]] format that is restricted to a maximum of 256 colors.

			The algorithm achieves dithering by pushing (adding) the residual [[quantization error]] of a [[pixel]] onto its neighboring pixels, to be dealt with later. It spreads the debt out according to the distribution (shown as a map of the neighboring pixels):
			: <math>
			\begin{bmatrix}
			& & & & \\
			& & * & \frac{\displaystyle 7}{\displaystyle 16} & ... \\
			... & \frac{\displaystyle 3}{\displaystyle 16} & \frac{\displaystyle 5}{\displaystyle 16} & \frac{\displaystyle 1}{\displaystyle 16} & ... \\
			\end{bmatrix}
			</math>
			The pixel indicated with a star (*) indicates the pixel currently being scanned, and the blank pixels are the previously-scanned pixels.
			The algorithm scans the image from left to right, top to bottom, quantizing pixel values one by one. Each time the quantization error is transferred to the neighboring pixels, while not affecting the pixels that already have been quantized. Hence, if a number of pixels have been rounded downwards, it becomes more likely that the next pixel is rounded upwards, such that on average, the quantization error is close to zero.

			The diffusion coefficients have the property that if the original pixel values are exactly halfway in between the nearest available colors, the dithered result is a checkerboard pattern. For example 50% grey data could be dithered as a black-and-white checkerboard pattern. For optimal dithering, the counting of quantization errors should be in sufficient accuracy to prevent rounding errors from affecting the result.

			In some implementations, the horizontal direction of scan alternates between lines; this is called "serpentine scanning" or [[boustrophedon transform]] dithering.

			In [[pseudocode]]:
			'''for each''' y from top to bottom
			'''for each''' x from left to right
			oldpixel := pixel[x][y]
			newpixel := find_closest_palette_color(oldpixel)
			pixel[x][y] := newpixel
			quant_error := oldpixel - newpixel
			pixel[x+1][y ] := pixel[x+1][y ] + 7/16 * quant_error
			pixel[x-1][y+1] := pixel[x-1][y+1] + 3/16 * quant_error
			pixel[x ][y+1] := pixel[x ][y+1] + 5/16 * quant_error
			pixel[x+1][y+1] := pixel[x+1][y+1] + 1/16 * quant_error

			For example, to convert 16 bit to 8 bit <code>find_closest_palette_color()</code> may perform just a simple rounding
			find_closest_palette_color(oldpixel) = oldpixel / 256

			==See also==
			*[[Error diffusion]]

			==References==
			*[http://www.visgraf.impa.br/Courses/ip00/proj/Dithering1/floyd_steinberg_dithering.html Floyd–Steinberg Dithering] (Graphics course project, Visgraf lab, Brazil)
			*R.W. Floyd, L. Steinberg, ''An adaptive algorithm for spatial grey scale''. Proceedings of the Society of Information Display '''17''', 75–77 (1976).

			==External links==
			*[http://www.2038bug.com/ptrans/ptrans.html PTRANS] Stand-alone ANSI-C programming language implementation.
			*[http://michal.is/projects/image-dithering-in-matlab/ Matlab implementation]

			{{DEFAULTSORT:Floyd-Steinberg dithering}}
			[[Category:Image processing]]
			[[Category:Articles with example pseudocode]]
			[[Category:Computer graphics algorithms]]

en>HRoestBot: r2.6.5) (Robot: Adding zh:算术集

2012-06-25T10:51:11Z

r2.6.5) (Robot: Adding zh:算术集

New page

Hi there, I am Andrew Berryhill. She works as a journey agent but quickly she'll be on her personal. Mississippi is where her house is but her spouse wants them to move. To play lacross is 1 of the issues she enjoys most.<br><br>My weblog - tarot readings ([http://gcjcteam.org/index.php?mid=etc_video&document_srl=696611&sort_index=regdate&order_type=desc click through the following website])