Gridded data interpolation
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Description
Use griddedInterpolant to perform interpolation on a 1-D, 2-D, 3-D, or N-D gridded data set. griddedInterpolant returns the interpolant F for the given data set. You can evaluate F at a set of query points, such as (xq,yq) in 2-D, to produce interpolated values vq = F(xq,yq).
Use scatteredInterpolant to perform interpolation with scattered data.
Creation
Syntax
F = griddedInterpolant
F = griddedInterpolant(x,v)
F = griddedInterpolant(X1,X2,...,Xn,V)
F = griddedInterpolant(V)
F = griddedInterpolant(gridVecs,V)
F = griddedInterpolant(___,Method)
F = griddedInterpolant(___,Method,ExtrapolationMethod)
Description
F = griddedInterpolant
example
F = griddedInterpolant(x,v)x and corresponding values v.
example
F = griddedInterpolant(X1,X2,...,Xn,V)n-dimensional arrays X1,X2,...,Xn. The V array contains the sample values associated with the point locations in X1,X2,...,Xn. Each of the arrays X1,X2,...,Xn must be the same size as V.
example
F = griddedInterpolant(V)griddedInterpolant defines the grid as a set of points whose spacing is 1 and range is [1, size(V,i)] in the ith dimension. Use this syntax when you want to conserve memory and are not concerned about the absolute distances between points.
example
F = griddedInterpolant(gridVecs,V)gridVecs that contains n grid vectors to describe an n-dimensional grid of sample points. Use this syntax when you want to use a specific grid and also conserve memory.
example
F = griddedInterpolant(___,Method)'linear', 'nearest', 'next', 'previous', 'pchip', 'cubic', 'makima', or 'spline'. You can specify Method as the last input argument in any of the previous syntaxes.
example
F = griddedInterpolant(___,Method,ExtrapolationMethod)griddedInterpolant uses ExtrapolationMethod to estimate the value when your query points fall outside the domain of your sample points.
Input Arguments
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x — Sample points
vector
Sample points, specified as a vector. x and v must be the same size. The sample points in x must be unique.
Data Types: single | double
v — Sample values
vector | matrix | multidimensional array
Sample values, specified as a vector, matrix, or multidimensional array. The elements of v are the values that correspond to the sample points in x.
To interpolate using a single set of values,
xandvmust be vectors of the same length.To interpolate using multiple sets of values,
vcan be an array with extra dimensions compared tox. The size of the first dimension ofvmust match the number of sample points inx, and each column invdefines a separate set of 1-D values. For example, ifxis a column vector with 10 elements, you can specifyvas a 10-by-4 matrix to interpolate using four different sets of values.
Data Types: single | double
X1, X2, Xn — Sample points in full grid form
arrays
Sample points in full grid form, specified as separate n-dimensional arrays. The sample points must be unique and sorted. You can create the arrays X1,X2,...,Xn using the ndgrid function. These arrays are all the same size, and each one is the same size as V.
Data Types: single | double
gridVecs — Sample points in grid vector form
cell array of grid vectors
Sample points in grid vector form, specified as a cell array of grid vectors {xg1,xg2,...,xgn}. The sample points must be unique and sorted. The vectors must specify a grid that is the same size as V. In other words, size(V) = [length(xg1) length(xg2),...,length(xgn)]. Use this form as an alternative to the full grid to save memory when your grid is very large.
Data Types: single | double
V — Sample values
array
Sample values, specified as an array. The elements of V are the values that correspond to the sample points. The first N dimensions of V must have the same sizes as the corresponding dimensions in the full grid of sample points, where N is the number of dimensions of the grid.
To interpolate using a single set of values, specify
Vas an array with the same size as the full grid of sample points. For example, if the sample points form a grid with size 100-by-100, you can specify the values with a matrix of the same size.To interpolate using multiple sets of values, specify
Vas an array with extra dimensions compared to the grid of sample points. The extra dimensions define multiple values at each sample point. For example, if the sample points form a grid with size 100-by-100, you can specify the values as an array with size 100-by-100-by-4 to interpolate using four different sets of 100-by-100 values.
