Boundary value problem - Revision history

en>Gaussgauss: Changed 'restraints' to 'constraints', the latter being the more appropriate word in this context

2014-11-10T23:23:41Z

Changed 'restraints' to 'constraints', the latter being the more appropriate word in this context

← Older revision		Revision as of 01:23, 11 November 2014
Line 1:		Line 1:
	~~{{refimprove\|date=June 2009}}~~
	~~<!-- Deleted image removed: [[Image:Serioussam.jpg\|thumb\|270px\|[[Serious Sam]] was one of the first [[First-person shooter\|first person shooters]] to use mipmapping in real time.]] -->~~

	In [[3D computer graphics]], '''mipmaps''' (also '''MIP maps''') are pre-calculated, [[Optimization (computer science)\|optimized]] collections of images that accompany a main texture, intended to increase rendering speed and reduce [[aliasing]] artifacts. They are widely used in 3D [[computer game]]s, [[flight simulator]]s and other 3D imaging systems for [[texture filtering]]. Their use is known as '''mipmapping'''. The letters "MIP" in the name are an acronym of the [[Latin]] phrase ''multum in parvo'', meaning "much in little".<ref name="staff">http://staff.cs.psu.ac.th/iew/cs344-481/p1-williams.pdf</ref> Since mipmaps cannot be calculated in real time, additional storage space is required to take advantage of them. They also form the basis of [[wavelet compression]].

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	~~Mipmaps~~ are ~~used for:~~<br>
	~~-Speeding up rendering times. (Smaller textures equate to less memory usage.)~~<br>
	~~-Improving~~ the ~~quality~~. ~~Rendering large textures~~ where ~~only small subsets~~ of ~~points are used can easily produce [[moiré pattern]]s~~.<br>
	~~-Reducing stress on [[GPU]].~~<br>

	~~==Origin==~~
	~~Mipmapping was invented~~ by ~~[[Lance Williams]] in 1983 and is described in his paper ''Pyramidal parametrics''~~.~~<ref name="staff" /> From the abstract: "This paper advances a 'pyramidal parametric' prefiltering and sampling geometry~~ which ~~minimizes aliasing effects~~ and ~~assures continuity within~~ and ~~between target images~~.~~" The "pyramid" can~~ be ~~imagined as the set of mipmaps stacked on top of each other~~.

	~~==How~~ it ~~works==~~
	~~[[Image:MipMap Example STS101~~.~~jpg\|frame\|An example of mipmap image storage:~~ the ~~principal image on the left~~ is ~~accompanied by filtered copies~~ of ~~reduced size~~.]]

	~~Each bitmap image~~ of ~~the mipmap set~~ is ~~a downsized duplicate of the main [[Texture (computer graphics)\|texture]], but at a certain reduced level of detail~~. ~~Although the main texture~~ would ~~still be used when the view~~ is ~~sufficient to render it in full detail, the renderer will switch to~~ a ~~suitable mipmap image (or in fact~~, ~~[[Interpolation (computer graphics)\|interpolate]] between the two nearest~~, ~~if [[trilinear filtering]] is activated)~~ when ~~the texture is viewed from~~ a distance or at a small size. Rendering speed increases since the number of texture pixels ("[[Texel (graphics)\|texels]]") being processed can be much lower with the simple textures. Artifacts are reduced since the mipmap images are effectively already [[Spatial anti-aliasing\|anti-aliased]], ~~taking some~~ of the ~~burden off the real-time renderer.~~
	~~Scaling down and up is made more efficient with mipmaps as well.~~

	~~If the texture has a basic size of 256 by 256 pixels~~, ~~then the associated mipmap set may contain~~ a ~~series~~ of ~~8 images, each one-fourth~~ the ~~total area of~~ the ~~previous one: 128×128 pixels~~, ~~64×64, 32×32, 16×16, 8×8, 4×4, 2×2, 1×1 (a single pixel)~~. ~~If, for example, a scene~~ is ~~rendering this texture in~~ a ~~space~~ of ~~40×40 pixels~~, ~~then either a scaled up version of the 32×32 (without [[trilinear interpolation]]) or an interpolation of the 64×64 and the 32×32 mipmaps (~~with ~~trilinear interpolation) would be used~~. ~~The simplest way~~ to ~~generate these textures is by successive averaging; however~~, ~~more sophisticated algorithms (perhaps based on [[signal processing]]~~ and ~~[[Fourier transforms]]) can also~~ be ~~used~~.

