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Multivariate interpolation

 

 
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Multivariate interpolation
 
 
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In numerical analysis
Numerical analysis

Numerical analysis is the study of algorithms for the problems of continuous mathematics .One of the earliest mathematical writings is the Babylonian tablet YBC 7289, which gives a sexagesimal numerical approximation of , the length of the diagonal in a unit square....
, multivariate interpolation or spatial interpolation is interpolation
Interpolation

In the mathematics subfield of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
 on functions of more than one variable.

The function to be interpolated is known at given points and the interpolation problem consist of yielding values at arbitrary points .

function values known on a prescribed uniform grid
Regular grid

A regular grid is a tessellation of the Euclidean plane by congruent rectangles or a Honeycomb of rectilinear parallelepipeds . Grids of this type appear on graph paper and may be used in finite element analysis as well as finite volume methods and finite difference methods....
, the following methods are available




Bitmap resampling
Resampling

Resampling is the digital process of changing the sample rate or dimensions of digital imagery or audio by temporally or areally analysing and sampling the original data....
 is the application of 2D multivariate interpolation in image processing
Image processing

In electrical engineering and computer science, image processing is any form of signal processing for which the input is an , such as photographs or video frame; the output of image processing can be either an image or a set of characteristics or parameters related to the image....
.

Three of the methods applied on the same dataset, from 16 values located at the black dots.






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In numerical analysis
Numerical analysis

Numerical analysis is the study of algorithms for the problems of continuous mathematics .One of the earliest mathematical writings is the Babylonian tablet YBC 7289, which gives a sexagesimal numerical approximation of , the length of the diagonal in a unit square....
, multivariate interpolation or spatial interpolation is interpolation
Interpolation

In the mathematics subfield of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
 on functions of more than one variable.

The function to be interpolated is known at given points and the interpolation problem consist of yielding values at arbitrary points .

Uniform grid

For function values known on a prescribed uniform grid
Regular grid

A regular grid is a tessellation of the Euclidean plane by congruent rectangles or a Honeycomb of rectilinear parallelepipeds . Grids of this type appear on graph paper and may be used in finite element analysis as well as finite volume methods and finite difference methods....
, the following methods are available

  • Nearest-neighbor interpolation (any dimension)


2 dimensions

  • Bilinear interpolation
    Bilinear interpolation

    In mathematics, bilinear interpolation is an extension of linear interpolation for interpolation functions of two variables on a regular grid. The key idea is to perform linear interpolation first in one direction, and then again in the other direction....
  • Bicubic interpolation
    Bicubic interpolation

    In mathematics, bicubic interpolation is an extension of cubic interpolation for interpolation data points on a two dimensional regular grid. The interpolated surface is Smooth function than corresponding surfaces obtained by bilinear interpolation or nearest-neighbor interpolation....
  • Bézier surface
    Bézier surface

    B?zier surfaces are a species of spline used in computer graphics, computer-aided design, and finite element modelling.As with the B?zier curve, a B?zier surface is defined by a set of control points....
  • Lanczos resampling
    Lanczos resampling

    Lanczos resampling is a multivariate interpolation method used to compute new values for any digitally sampled data. It is often used to resize digital images, but could be used for any other digital signal....


Bitmap resampling
Resampling

Resampling is the digital process of changing the sample rate or dimensions of digital imagery or audio by temporally or areally analysing and sampling the original data....
 is the application of 2D multivariate interpolation in image processing
Image processing

In electrical engineering and computer science, image processing is any form of signal processing for which the input is an , such as photographs or video frame; the output of image processing can be either an image or a set of characteristics or parameters related to the image....
.

Three of the methods applied on the same dataset, from 16 values located at the black dots. The colours represent the interpolated values.

See also Padua points
Padua points

In polynomial interpolation of multivariate interpolation, the Padua points are the first known example of unisolvent_functions point set with minimal growth of their Lebesgue constant , proved to be O...
, for polynomial interpolation
Polynomial interpolation

In the mathematics subfield of numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial. In other words, given some data points , the aim is to find a polynomial which goes exactly through these points....
 in two variables.

3 dimensions

  • Trilinear interpolation
    Trilinear interpolation

    Trilinear interpolation is a method of multivariate interpolation on a Three dimensional space regular grid. It approximates the value of an intermediate point within the local axial rectangular prism linearly, using data on the lattice points....
  • Tricubic interpolation
    Tricubic interpolation

    In the mathematical subfield numerical analysis, tricubic interpolation is a method for obtaining values at arbitrary points in Three-dimensional space of a function defined on a regular grid....
See also bitmap resampling
Resampling

Resampling is the digital process of changing the sample rate or dimensions of digital imagery or audio by temporally or areally analysing and sampling the original data....


Non-uniform grid

Schemes defined on a non-uniform grid
Unstructured grid

An unstructured grid is a tessellation of a part of the Euclidean plane or Euclidean space by simple shapes, such as triangles or tetrahedron, in an irregular pattern....
 should all work on a uniform grid, typically reducing to another known method.
  • Nearest-neighbor interpolation
  • Natural neighbor
    Natural neighbor

    Natural neighbor interpolation is a method of spatial interpolation, developed by Robin Sibson. The method is based on Voronoi diagram of a discrete set of spatial points....
  • Inverse distance weighting
    Inverse distance weighting

    Inverse distance weighting is a method for multivariate interpolation, a process of assigning values to unknown points by using values from usually scattered set of known points....
  • Kriging
    Kriging

    Kriging is a group of geostatistics techniques to interpolation the value of a random field at an unobserved location from observations of its value at nearby locations....