Data Types: single | double
Method — Interpolation method
'linear' (default) | 'nearest' | 'next' | 'previous' | 'pchip' | 'cubic' | 'spline' | 'makima'
Interpolation method, specified as one of the options in this table.
| Method | Description | Continuity | Comments |
|---|---|---|---|
'linear' (default) | Linear interpolation. The interpolated value at a query point is based on linear interpolation of the values at neighboring grid points in each respective dimension. | C0 |
|
'nearest' | Nearest neighbor interpolation. The interpolated value at a query point is the value at the nearest sample grid point. | Discontinuous |
|
'next' | Next neighbor interpolation (for 1-D only). The interpolated value at a query point is the value at the next sample grid point. | Discontinuous |
|
'previous' | Previous neighbor interpolation (for 1-D only). The interpolated value at a query point is the value at the previous sample grid point. | Discontinuous |
|
'pchip' | Shape-preserving piecewise cubic interpolation (for 1-D only). The interpolated value at a query point is based on a shape-preserving piecewise cubic interpolation of the values at neighboring grid points. | C1 |
|
'cubic' | Cubic interpolation. The interpolated value at a query point is based on a cubic interpolation of the values at neighboring grid points in each respective dimension. The interpolation is based on a cubic convolution. | C1 |
|
'makima' | Modified Akima cubic Hermite interpolation. The interpolated value at a query point is based on a piecewise function of polynomials with degree at most three evaluated using the values of neighboring grid points in each respective dimension. The Akima formula is modified to avoid overshoots. | C1 |
|
'spline' | Cubic spline interpolation. The interpolated value at a query point is based on a cubic interpolation of the values at neighboring grid points in each respective dimension. The interpolation is based on a cubic spline using not-a-knot end conditions. | C2 |
|
ExtrapolationMethod — Extrapolation method
'linear' (default) | 'nearest' | 'next' | 'previous' | 'pchip' | 'cubic' | 'spline' | 'makima' | 'none'
Extrapolation method, specified as 'linear', 'nearest', 'next', 'previous', 'pchip', 'cubic', 'spline', or 'makima'. In addition, you can specify 'none' if you want queries outside the domain of your grid to return NaN values.
If you omit ExtrapolationMethod, the default is the value you specify for Method. If you omit both the Method and ExtrapolationMethod arguments, both values default to 'linear'.
Properties
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GridVectors — Grid vectors
cell array
Grid vectors, specified as a cell array {xg1,xg2,...,xgn}. These vectors specify the grid points (locations) for the values in Values. The grid points must be unique.
Data Types: cell
Values — Function values at sample points
array
Function values at sample points, specified as an array of values associated with the grid points in GridVectors.
Data Types: single | double
Method — Interpolation method
'linear' (default) | 'nearest' | 'next' | 'previous' | 'pchip' | 'cubic' | 'spline' | 'makima'
Interpolation method, specified as a character vector. Method can be: 'linear', 'nearest', 'next', 'previous', 'pchip', 'cubic', 'spline', or 'makima'. See Method for descriptions of these methods.
Data Types: char
ExtrapolationMethod — Extrapolation method
'linear' | 'nearest' | 'next' | 'previous' | 'pchip' | 'cubic' | 'spline' | 'makima' | 'none'
Extrapolation method, specified as a character vector. ExtrapolationMethod can be: 'linear', 'nearest', 'next', 'previous', 'pchip', 'cubic', 'spline', 'makima', or 'none'. A value of 'none' indicates that extrapolation is disabled. The default value is the value of Method.
Data Types: char
Usage
Syntax
Vq = F(Xq)
Vq = F(xq1,xq2,...,xqn)
Vq = F(Xq1,Xq2,...,Xqn)
Vq = F({xgq1,xgq2,...,xgqn})
Description
Use griddedInterpolant to create the interpolant, F. Then you can evaluate F at specific query points using any of the following syntaxes:
Vq = F(Xq) specifies the query points in the matrix Xq. Each row of Xq contains the coordinates of a query point.
Vq = F(xq1,xq2,...,xqn) specifies the query points xq1,xq2,...,xqn as column vectors of length m, representing m points scattered in n-dimensional space.
Vq = F(Xq1,Xq2,...,Xqn) specifies the query points using the n-dimensional arrays Xq1,Xq2,...,Xqn, which define a full grid of points.
Vq = F({xgq1,xgq2,...,xgqn}) specifies the query points as grid vectors. Use this syntax to conserve memory when you want to query a large grid of points.
Examples
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1-D Interpolation
Open Live Script
Use griddedInterpolant to interpolate a 1-D data set.
Create a vector of scattered sample points v. The points are sampled at random 1-D locations between 0 and 20.
x = sort(20*rand(100,1));v = besselj(0,x);
Create a gridded interpolant object for the data. By default, griddedInterpolant uses the 'linear' interpolation method.
