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en>Monkbot: /* References */Task 5: Fix CS1 deprecated coauthor parameter errors

2014-07-24T00:26:37Z

References: Task 5: Fix CS1 deprecated coauthor parameter errors

← Older revision		Revision as of 02:26, 24 July 2014
Line 1:		Line 1:
	~~In [[numerical analysis]]~~, ~~'''multivariate interpolation''' or '''spatial interpolation'''~~ is ~~[[interpolation]] on functions of more than one variable~~.		Hi there. Let me start by introducing the writer, her name is Sophia Boon but she never truly liked that title. Some time in the past she chose to reside in Alaska and her mothers and fathers reside close by. For many years she's been operating as a journey agent. To climb is some thing I really enjoy doing.<br><br>Take a look at my weblog: [https://www.machlitim.org.il/subdomain/megila/end/node/12300 real psychic readings]

	~~The function~~ to ~~be interpolated is known at given points <math>(x_i, y_i, z_i, \dots)</math>~~ and ~~the interpolation problem consist of yielding values at arbitrary points <math>(x,y,z,\dots)</math>~~.

	~~==Regular grid==~~
	For ~~function values known on~~ a ~~[[regular grid]] (having predetermined, not necessarily uniform, spacing), the following methods are available~~.

	~~===Any dimension===~~
	* [[Nearest-neighbor interpolation]]

	~~===2 dimensions===~~
	* [[Barnes interpolation]]
	* [[Bilinear interpolation]]
	* [[Bicubic interpolation]]
	* [[Bézier surface]]
	* [[Lanczos resampling]]
	* [[Delaunay triangulation]]
	* [[Inverse distance weighting]]
	* [[Kriging]]
	* [[Natural neighbor]]
	* [[Spline interpolation]]

	~~[[Resampling (bitmap)\|Bitmap resampling]]~~ is ~~the application of 2D multivariate interpolation in [[image processing]].~~

	~~Three of the methods applied on the same dataset, from 16 values located at the black dots. The colours represent the interpolated values.~~
	~~<gallery>~~
	~~Image:Nearest2DInterpolExample.png\|Nearest neighbor~~
	~~Image:BilinearInterpolExample.png\|Bilinear~~
	~~Image:BicubicInterpolationExample.png\|Bicubic~~
	~~</gallery>~~

	~~See also [[Padua points]], for [[polynomial interpolation]] in two variables.~~

	~~===3 dimensions===~~
	* [[Trilinear interpolation]]
	* [[Tricubic interpolation]]

	~~See also [[Resampling (bitmap)\|bitmap resampling]].~~

	~~===Tensor product splines for ''N'' dimensions===~~

	~~Catmull-Rom splines can be easily generalized to any number of dimensions~~.
	~~The [[cubic Hermite spline]] article will remind you that <math>\mathrm{CINT}_x(f_{-1}, f_0, f_1, f_2) = \mathbf{b}(x) \cdot \left( f_{-1} f_0 f_1 f_2 \right)</math> for some 4-vector~~ <~~math~~>~~\mathbf{b}(x)~~<~~/math~~> ~~which is~~ a ~~function of ''x'' alone, where <math>f_j</math> is the value~~ at ~~<math>j</math> of the function to be interpolated.~~
	~~Rewrite this approximation as~~
	:~~<math>~~
	~~\mathrm{CR}(x) = \sum_{i=-1}^2 f_i b_i(x)~~
	~~</math>~~
	~~This formula can be directly generalized to N dimensions:<ref>~~[~~http~~://~~arxiv~~.org~~/abs/0905~~.~~3564 Two hierarchies of spline interpolations. Practical algorithms for multivariate higher order splines]<~~/~~ref>~~
	~~:<math>~~
	~~\mathrm{CR}(x_1,\dots,x_N) = \sum_{i_1,\dots,i_N=-1}^2 f_{i_1\dots i_N} \prod_{j=1}^N b_{i_j}(x_j)~~
	~~</math>~~
	~~Note that similar generalizations can be made for other types of spline interpolations, including Hermite splines.~~
	In regards to efficiency, the general formula can in fact be computed as a composition of successive <math>\mathrm{CINT}</math>-type operations for any type of tensor product splines, as explained in the [[tricubic interpolation]] article.
	~~However, the fact remains that if there are <math>n<~~/~~math> terms in the 1-dimensional <math>\mathrm{CR}<~~/~~math>-like summation, then there will be <math>n^N<~~/~~math> terms in the <math>N<~~/~~math>-dimensional summation.~~

	~~== Irregular grid (scattered data) ==~~
	~~Schemes defined for scattered data on an [[irregular grid]] should all work on a regular grid, typically reducing to another known method.~~
	* [[Nearest-neighbor interpolation]]
	* [[Triangulated irregular network]]-based [[natural neighbor]]
	* [[Triangulated irregular network]]-based [[linear interpolation]] (a type of [[piecewise linear function]])
	* [[Inverse distance weighting]]
	* [[Kriging]]
	* [[Radial basis function]]
	* [[Thin plate spline]]
	* [[Polyharmonic spline]] (the thin-plate-spline is a special case of a polyharmonic spline)
	* Least-squares [[spline (mathematics)\|spline]]

	~~==Notes==~~
	~~<references />~~

	~~==External links==~~
	* [http://chichi.lalescu.ro/splines.html Example C++ code for several 1D, 2D and 3D spline interpolations (including Catmull-Rom splines).]
	* [http://web.archive.org/web/20060915111500/http://www.ices.utexas.edu/CVC/papers/multidim.pdf Multi-dimensional Hermite Interpolation and Approximation], Prof. Chandrajit Bajaja, [[Purdue University]]

	~~[[Category:Interpolation]]~~
	~~[[Category:Multivariate interpolation\| ]~~]