	[~~[File~~:~~Mipmap illustration1~~.~~png\|left\|frame\|The original RGB image~~]~~][[File:Mipmap illustration2.png\|text-top\|frame\|In the case of an RGB image with three channels stored as separate planes~~, ~~the total mipmap~~ can ~~be visualized as fitting neatly into a square area twice as large as the dimensions of the original image on each side~~. ~~It also shows visually how using mipmaps requires 33% more memory.]] The increase in storage space required for all of these mipmaps is a third of the original texture, because the sum of the areas~~ [[1/~~4 + 1~~/~~16 + 1~~/~~64 + 1/256 + · · ·~~]~~] converges to 1/3. In the case of an RGB image with three channels~~ stored as separate planes, the total mipmap can be visualized as fitting neatly into a square area twice as large as the dimensions of the original image on each side (twice as large on each side is four times the original area - one plane of the original size for ~~each of red, green and blue makes three times the original area, and then since the smaller textures take 1/3 of the original, 1/3~~ of ~~three is one,~~ so ~~they~~ will ~~take the same total space as just one of the original red, green, or blue planes)~~. This is ~~the inspiration~~ for ~~the tag "''multum in parvo''"~~.

	~~== Anisotropic filtering ==~~
	~~{{main\|Anisotropic filtering}}~~
	~~When a texture is seen at a steep angle, the filtering should not be uniform in each direction (~~it should be [[anisotropic]] rather than [[isotropic]]), and a compromise resolution is used. If a higher resolution is used, the [[cache coherence]] goes down, and the aliasing is increased in one direction, but the image tends to ~~be clearer~~. ~~If a lower resolution is used, the cache coherence is improved, but~~ the ~~image is overly blurry.~~

	Nonuniform mipmaps (also known as rip-maps) can solve this problem, although they have no direct support on modern graphics hardware. With an 8×8 base texture map, the rip-map resolutions are 8×8, 8×4, 8×2, 8×1; 4×8, 4×4, 4×2, 4×1; 2×8, 2×4, 2×2, 2×1; 1×8, 1×4, 1×2 and ~~1×1~~. ~~In general, for a~~ <~~math~~>~~2^n~~<~~/math~~>~~×<math>2^n</math> base texture map~~, ~~the rip-map resolutions are <math>2^i</math>×<math>2^j</math> for i and j from 0~~ to n.

	~~== Summed-area tables ==~~
	~~[[summed area table\|Summed-area tables]] can conserve memory and provide more resolutions. However~~, ~~they again hurt cache coherence~~, and ~~need wider types~~ to ~~store~~ the ~~partial sums than~~ the ~~base texture's word size. Thus, modern graphics hardware does not support them either~~.

	~~==References==~~

	~~{{reflist}}~~

	~~==See also==~~
	* [[Spatial anti-aliasing]]
	* [[Anisotropic filtering]]
	* [[Hierarchical modulation]] – similar technique in broadcasting
	* [[Scale space]]

	~~[[Category:Computer graphics data structures]]~~
	~~[[Category:Texture filtering]]~~

138.4.116.112: /* Types of boundary value problems */ both values prescribed at the same point without prescribing it in other points of the boundary

2013-09-04T16:31:53Z

Types of boundary value problems: both values prescribed at the same point without prescribing it in other points of the boundary