F = griddedInterpolant(x,v)
F = griddedInterpolant with properties: GridVectors: {[100x1 double]} Values: [100x1 double] Method: 'linear' ExtrapolationMethod: 'linear'Query the interpolant F at 500 uniformly spaced points between 0 and 20. Plot the interpolated results (xq,vq) on top of the original data (x,v).
xq = linspace(0,20,500);vq = F(xq);plot(x,v,'ro')hold onplot(xq,vq,'.')legend('Sample Points','Interpolated Values')
3-D Interpolation Using Full Grid vs. Grid Vectors
Open Live Script
Interpolate 3-D data using two methods to specify the query points.
Create and plot a 3-D data set representing the function evaluated at a set of gridded sample points in the range [-5,5].
[x,y] = ndgrid(-5:0.8:5);z = sin(x.^2 + y.^2) ./ (x.^2 + y.^2);surf(x,y,z)
Create a gridded interpolant object for the data.
F = griddedInterpolant(x,y,z);
Use a finer mesh to query the interpolant and improve the resolution.
[xq,yq] = ndgrid(-5:0.1:5);vq = F(xq,yq);surf(xq,yq,vq)
In cases where there are a lot of sample points or query points, and where memory usage becomes a concern, you can use grid vectors to improve memory usage.
When you specify grid vectors instead of using
ndgridto create the full grid,griddedInterpolantavoids forming the full query grid to carry out the calculations.When you pass grid vectors, they are normally grouped together as cells in a cell array,
{xg1, xg2, ..., xgn}. The grid vectors are a compact way to represent the points of the full grid.
Alternatively, execute the previous commands using grid vectors.
x = -5:0.8:5;y = x';z = sin(x.^2 + y.^2) ./ (x.^2 + y.^2);F = griddedInterpolant({x,y},z);xq = -5:0.1:5;yq = xq';vq = F({xq,yq});surf(xq,yq,vq)
Interpolation with Default Grid
Open Live Script
Use the default grid to perform a quick interpolation on a set of sample points. The default grid uses unit-spaced points, so this interpolation is useful when the exact xy spacing between the sample points is not important.
Create a matrix of sample function values and plot them against the default grid.
x = (1:0.3:5)';y = x';V = cos(x) .* sin(y);n = length(x);surf(1:n,1:n,V)
Interpolate the data using the default grid.
F = griddedInterpolant(V)
F = griddedInterpolant with properties: GridVectors: {[1 2 3 4 5 6 7 8 9 10 11 12 13 14] [1 2 3 4 5 6 7 8 9 10 11 12 13 14]} Values: [14x14 double] Method: 'linear' ExtrapolationMethod: 'linear'Query the interpolant and plot the results.
[xq,yq] = ndgrid(1:0.2:n);Vq = F(xq,yq);surf(xq',yq',Vq)
2-D Interpolation over Finer Grid
Open Live Script
Interpolate coarsely sampled data using a full grid with spacing of 0.5.
Define the sample points as a full grid with range [1, 10] in both dimensions.
[X,Y] = ndgrid(1:10,1:10);
Sample at the grid points.
V = X.^2 + Y.^2;
Create the interpolant, specifying cubic interpolation.
F = griddedInterpolant(X,Y,V,'cubic');Define a full grid of query points with 0.5 spacing and evaluate the interpolant at those points. Then plot the result.
[Xq,Yq] = ndgrid(1:0.5:10,1:0.5:10);Vq = F(Xq,Yq);mesh(Xq,Yq,Vq);
1-D Extrapolation
Open Live Script
Compare results of querying the interpolant outside the domain of F using the 'pchip' and 'nearest' extrapolation methods.
Create the interpolant, specifying 'pchip' as the interpolation method and 'nearest' as the extrapolation method.
x = [1 2 3 4 5];v = [12 16 31 10 6];F = griddedInterpolant(x,v,'pchip','nearest')
F = griddedInterpolant with properties: GridVectors: {[1 2 3 4 5]} Values: [12 16 31 10 6] Method: 'pchip' ExtrapolationMethod: 'nearest'Query the interpolant, and include points outside the domain of F.
xq = 0:0.1:6;vq = F(xq);figureplot(x,v,'o',xq,vq,'-b');legend ('v','vq')
Query the interpolant at the same points again, this time using the 'pchip' extrapolation method.
