# Documentation

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# scatteredInterpolant

Interpolate 2-D or 3-D scattered data

## Description

Use scatteredInterpolant to perform interpolation on a 2-D or 3-D data set of scattered data. scatteredInterpolant 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 griddedInterpolant to perform interpolation with gridded data.

## Create Scattered Interpolant

### Syntax

F = scatteredInterpolant
F = scatteredInterpolant(x,y,v)
F = scatteredInterpolant(x,y,z,v)
F = scatteredInterpolant(P,v)
F = scatteredInterpolant(___,Method)
F = scatteredInterpolant(___,Method,ExtrapolationMethod)

### Description

F = scatteredInterpolant creates an empty scattered data interpolant object.

example

F = scatteredInterpolant(x,y,v) creates an interpolant that fits a surface of the form v = F(x,y). Vectors x and y specify the (x,y) coordinates of the sample points. v is a vector that contains the sample values associated with the points (x,y).

example

F = scatteredInterpolant(x,y,z,v) creates a 3-D interpolant of the form v = F(x,y,z).

example

F = scatteredInterpolant(P,v) specifies the coordinates of the sample points as an array. The rows of P contain the (x, y) or (x, y, z) coordinates for the values in v.

example

F = scatteredInterpolant(___,Method) specifies an interpolation method: 'nearest', 'linear', or 'natural'. Specify Method as the last input argument in any of the first three syntaxes.

example

F = scatteredInterpolant(___,Method,ExtrapolationMethod) specifies both the interpolation and extrapolation methods. Pass Method and ExtrapolationMethod together as the last two input arguments in any of the first three syntaxes.

• Method can be: 'nearest', 'linear', or 'natural'.

• ExtrapolationMethod can be: 'nearest', 'linear', or 'none'.

### Input Arguments

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Sample points, specified as vectors of the same size as v. The sample points should be unique. However, if the sample points contain duplicates, scatteredInterpolant displays a warning and merges the duplicates into a single point.

Data Types: single | double

Sample points array, specified as an m-by-n matrix, where m is the number of points and n is the dimension of the space where the points reside. Each row of P contains the (x, y) or (x, y, z) coordinates of a sample point. The sample points should be unique. However, if the sample points contain duplicates, scatteredInterpolant displays a warning and merges the duplicates into a single point.

Data Types: single | double

Sample values, specified as a vector that defines the function values at the sample points, v = F(x,y,z).

Data Types: single | double

Interpolation method, specified as one of these options.

MethodDescriptionContinuity
'linear' (default)

Linear interpolation

C0
'nearest'

Nearest neighbor interpolation

Discontinuous
'natural'

Natural neighbor interpolation

C1 (except at sample points)

Extrapolation method, specified as one of these options.

ExtrapolationMethodDescription
'linear'

Linear extrapolation based on boundary gradients. Default when Method is 'linear' or 'natural'.

'nearest'

Nearest neighbor extrapolation. This method evaluates to the value of the nearest neighbor on the boundary. Default when Method = 'nearest'.

'none'

No extrapolation. Any queries outside the convex hull of F.Points return NaN.

## Evaluate Interpolant at Specific Points

Use scatteredInterpolant to create the interpolant, F. Then you can evaluate F at specific points using any of the following syntaxes:

• Vq = F(Pq) specifies the query points in the matrix Pq. Each row in Pq contains the coordinates of a query point.

• Vq = F(Xq,Yq) and Vq = F(Xq,Yq,Zq) specify the query points as two or three matrices of equal size.

• Vq = F({xq,yq}) and Vq = F({xq,yq,zq}) specify the query points as grid vectors. Use this syntax to conserve memory when you want to query a large grid of points.

## Properties

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Sample points, specified as a matrix. The size of the matrix is m-by-2 or m-by-3 to represent m points in 2-D or 3-D space. Each row of F.Points contains the (x, y) or (x, y, z) coordinates of a unique sample point. The rows in F.Points correspond to the function values in F.Values.

Data Types: single | double

Function values at sample points, specified as a vector of values associated with each point in F.Points.

Data Types: single | double

Interpolation method, specified as a character vector for the method used to interpolate the data: 'linear','nearest' , or 'natural'.

Extrapolation method, specified as a character vector for the method used to extrapolate the data: 'nearest', 'linear', or 'none'.

Example:

Data Types: single | double

## Examples

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Define some sample points and calculate the value of a trigonometric function at those locations. These points are the sample values for the interpolant.

t = linspace(3/4*pi,2*pi,50)';
x = [3*cos(t); 2*cos(t); 0.7*cos(t)];
y = [3*sin(t); 2*sin(t); 0.7*sin(t)];
v = repelem([-0.5; 1.5; 2],length(t));

Create the interpolant.

F = scatteredInterpolant(x,y,v);

Evaluate the interpolant at query locations (xq,yq).

tq = linspace(3/4*pi+0.2,2*pi-0.2,40)';
xq = [2.8*cos(tq); 1.7*cos(tq); cos(tq)];
yq = [2.8*sin(tq); 1.7*sin(tq); sin(tq)];
vq = F(xq,yq);

Plot the result.

plot3(x,y,v,'.',xq,yq,vq,'.'), grid on
title('Linear Interpolation')
xlabel('x'), ylabel('y'), zlabel('Values')
legend('Sample data','Interpolated query data','Location','Best')

Create an interpolant for a set of scattered sample points, then evaluate the interpolant at a set of 3-D query points.