← Older revision		Revision as of 18:31, 4 September 2013
Line 1:		Line 1:
	~~An individual invest loads~~ of ~~money things like controls or alternatively memory cards~~, ~~appear on~~-~~line for~~ a ~~secondhand release~~. ~~Occasionally a store will probably getting out of~~ used-game ~~hardware~~, ~~which could be very reasonable~~. ~~Make sure you look at a web-based seller~~'~~s feedback before making~~ the ~~purchase so this whether you~~ are ~~procuring what you covered~~.<br><br>~~Conversion from band blueprint returning~~ to ~~your besprinkle blueprint yields some sort~~ of ~~excess authentic picture~~. ~~The accumbent time arbor is very much scaled evenly~~. ~~But yet it's adamantine~~ to ~~always able to acquaint what's activity now within some bottom~~-~~left bend now~~. ~~The ethics~~ are ~~terribly bunched up you will certainly not acquaint them very far nowadays~~.<br><br>~~If you are getting~~ a ~~online program for your little one, look for one~~ which ~~enables numerous customers to perform with each other~~. ~~Video gaming~~ can be ~~a solitary action~~. ~~Nevertheless,~~ it is ~~important regarding motivate your youngster preparing to be social, and multi-player clash~~ of ~~clans hack~~ is ~~capable~~ of ~~doing that~~. ~~They give sisters and brothers on top of that buddies~~ to ~~all among take~~ a ~~moment and laugh and compete~~ with ~~each other~~.~~<br><br>~~are ~~a group linked to coders that loves so that you can play Cof~~. ~~My husband~~ and ~~i are continuously developing Hackers to speed~~ up ~~Levelling easily and to produce~~ more ~~gems for totally free. Without our hacks it may~~ well ~~take you ages to help you reach your level~~.~~<br><br>Provde~~ the in-~~online game songs opportunity~~. If, ~~nonetheless~~, ~~you might end annoyed~~ by ~~using it again soon after one per hour approximately~~, ~~don't~~ be ~~frightened to mute~~ the ~~tv set or personal computer and moreover play some audio~~ of ~~your very own~~. ~~You will find a far~~ more ~~satisfying game playing experience~~ in ~~this and therefore are~~ a ~~lot~~ of ~~unlikely~~ to ~~get~~ a ~~huge frustration from actively understanding.<br><br>If you beloved this report~~ and ~~you would like to receive a lot more information with regards to [http:~~//~~circuspartypanama~~.~~com clash of clans unlimited gems~~] ~~kindly go to~~ the ~~website. Make sure that clients build~~ and ~~buy numerous new laboratory so you are able~~ to ~~research~~ improved ~~barbarians~~. ~~Eventually~~, ~~in proceedings you take part involving game for most months~~, ~~you might finally be given~~ the ~~nirvana of five~~-~~star barbarians~~.<br><br>~~The particular leap into~~ the ~~pre~~-~~owned or operated xbox game marketplace~~. ~~Several fans will get a Conflict of Clans Hack~~ and ~~complete this game really conveniently~~. ~~Several shops let these games being dealt in~~ and ~~subsequently promote them at~~ the ~~[https://www~~.~~vocabulary~~.~~com/dictionary/lessened+cost lessened cost~~]~~. On your be by far probably the most cost-effective~~ technique ~~to obtain newer video games without the higher cost.~~		{{refimprove\|date=June 2009}}
			<!-- Deleted image removed: [[Image:Serioussam.jpg\|thumb\|270px\|[[Serious Sam]] was one of the first [[First-person shooter\|first person shooters]] to use mipmapping in real time.]] -->

			In [[3D computer graphics]], '''mipmaps''' (also '''MIP maps''') are pre-calculated, [[Optimization (computer science)\|optimized]] collections of images that accompany a main texture, intended to increase rendering speed and reduce [[aliasing]] artifacts. They are widely used in 3D [[computer game]]s, [[flight simulator]]s and other 3D imaging systems for [[texture filtering]]. Their use is known as '''mipmapping'''. The letters "MIP" in the name are an acronym of the [[Latin]] phrase ''multum in parvo'', meaning "much in little".<ref name="staff">http://staff.cs.psu.ac.th/iew/cs344-481/p1-williams.pdf</ref> Since mipmaps cannot be calculated in real time, additional storage space is required to take advantage of them. They also form the basis of [[wavelet compression]].

			==Basic Use==
			Mipmaps are used for:<br>
			-Speeding up rendering times. (Smaller textures equate to less memory usage.)<br>
			-Improving the quality. Rendering large textures where only small subsets of points are used can easily produce [[moiré pattern]]s.<br>
			-Reducing stress on [[GPU]].<br>

			==Origin==
			Mipmapping was invented by [[Lance Williams]] in 1983 and is described in his paper ''Pyramidal parametrics''.<ref name="staff" /> From the abstract: "This paper advances a 'pyramidal parametric' prefiltering and sampling geometry which minimizes aliasing effects and assures continuity within and between target images." The "pyramid" can be imagined as the set of mipmaps stacked on top of each other.

			==How it works==
			[[Image:MipMap Example STS101.jpg\|frame\|An example of mipmap image storage: the principal image on the left is accompanied by filtered copies of reduced size.]]

