Interpolation for 2-D gridded data in meshgrid format
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Syntax
Vq = interp2(X,Y,V,Xq,Yq)
Vq = interp2(V,Xq,Yq)
Vq = interp2(V)
Vq = interp2(V,k)
Vq = interp2(___,method)
Vq = interp2(___,method,extrapval)
Description
example
Vq = interp2(X,Y,V,Xq,Yq) returnsinterpolated values of a function of two variables at specific querypoints using linear interpolation. The results always pass throughthe original sampling of the function. X and Y containthe coordinates of the sample points. V containsthe corresponding function values at each sample point. Xq and Yq containthe coordinates of the query points.
Vq = interp2(V,Xq,Yq) assumes a default grid of sample points. The default grid points cover the rectangular region, X=1:n and Y=1:m, where [m,n] = size(V). Use this syntax when you want to conserve memory and are not concerned about the absolute distances between points.
Vq = interp2(V) returnsthe interpolated values on a refined grid formed by dividing the intervalbetween sample values once in each dimension.
example
Vq = interp2(V,k) returns the interpolated values on a refined grid formed by repeatedlyhalving the intervals k times in each dimension.This results in 2^k-1 interpolated points betweensample values.
example
Vq = interp2(___,method) specifies an alternative interpolation method: 'linear', 'nearest', 'cubic', 'makima', or 'spline'. The default method is 'linear'.
example
Vq = interp2(___,method,extrapval) alsospecifies extrapval, a scalar value that is assignedto all queries that lie outside the domain of the sample points.
If you omit the extrapval argument for queriesoutside the domain of the sample points, then based on the method argument interp2 returnsone of the following:
Extrapolated values for the
'spline'and'makima'methodsNaNvalues for other interpolation methods
Examples
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Interpolate over a Grid Using Default Method
Open Live Script
Coarsely sample the peaks function.
[X,Y] = meshgrid(-3:3);V = peaks(X,Y);
Plot the coarse sampling.
figuresurf(X,Y,V)title('Original Sampling');
Create the query grid with spacing of 0.25.
[Xq,Yq] = meshgrid(-3:0.25:3);
Interpolate at the query points.
Vq = interp2(X,Y,V,Xq,Yq);
Plot the result.
figuresurf(Xq,Yq,Vq);title('Linear Interpolation Using Finer Grid');
Interpolate over a Grid Using Cubic Method
Open Live Script
Coarsely sample the peaks function.
[X,Y] = meshgrid(-3:3);V = peaks(7);
Plot the coarse sampling.
figuresurf(X,Y,V)title('Original Sampling');
Create the query grid with spacing of 0.25.
[Xq,Yq] = meshgrid(-3:0.25:3);
Interpolate at the query points, and specify cubic interpolation.
Vq = interp2(X,Y,V,Xq,Yq,'cubic');Plot the result.
figuresurf(Xq,Yq,Vq);title('Cubic Interpolation Over Finer Grid');
Refine Grayscale Image
Open Live Script
Load some image data into the workspace.
load flujet.matcolormap gray
Isolate a small region of the image and cast it to single-precision.
V = single(X(200:300,1:25));
Display the image region.
imagesc(V);axis offtitle('Original Image')
Insert interpolated values by repeatedly dividing the intervals between points of the refined grid five times in each dimension.
Vq = interp2(V,5);
Display the result.
imagesc(Vq);axis offtitle('Linear Interpolation')
Evaluate Outside the Domain of X and Y
Open Live Script
Coarsely sample a function over the range, [-2, 2] in both dimensions.
[X,Y] = meshgrid(-2:0.75:2);R = sqrt(X.^2 + Y.^2)+ eps;V = sin(R)./(R);
Plot the coarse sampling.
figuresurf(X,Y,V)xlim([-4 4])ylim([-4 4])title('Original Sampling')
Create the query grid that extends beyond the domain of X and Y.
[Xq,Yq] = meshgrid(-3:0.2:3);
Perform cubic interpolation within the domain of X and Y, and assign all queries that fall outside to zero.
Vq = interp2(X,Y,V,Xq,Yq,'cubic',0);Plot the result.
figuresurf(Xq,Yq,Vq)title('Cubic Interpolation with Vq=0 Outside Domain of X and Y');
Input Arguments
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X,Y — Sample grid points
matrices | vectors
Sample grid points, specified as real matrices or vectors. Thesample grid points must be unique.
