Interpolation for 2-D gridded data in meshgrid format (2024)

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' methods

  • NaN values 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');

Interpolation for 2-D gridded data in meshgrid format (1)

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');

Interpolation for 2-D gridded data in meshgrid format (2)

Interpolate over a Grid Using Cubic Method

Open Live Script

Coarsely sample the peaks function.

Plot the coarse sampling.

figuresurf(X,Y,V)title('Original Sampling');

Interpolation for 2-D gridded data in meshgrid format (3)

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');

Interpolation for 2-D gridded data in meshgrid format (4)

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')

Interpolation for 2-D gridded data in meshgrid format (5)

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')

Interpolation for 2-D gridded data in meshgrid format (6)

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')

Interpolation for 2-D gridded data in meshgrid format (7)

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');

Interpolation for 2-D gridded data in meshgrid format (8)

Input Arguments

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X,YSample grid points
matrices | vectors

Sample grid points, specified as real matrices or vectors. Thesample grid points must be unique.

  • If X and Y arematrices, then they contain the coordinates of a full grid (in meshgrid format).Use the meshgrid function tocreate the X and Y matricestogether. Both matrices must be the same size.

  • If X and Y are 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

VSample values
matrix

Sample values, specified as a real or complex matrix. The sizerequirements for V depend on the size of X and Y:

  • If X and Y arematrices representing a full grid (in meshgrid format),then V must be the same size as X and Y.

  • If X and Y aregrid vectors, then V must be a matrix containing length(Y) rowsand length(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,YqQuery points
scalars | vectors | matrices | arrays

Query points, specified as a real scalars, vectors, matrices,or arrays.

  • If Xq and Yq arescalars, then they are the coordinates of a single query point.

  • If Xq and Yq arevectors of different orientations, then Xq and Yq aretreated as grid vectors.

  • If Xq and Yq arevectors of the same size and orientation, then Xq and Yq aretreated as scatteredpoints in 2-D space.

  • If Xq and Yq arematrices, then they represent either a full grid of query points (in meshgrid format)or scattered points.

  • If Xq and Yq areN-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

kRefinement 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.

Interpolation for 2-D gridded data in meshgrid format (9)

Example: interp2(V,2)

Data Types: single | double

methodInterpolation method
'linear' (default) | 'nearest' | 'cubic' | 'spline' | 'makima'

Interpolation method, specified as one of the options in this table.

MethodDescriptionContinuityComments
'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
  • Requires at least two grid points in each dimension

  • Requires more memory than 'nearest'

'nearest'The interpolated value at a query point is the value at the nearest sample grid point. Discontinuous
  • Requires two grid points in each dimension.

  • Fastest computation with modest memory requirements

'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
  • Grid must have uniform spacing in each dimension, but the spacing does not have to be the same for all dimensions

  • Requires at least four points in each dimension

  • Requires more memory and computation time than 'linear'

'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
  • Requires at least 2 points in each dimension

  • Produces fewer undulations than 'spline'

  • Computation time is typically less than 'spline', but the memory requirements are similar

'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
  • Requires four points in each dimension

  • Requires more memory and computation time than 'cubic'

extrapvalFunction 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.

SyntaxesSpecialConditionsSize of VqExample
interp2(X,Y,V,Xq,Yq)
interp2(V,Xq,Yq)
and variations of these syntaxes that include method or extrapval
Xq, Yq are scalarsScalarsize(Vq) = [1 1] when you pass Xq and Yq asscalars.
Same as aboveXq, Yq are vectors ofthe same size and orientationVector of same size and orientation as Xq and YqIf size(Xq) = [100 1]
and size(Yq)= [100 1],
then size(Vq) = [1001].
Same as aboveXq, Yq are vectors ofmixed orientationMatrix 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 aboveXq, Yq are matrices orarrays of the same sizeMatrix or array of the same size as Xq and YqIf 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:
2^k* (size(V,1)-1)+1,

and thenumber of columns is:
2^k * (size(V,2)-1)+1

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

Interpolation for 2-D gridded data in meshgrid format (10)

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.

Interpolation for 2-D gridded data in meshgrid format (11)

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

Version History

Introduced before R2006a

See Also

griddata | interp1 | interp3 | interpn | meshgrid | griddedInterpolant | scatteredInterpolant

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Interpolation for 2-D gridded data in meshgrid format (12)

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Interpolation for 2-D gridded data in meshgrid format (2024)
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