All Topics  
Trilinear interpolation

 

 
   Email Print
   Bookmark   Link
 

 

Trilinear interpolation
 
 
  Topics Trilinear interpolation Topic Home

Trilinear interpolation is a method of multivariate interpolation
Multivariate interpolation

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 and the interpolation problem consist of yielding values at arbitrary points ....
 on a 3-dimensional regular 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....
. It approximates the value of an intermediate point within the local axial rectangular prism
Prism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
 linearly, using data on the lattice points. For an arbitrary, unstructured mesh
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....
 (as used in finite element analysis), other methods of interpolation must be used; if all the mesh elements are tetrahedra
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
 (3D simplices
Simplex

In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of affine transformation Point s in some Euclidean space of dimension n or higher ....
), then barycentric coordinates
Barycentric coordinates (mathematics)

In mathematics, barycentric coordinates are coordinates defined by the vertices of a simplex . Barycentric coordinates are a form of homogeneous coordinates....
 provide a straightforward procedure.

Trilinear interpolation is frequently used 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....
, data analysis
Data analysis

Data analysis is a process of gathering, modeling, and transforming data with the goal of highlighting useful information, suggesting conclusions, and supporting decision making....
, and computer graphics
Computer graphics

Computer graphics are graphics created by computers and, more generally, the representation and manipulation of pictorial data by a computer....
.

Compared to linear and bilinear interpolation
Trilinear interpolation is the extension of linear interpolation
Linear interpolation

Linear interpolation is a method of curve fitting using linear polynomials. It is heavily employed in mathematics , and numerous applications including computer graphics....
, which operates in spaces with dimension
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
 , and 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....
, which operates with dimension , to dimension .






Discussion
Ask a question about 'Trilinear interpolation'
Start a new discussion about 'Trilinear interpolation'
Answer questions from other users
Full Discussion Forum



Encyclopedia


Trilinear interpolation is a method of multivariate interpolation
Multivariate interpolation

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 and the interpolation problem consist of yielding values at arbitrary points ....
 on a 3-dimensional regular 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....
. It approximates the value of an intermediate point within the local axial rectangular prism
Prism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
 linearly, using data on the lattice points. For an arbitrary, unstructured mesh
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....
 (as used in finite element analysis), other methods of interpolation must be used; if all the mesh elements are tetrahedra
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
 (3D simplices
Simplex

In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of affine transformation Point s in some Euclidean space of dimension n or higher ....
), then barycentric coordinates
Barycentric coordinates (mathematics)

In mathematics, barycentric coordinates are coordinates defined by the vertices of a simplex . Barycentric coordinates are a form of homogeneous coordinates....
 provide a straightforward procedure.

Trilinear interpolation is frequently used 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....
, data analysis
Data analysis

Data analysis is a process of gathering, modeling, and transforming data with the goal of highlighting useful information, suggesting conclusions, and supporting decision making....
, and computer graphics
Computer graphics

Computer graphics are graphics created by computers and, more generally, the representation and manipulation of pictorial data by a computer....
.

Compared to linear and bilinear interpolation


Trilinear interpolation is the extension of linear interpolation
Linear interpolation

Linear interpolation is a method of curve fitting using linear polynomials. It is heavily employed in mathematics , and numerous applications including computer graphics....
, which operates in spaces with dimension
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
 , and 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....
, which operates with dimension , to dimension . The order of accuracy is 1 for all these interpolation schemes, and it requires adjacent pre-defined values surrounding the interpolation point. There are several ways to arrive at trilinear interpolation, it is equivalent to 3-dimensional tensor
Tensor

A tensor is an object which extends the notion of Scalar , Vector , and Matrix . The term has slightly different meanings in mathematics and physics....
 B-spline
B-spline

In the mathematics subfield of numerical analysis, a B-spline is a spline function that has minimal Support with respect to a given Degree of a polynomial, Smooth function, and Domain partition....
 interpolation of order 1, and the trilinear interpolation operator is also a tensor product of 3 linear interpolation operators

Method


On a periodic and cubic lattice with spacing 1, let , , and be the differences between each of , , and the smaller coordinate related, that is:

First we interpolate along (imagine we are pushing the front face of the cube to the back), giving:

Then we interpolate these values (along , as we were pushing the top edge to the bottom), giving:

Finally we interpolate these values along (walking through a line):

This gives us a predicted value for the point.

The result of trilinear interpolation is independent of the order of the interpolation steps along the three axes: any other order, for instance along , then along , and finally along , produces the same value.

The above operations can be visualized as follows: First we find the eight corners of a cube that surround our point of interest. These corners have the values C000, C100, C010, C110, C001, C101, C011, C111.

Next, we perform linear interpolation between C000 and C100 to find C00, C001 and C101 to find C01, C011 and C111 to find C11, C010 and C110 to find C10.

Now we do interpolation between C00 and C10 to find C0, C01 and C11 to find C1. Finally, we calculate the value C via linear interpolation of C0 and C1

In practice, a trilinear interpolation is identical to three successive linear interpolation
Linear interpolation

Linear interpolation is a method of curve fitting using linear polynomials. It is heavily employed in mathematics , and numerous applications including computer graphics....
s, or one 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....
s combined with a linear interpolation:

See also

  • Linear interpolation
    Linear interpolation

    Linear interpolation is a method of curve fitting using linear polynomials. It is heavily employed in mathematics , and numerous applications including computer graphics....
  • 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....
  • 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....


External links

  • , describes an interative inverse trilinear interpolation (given the vertices and the value of C find Xd, Yd and Zd).
  • Paul Bourke, , 1999. Contains a very clever and simple method to find trilinear interpolation that is based on binary logic and can be extended to any dimension (Tetralinear, Pentalinear, ...).