If
XandYarematrices, then they contain the coordinates of a full grid (in meshgrid format).Use the meshgrid function tocreate theXandYmatricestogether. Both matrices must be the same size.If
XandYare vectors, then they are treated as grid vectors. The values in both vectors must be strictly monotonic, either increasing or decreasing.
Example: [X,Y] = meshgrid(1:30,-10:10)
Data Types: single | double
V — Sample values
matrix
Sample values, specified as a real or complex matrix. The sizerequirements for V depend on the size of X and Y:
If
XandYarematrices representing a full grid (inmeshgridformat),thenVmust be the same size asXandY.If
XandYaregrid vectors, thenVmust be a matrix containinglength(Y)rowsandlength(X)columns.
If V contains complex numbers, then interp2 interpolatesthe real and imaginary parts separately.
Example: rand(10,10)
Data Types: single | double
Complex Number Support: Yes
Xq,Yq — Query points
scalars | vectors | matrices | arrays
Query points, specified as a real scalars, vectors, matrices,or arrays.
If
XqandYqarescalars, then they are the coordinates of a single query point.If
XqandYqarevectors of different orientations, thenXqandYqaretreated as grid vectors.If
XqandYqarevectors of the same size and orientation, thenXqandYqaretreated as scatteredpoints in 2-D space.If
XqandYqarematrices, then they represent either a full grid of query points (inmeshgridformat)or scattered points.If
XqandYqareN-D arrays, then they represent scattered points in 2-D space.
Example: [Xq,Yq] = meshgrid((1:0.1:10),(-5:0.1:0))
Data Types: single | double
k — Refinement factor
1 (default) | real, nonnegative, integer scalar
Refinement factor, specified as a real, nonnegative, integerscalar. This value specifies the number of times to repeatedly dividethe intervals of the refined grid in each dimension. This resultsin 2^k-1 interpolated points between sample values.
If k is 0, then Vq isthe same as V.
interp2(V,1) is the same as interp2(V).
The following illustration shows the placement of interpolatedvalues (in red) among nine sample values (in black) for k=2.
Example: interp2(V,2)
Data Types: single | double
method — Interpolation method
'linear' (default) | 'nearest' | 'cubic' | 'spline' | 'makima'
Interpolation method, specified as one of the options in this table.
| Method | Description | Continuity | Comments |
|---|---|---|---|
'linear' | The interpolated value at a query point is based on linear interpolation of the values at neighboring grid points in each respective dimension. This is the default interpolation method. | C0 |
|
'nearest' | The interpolated value at a query point is the value at the nearest sample grid point. | Discontinuous |
|
'cubic' | 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' | 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 |
|
extrapval — Function value outside domain of X and Y
scalar
Function value outside domain of X and Y,specified as a real or complex scalar. interp2 returnsthis constant value for all points outside the domain of X and Y.
Example: 5
Example: 5+1i
Data Types: single | double
Complex Number Support: Yes
Output Arguments
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Vq — Interpolated values
scalar | vector | matrix
Interpolated values, returned as a real or complex scalar, vector,or matrix. The size and shape of Vq depends onthe syntax you use and, in some cases, the size and value of the inputarguments.
| Syntaxes | SpecialConditions | Size of Vq | Example |
|---|---|---|---|
interp2(X,Y,V,Xq,Yq)interp2(V,Xq,Yq)and variations of these syntaxes that include method or extrapval | Xq, Yq are scalars | Scalar | size(Vq) = [1 1] when you pass Xq and Yq asscalars. |
| Same as above | Xq, Yq are vectors ofthe same size and orientation | Vector of same size and orientation as Xq and Yq | If size(Xq) = [100 1]and size(Yq)= [100 1], then size(Vq) = [1001]. |
| Same as above | Xq, Yq are vectors ofmixed orientation | Matrix in which the number of rows is length(Yq),and the number of columns is length(Xq) | If size(Xq) = [1 100]and size(Yq)= [50 1], then size(Vq) = [50 100]. |
| Same as above | Xq, Yq are matrices orarrays of the same size | Matrix or array of the same size as Xq and Yq | If size(Xq) = [50 25]and size(Yq)= [50 25], then size(Vq) = [5025]. |
interp2(V,k)and variationsof this syntax that include method or extrapval | None | Matrix in which the number of rows is: and thenumber of columns is: | If size(V) = [10 20]and k= 2, then size(Vq) = [37 77]. |
More About
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Strictly Monotonic
A set of values that are always increasingor decreasing, without reversals. For example, the sequence, a= [2 4 6 8] is strictly monotonic and increasing. The sequence, b= [2 4 4 6 8] is not strictly monotonic because there isno change in value between b(2) and b(3).The sequence, c = [2 4 6 8 6] contains a reversalbetween c(4) and c(5), so itis not monotonic at all.