93.232.197.108: /* Properties */ V-ring links to something complete different

2013-10-07T21:59:30Z

Properties: V-ring links to something complete different

← Older revision		Revision as of 23:59, 7 October 2013
Line 1:		Line 1:
	~~Friends contact him Royal~~. ~~Delaware has usually been my residing location~~ and ~~will by no means move~~. The ~~occupation I~~'~~ve been occupying~~ for ~~many years~~ is a ~~bookkeeper but I~~'~~ve already applied~~ for ~~an additional one~~. ~~One~~ of the ~~issues I adore most is greeting card collecting but I don't have~~ the ~~time lately.~~<br><br>~~Have~~ a ~~look at my web page extended auto warranty~~ - [http://~~Nauri~~.~~biz~~/xe/~~test~~/~~937675 listen to this podcast~~],		In [[numerical analysis]], '''multivariate interpolation''' or '''spatial interpolation''' is [[interpolation]] on functions of more than one variable.

			The function to be interpolated is known at given points <math>(x_i, y_i, z_i, \dots)</math> and the interpolation problem consist of yielding values at arbitrary points <math>(x,y,z,\dots)</math>.

			==Regular grid==
			For function values known on a [[regular grid]] (having predetermined, not necessarily uniform, spacing), the following methods are available.

			===Any dimension===
			* [[Nearest-neighbor interpolation]]

			===2 dimensions===
			* [[Barnes interpolation]]
			* [[Bilinear interpolation]]
			* [[Bicubic interpolation]]
			* [[Bézier surface]]
			* [[Lanczos resampling]]
			* [[Delaunay triangulation]]
			* [[Inverse distance weighting]]
			* [[Kriging]]
			* [[Natural neighbor]]
			* [[Spline interpolation]]

			[[Resampling (bitmap)\|Bitmap resampling]] is the application of 2D multivariate interpolation in [[image processing]].

			Three of the methods applied on the same dataset, from 16 values located at the black dots. The colours represent the interpolated values.
			<gallery>
			Image:Nearest2DInterpolExample.png\|Nearest neighbor
			Image:BilinearInterpolExample.png\|Bilinear
			Image:BicubicInterpolationExample.png\|Bicubic
			</gallery>

			See also [[Padua points]], for [[polynomial interpolation]] in two variables.

			===3 dimensions===
			* [[Trilinear interpolation]]
			* [[Tricubic interpolation]]

			See also [[Resampling (bitmap)\|bitmap resampling]].

			===Tensor product splines for ''N'' dimensions===

			Catmull-Rom splines can be easily generalized to any number of dimensions.
			The [[cubic Hermite spline]] article will remind you that <math>\mathrm{CINT}_x(f_{-1}, f_0, f_1, f_2) = \mathbf{b}(x) \cdot \left( f_{-1} f_0 f_1 f_2 \right)</math> for some 4-vector <math>\mathbf{b}(x)</math> which is a function of ''x'' alone, where <math>f_j</math> is the value at <math>j</math> of the function to be interpolated.
			Rewrite this approximation as
			:<math>
			\mathrm{CR}(x) = \sum_{i=-1}^2 f_i b_i(x)
			</math>
			This formula can be directly generalized to N dimensions:<ref>[http://arxiv.org/abs/0905.3564 Two hierarchies of spline interpolations. Practical algorithms for multivariate higher order splines]</ref>
			:<math>
			\mathrm{CR}(x_1,\dots,x_N) = \sum_{i_1,\dots,i_N=-1}^2 f_{i_1\dots i_N} \prod_{j=1}^N b_{i_j}(x_j)
			</math>
			Note that similar generalizations can be made for other types of spline interpolations, including Hermite splines.
			In regards to efficiency, the general formula can in fact be computed as a composition of successive <math>\mathrm{CINT}</math>-type operations for any type of tensor product splines, as explained in the [[tricubic interpolation]] article.
			However, the fact remains that if there are <math>n</math> terms in the 1-dimensional <math>\mathrm{CR}</math>-like summation, then there will be <math>n^N</math> terms in the <math>N</math>-dimensional summation.

			== Irregular grid (scattered data) ==
			Schemes defined for scattered data on an [[irregular grid]] should all work on a regular grid, typically reducing to another known method.
			* [[Nearest-neighbor interpolation]]
			* [[Triangulated irregular network]]-based [[natural neighbor]]
			* [[Triangulated irregular network]]-based [[linear interpolation]] (a type of [[piecewise linear function]])
			* [[Inverse distance weighting]]
			* [[Kriging]]
			* [[Radial basis function]]
			* [[Thin plate spline]]
			* [[Polyharmonic spline]] (the thin-plate-spline is a special case of a polyharmonic spline)
			* Least-squares [[spline (mathematics)\|spline]]

			==Notes==
			<references />

			==External links==
			* [http://chichi.lalescu.ro/splines.html Example C++ code for several 1D, 2D and 3D spline interpolations (including Catmull-Rom splines).]
			* [http://web.archive.org/web/20060915111500/http://www.ices.utexas.edu/CVC/papers/multidim.pdf Multi-dimensional Hermite Interpolation and Approximation], Prof. Chandrajit Bajaja, [[Purdue University]]

			[[Category:Interpolation]]
			[[Category:Multivariate interpolation\| ]]

en>Qetuth: more specific stub type

2012-01-01T08:00:08Z

more specific stub type

New page

Friends contact him Royal. Delaware has usually been my residing location and will by no means move. The occupation I've been occupying for many years is a bookkeeper but I've already applied for an additional one. One of the issues I adore most is greeting card collecting but I don't have the time lately.<br><br>Have a look at my web page extended auto warranty - [http://Nauri.biz/xe/test/937675 listen to this podcast],