Define 200 random points and sample a trigonometric function. These points are the sample values for the interpolant.

P = -2.5 + 5*gallery('uniformdata',[200 3],0);
v = sin(P(:,1).^2 + P(:,2).^2 + P(:,3).^2)./(P(:,1).^2+P(:,2).^2+P(:,3).^2);

Create the interpolant.

F = scatteredInterpolant(P,v);

Evaluate the interpolant at query locations (xq,yq,zq).

[xq,yq,zq] = meshgrid(-2:0.25:2);
vq = F(xq,yq,zq);

Plot slices of the result.

xslice = [-.5,1,2];
yslice = [0,2];
zslice = [-2,0];
slice(xq,yq,zq,vq,xslice,yslice,zslice)

Replace the elements in the Values property when you want to change the values at the sample points. You get immediate results when you evaluate the new interpolant because the original triangulation does not change.

Create 50 random points and sample an exponential function. These points are the sample values for the interpolant.

x = -2.5 + 5*gallery('uniformdata',[50 1],0);
y = -2.5 + 5*gallery('uniformdata',[50 1],1);
v = x.*exp(-x.^2-y.^2);

Create the interpolant.

F = scatteredInterpolant(x,y,v)
F =
scatteredInterpolant with properties:

Points: [50×2 double]
Values: [50×1 double]
Method: 'linear'
ExtrapolationMethod: 'linear'

Evaluate the interpolant at (1.40,1.90).

F(1.40,1.90)
ans = 0.0029

Change the interpolant sample values and reevaluate the interpolant at the same point.

vnew = x.^2 + y.^2;
F.Values = vnew;
F(1.40,1.90)
ans = 6.1109

Compare the results of several different interpolation algorithms offered by scatteredInterpolant.

Create a sample data set of 50 scattered points. The number of points is artificially small to highlight the differences between the interpolation methods.

x = -3 + 6*rand(50,1);
y = -3 + 6*rand(50,1);
v = sin(x).^4 .* cos(y);

Create the interpolant and a grid of query points.

F = scatteredInterpolant(x,y,v);
[xq,yq] = meshgrid(-3:0.1:3);

Plot the results using the 'nearest', 'linear', and 'natural' methods. Each time the interpolation method changes, you need to requery the interpolant to get the updated results.

F.Method = 'nearest';
vq1 = F(xq,yq);
plot3(x,y,v,'mo')
hold on
mesh(xq,yq,vq1)
title('Nearest Neighbor')
legend('Sample Points','Interpolated Surface','Location','NorthWest')

F.Method = 'linear';
vq2 = F(xq,yq);
figure
plot3(x,y,v,'mo')
hold on
mesh(xq,yq,vq2)
title('Linear')
legend('Sample Points','Interpolated Surface','Location','NorthWest')

F.Method = 'natural';
vq3 = F(xq,yq);
figure
plot3(x,y,v,'mo')
hold on
mesh(xq,yq,vq3)
title('Natural Neighbor')
legend('Sample Points','Interpolated Surface','Location','NorthWest')

Plot the exact solution.

figure
plot3(x,y,v,'mo')
hold on
mesh(xq,yq,sin(xq).^4 .* cos(yq))
title('Exact Solution')
legend('Sample Points','Exact Surface','Location','NorthWest')

Query an interpolant at a single point outside the convex hull using nearest neighbor extrapolation.

Define a matrix of 200 random points and sample an exponential function. These points are the sample values for the interpolant.

P = -2.5 + 5*gallery('uniformdata',[200 2],0);
x = P(:,1);
y = P(:,2);
v = x.*exp(-x.^2-y.^2);

Create the interpolant, specifying linear interpolation and nearest neighbor extrapolation.

F = scatteredInterpolant(P,v,'linear','nearest')
F =
scatteredInterpolant with properties:

Points: [200×2 double]
Values: [200×1 double]
Method: 'linear'
ExtrapolationMethod: 'nearest'

Evaluate the interpolant outside the convex hull.

vq = F(3.0,-1.5)
vq = 0.0031

Disable extrapolation and evaluate F at the same point.

F.ExtrapolationMethod = 'none';
vq = F(3.0,-1.5)
vq = NaN

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## Tips

• It is quicker to evaluate a scatteredInterpolant object F at many different sets of query points than it is to compute the interpolations separately using the functions griddata or griddatan. For example:

% Fast to create interpolant F and evaluate multiple times
F = scatteredInterpolant(X,Y,V)
v1 = F(Xq1,Yq1)
v2 = F(Xq2,Yq2)

% Slower to compute interpolations separately using griddata
v1 = griddata(X,Y,V,Xq1,Yq1)
v2 = griddata(X,Y,V,Xq2,Yq2)

• To change the interpolation sample values or interpolation method, it is more efficient to update the properties of the interpolant object F than it is to create a new scatteredInterpolant object. When you update F.Values or F.Method, the underlying Delaunay triangulation of the input data does not change, so you can compute new results quickly.

## Algorithms

scatteredInterpolant uses a Delaunay triangulation of the scattered sample points to perform interpolation [1].

## References

[1] Amidror, Isaac. "Scattered data interpolation methods for electronic imaging systems: a survey." Journal of Electronic Imaging. Vol. 11, No. 2, April 2002, pp. 157–176.