Full Grid (in meshgrid Format)
For interp2, the fullgrid is a pair of matrices whose elements represent a grid of pointsover a rectangular region. One matrix contains the x-coordinates,and the other matrix contains the y-coordinates.The values in the x-matrix are strictly monotonic and increasingalong the rows. The values along its columns are constant. The valuesin the y-matrix are strictly monotonic and increasingalong the columns. The values along its rows are constant. Use the meshgrid function to create a full gridthat you can pass to interp2.
For example, the following code creates a full grid for theregion, –1 ≤ x ≤ 3 and 1 ≤ y ≤4:
[X,Y] = meshgrid(-1:3,(1:4))
X = -1 0 1 2 3 -1 0 1 2 3 -1 0 1 2 3 -1 0 1 2 3Y = 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4
Grid vectors are a more compact format to represent a grid than the full grid. The relation between the two formats and the matrix of sample values V is
Grid Vectors
For interp2, grid vectors consist of a pair of vectors that define the x- and y-coordinates in a grid. The row vector defines x-coordinates, and the column vector defines y-coordinates.
For example, the following code creates the grid vectors that specify the region, –1 ≤ x ≤ 3 and 1 ≤ y ≤ 4:
x = -1:3;y = (1:4)';
Scattered Points
For interp2, scatteredpoints consist of a pair of arrays that define a collection of pointsscattered in 2-D space. One array contains the x-coordinates,and the other contains the y-coordinates.
For example, the following code specifies the points, (2,7),(5,3), (4,1), and (10,9):
x = [2 5; 4 10];y = [7 3; 1 9];
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
XqandYqmustbe the same size. Usemeshgridto evaluate ona grid.For best results, provide
XandYas vectors. The values in these vectors must be strictly monotonic and increasing.Code generation does not support the
'makima'interpolation method.For the
'cubic'interpolation method, if the grid does not have uniform spacing, an error results. In this case, use the'spline'interpolation method.For best results when you use the
'spline'interpolation method:Use
meshgridto create the inputsXqandYq.Use a small number of interpolation points relative to the dimensions of
V. Interpolating over a large set of scattered points can be inefficient.
GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.
Usage notes and limitations:
XqandYqmust be the same size. Usemeshgridto evaluate on a grid.For best results, provide
XandYas vectors. The values in these vectors must be strictly monotonic and increasing.Code generation does not support the
'makima'interpolation method.For the
'cubic'interpolation method, if the grid does not have uniform spacing, an error results. In this case, use the'spline'interpolation method.For best results when you use the
'spline'interpolation method:Use
meshgridto create the inputsXqandYq.Use a small number of interpolation points relative to the dimensions of
V. Interpolating over a large set of scattered points can be inefficient.
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:
Vmust be a double or single 2-D array.Vcan be real or complex.Vcannot be a vector.XandYmust:Have the same type (double or single).
Be finite vectors or 2-D arrays with increasing and nonrepeating elements in corresponding dimensions.
Align with Cartesian axes when
XandYare nonvector 2-D arrays (as if they were produced bymeshgrid).Have dimensions consistent with
V.
XqandYqmust be vectors or arrays of the same type (double or single). IfXqandYqare arrays, then they must have the same size. If they are vectors with different lengths, then they must have different orientations.methodmust be'linear','nearest', or'cubic'.The extrapolation for the out-of-boundary input is not supported.
For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Distributed Arrays
Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.
This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).
Version History
Introduced before R2006a
See Also
griddata | interp1 | interp3 | interpn | meshgrid | griddedInterpolant | scatteredInterpolant
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