F.ExtrapolationMethod = 'pchip';figurevq = F(xq);plot(x,v,'o',xq,vq,'-b');legend ('v','vq')
Interpolate Multiple Sets of Values on Same Grid
Open Live Script
Use griddedInterpolant to interpolate three different sets of values at the same query points.
Create a grid of sample points with and .
gx = -5:5;gy = -3:3;[X,Y] = ndgrid(gx,gy);
Evaluate three different functions at the query points, and then concatenate the values into a 3-D array. The size of V is the same as the X and Y grids in the first two dimensions, but the size of the extra dimension reflects the number of values associated with each sample point (in this case, three).
f1 = X.^2 + Y.^2;f2 = X.^3 + Y.^3;f3 = X.^4 + Y.^4;V = cat(3,f1,f2,f3);
Create an interpolant using the sample points and associated values.
F = griddedInterpolant(X,Y,V);
Create a grid of query points with a finer mesh size compared to the sample points.
qx = -5:0.4:5;qy = -3:0.4:3;[XQ,YQ] = ndgrid(qx,qy);
Interpolate all three sets of values at the query points.
VQ = F(XQ,YQ);
Compare the original data with the interpolated results.
tiledlayout(3,2)nexttilesurf(X,Y,f1)title('f1')nexttilesurf(XQ,YQ,VQ(:,:,1))title('Interpolated f1')nexttilesurf(X,Y,f2)title('f2')nexttilesurf(XQ,YQ,VQ(:,:,2))title('Interpolated f2')nexttilesurf(X,Y,f3)title('f3')nexttilesurf(XQ,YQ,VQ(:,:,3))title('Interpolated f3')
More About
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Interpolant
Interpolating function that you can evaluate at query points.
Gridded Data
A set of points that are axis-aligned and ordered.
Scattered Data
A set of points that have no structure among their relative locations.
Full Grid
A grid represented as a set of arrays. For example, you can create a full grid using ndgrid.
Grid Vectors
A set of vectors that serve as a compact representation of a grid in ndgrid format.
For example, [X,Y] = ndgrid(xg,yg) returns a full grid in the matrices X and Y. You can represent the same grid using the grid vectors xg and yg.
Tips
It is quicker to evaluate a
griddedInterpolantobjectFat many different sets of query points than it is to compute the interpolations separately usinginterp1,interp2,interp3, orinterpn. For example:% Fast to create interpolant F and evaluate multiple timesF = griddedInterpolant(X1,X2,V)v1 = F(Xq1)v2 = F(Xq2)% Slower to compute interpolations separately using interp2v1 = interp2(X1,X2,V,Xq1)v2 = interp2(X1,X2,V,Xq2)
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
Dimension of the query points and dimensions of the input gridded data must be compile time constant.
MethodandExtrapolationMethodmust be constants (known values when code is generated).Code generation does not support the makima function for 2-D, 3-D, or N-D interpolations.
Empty interpolant objects are not supported for code generation.
If the
Valuesproperty ofgriddedInterpolantis variable-size, is not a variable-length vector, and becomes a row vector at run time, then an error occurs.If the sample values or query points contain
Infor-Inf, the output of the generated code might not match the output in MATLAB®.
Thread-Based Environment
Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.
This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
Usage notes and limitations:
For 1-D interpolants, supported interpolation methods are
'cubic','linear','nearest','next','previous', and'spline'.For 2-D interpolants, supported interpolation methods are
'cubic','linear', and'nearest'.For 3-D and higher dimensional interpolants, supported interpolation methods are
'linear'and'nearest'.6-D or higher dimensional interpolants are not supported.
Interpolation method
'spline'is not supported when sample values argumentVcontainsNaNvalues.Extrapolation method
'spline'is only supported when using the'spline'interpolation method.Extrapolation method
'cubic'is only supported when using'cubic'or'spline'interpolation methods.
For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Version History
Introduced in R2011b
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R2021a: Interpolate multiple data sets simultaneously
Support added to interpolate multiple data sets on the same grid at the same query points. For example, if you specify a 2-D grid, a 3-D array of values at the grid points, and a 2-D collection of query points, then griddedInterpolant returns the interpolated values at the query points for each 2-D page in the 3-D array of values.
Previously, this functionality was available in interp1 for 1-D interpolation, but this improvement to griddedInterpolant adds support for N-D multivalued interpolation.
See Also
scatteredInterpolant | interp1 | interp2 | interp3 | interpn | ndgrid | meshgrid | fillmissing | filloutliers
Topics
- Resample Image with Gridded Interpolation
- Interpolating Gridded Data
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