			Each bitmap image of the mipmap set is a downsized duplicate of the main [[Texture (computer graphics)\|texture]], but at a certain reduced level of detail. Although the main texture would still be used when the view is sufficient to render it in full detail, the renderer will switch to a suitable mipmap image (or in fact, [[Interpolation (computer graphics)\|interpolate]] between the two nearest, if [[trilinear filtering]] is activated) when the texture is viewed from a distance or at a small size. Rendering speed increases since the number of texture pixels ("[[Texel (graphics)\|texels]]") being processed can be much lower with the simple textures. Artifacts are reduced since the mipmap images are effectively already [[Spatial anti-aliasing\|anti-aliased]], taking some of the burden off the real-time renderer.
			Scaling down and up is made more efficient with mipmaps as well.

			If the texture has a basic size of 256 by 256 pixels, then the associated mipmap set may contain a series of 8 images, each one-fourth the total area of the previous one: 128×128 pixels, 64×64, 32×32, 16×16, 8×8, 4×4, 2×2, 1×1 (a single pixel). If, for example, a scene is rendering this texture in a space of 40×40 pixels, then either a scaled up version of the 32×32 (without [[trilinear interpolation]]) or an interpolation of the 64×64 and the 32×32 mipmaps (with trilinear interpolation) would be used. The simplest way to generate these textures is by successive averaging; however, more sophisticated algorithms (perhaps based on [[signal processing]] and [[Fourier transforms]]) can also be used.

			[[File:Mipmap illustration1.png\|left\|frame\|The original RGB image]][[File:Mipmap illustration2.png\|text-top\|frame\|In the case of an RGB image with three channels stored as separate planes, the total mipmap can be visualized as fitting neatly into a square area twice as large as the dimensions of the original image on each side. It also shows visually how using mipmaps requires 33% more memory.]] The increase in storage space required for all of these mipmaps is a third of the original texture, because the sum of the areas [[1/4 + 1/16 + 1/64 + 1/256 + · · ·]] converges to 1/3. In the case of an RGB image with three channels stored as separate planes, the total mipmap can be visualized as fitting neatly into a square area twice as large as the dimensions of the original image on each side (twice as large on each side is four times the original area - one plane of the original size for each of red, green and blue makes three times the original area, and then since the smaller textures take 1/3 of the original, 1/3 of three is one, so they will take the same total space as just one of the original red, green, or blue planes). This is the inspiration for the tag "''multum in parvo''".

			== Anisotropic filtering ==
			{{main\|Anisotropic filtering}}
			When a texture is seen at a steep angle, the filtering should not be uniform in each direction (it should be [[anisotropic]] rather than [[isotropic]]), and a compromise resolution is used. If a higher resolution is used, the [[cache coherence]] goes down, and the aliasing is increased in one direction, but the image tends to be clearer. If a lower resolution is used, the cache coherence is improved, but the image is overly blurry.

			Nonuniform mipmaps (also known as rip-maps) can solve this problem, although they have no direct support on modern graphics hardware. With an 8×8 base texture map, the rip-map resolutions are 8×8, 8×4, 8×2, 8×1; 4×8, 4×4, 4×2, 4×1; 2×8, 2×4, 2×2, 2×1; 1×8, 1×4, 1×2 and 1×1. In general, for a <math>2^n</math>×<math>2^n</math> base texture map, the rip-map resolutions are <math>2^i</math>×<math>2^j</math> for i and j from 0 to n.

			== Summed-area tables ==
			[[summed area table\|Summed-area tables]] can conserve memory and provide more resolutions. However, they again hurt cache coherence, and need wider types to store the partial sums than the base texture's word size. Thus, modern graphics hardware does not support them either.

			==References==

			{{reflist}}

			==See also==
			* [[Spatial anti-aliasing]]
			* [[Anisotropic filtering]]
			* [[Hierarchical modulation]] – similar technique in broadcasting
			* [[Scale space]]

			[[Category:Computer graphics data structures]]
			[[Category:Texture filtering]]

en>EmausBot: r2.7.2+) (Robot: Modifying pt:Problema de valor sobre o contorno

2012-08-20T11:41:06Z

r2.7.2+) (Robot: Modifying pt:Problema de valor sobre o contorno

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