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.

Creation

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.

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).

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

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.

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.

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

x, y, z — Sample points

vector

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: double

P — Sample points array

matrix

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: double

v — Sample values

vector

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

Data Types: double

Method — Interpolation method

'linear' (default) | 'nearest' | 'natural'

Interpolation method, specified as one of these options.

Method	Description	Continuity
'linear' (default)	Linear interpolation	C0
'nearest'	Nearest neighbor interpolation	Discontinuous
'natural'	Natural neighbor interpolation	C1 (except at sample points)

ExtrapolationMethod — Extrapolation method

'linear' | 'nearest' | 'none'

Extrapolation method, specified as one of these options.

ExtrapolationMethod	Description
'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 is 'nearest'.
'none'	No extrapolation. Any queries outside the convex hull of Points return NaN.

Properties

Points — Sample points

matrix

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 Points contains the (x, y) or (x, y, z) coordinates of a unique sample point. The rows in Points correspond to the function values in Values.

Data Types: double

Values — Function values at sample points

vector

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

Data Types: double

Method — Interpolation method

'linear' (default) | 'nearest' | 'natural'

Interpolation method, specified as 'linear','nearest' , or 'natural'. See Method for descriptions of these methods.

ExtrapolationMethod — Extrapolation method

'linear' | 'nearest' | 'none'

Extrapolation method, specified as 'nearest', 'linear', or 'none'. See ExtrapolationMethod for descriptions of these methods.

Data Types: double

Usage

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

Vq = F(Pq)
Vq = F(Xq,Yq)
Vq = F(Xq,Yq,Zq)
Vq = F({xq,yq})
Vq = F({xq,yq,zq})


• 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.

Examples

2-D Interpolation

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')

3-D Interpolation

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.

rng default;
P = -2.5 + 5*rand([200 3]);
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)

Replacement of Sample Values

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.

rng('default')
x = -2.5 + 5*rand([50 1]);
y = -2.5 + 5*rand([50 1]);
v = x.*exp(-x.^2-y.^2);

Create the interpolant.

F = scatteredInterpolant(x,y,v)

F = 
  scatteredInterpolant with properties:

                 Points: [50x2 double]
                 Values: [50x1 double]
                 Method: 'linear'
    ExtrapolationMethod: 'linear'

Evaluate the interpolant at (1.40,1.90).

F(1.40,1.90)

ans = 0.0069

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 = 5.6491

Eliminate Duplicate Sample Points

Use groupsummary to eliminate duplicate sample points and control how they are combined prior to calling scatteredInterpolant.

Create a 200-by-3 matrix of sample point locations. Add duplicate points in the last five rows.

P = -2.5 + 5*rand(200,3);
P(197:200,:) = repmat(P(196,:),4,1);

Create a vector of random values at the sample points.

V = rand(size(P,1),1);

If you attempt to use scatteredInterpolant with duplicate sample points, it throws a warning and averages the corresponding values in V to produce a single unique point. However, you can use groupsummary to eliminate the duplicate points prior to creating the interpolant. This is particularly useful if you want to combine the duplicate points using a method other than averaging.

Use groupsummary to eliminate the duplicate sample points and preserve the maximum value in V at the duplicate sample point location. Specify the sample points matrix as the grouping variable and the corresponding values as the data.

[V_unique,P_unique] = groupsummary(V,P,@max);

Since the grouping variable has three columns, groupsummary returns the unique groups P_unique as a cell array. Convert the cell array back into a matrix.

P_unique = [P_unique{:}];

Create the interpolant. Since the sample points are now unique, scatteredInterpolant does not throw a warning.

I = scatteredInterpolant(P_unique,V_unique);

Compare Scattered Data Interpolation Methods

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')

2-D Extrapolation

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.

rng('default')
P = -2.5 + 5*rand([200 2]);
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: [200x2 double]
                 Values: [200x1 double]
                 Method: 'linear'
    ExtrapolationMethod: 'nearest'

Evaluate the interpolant outside the convex hull.

vq = F(3.0,-1.5)

vq = 0.0029

Disable extrapolation and evaluate F at the same point.

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

vq = NaN

More About

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 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 Values or Method, the underlying Delaunay triangulation of the input data does not change, so you can compute new results quickly.

• Scattered data interpolation with scatteredInterpolant uses a Delaunay triangulation of the data, so can be sensitive to scaling issues in the sample points x, y, z, or P. When this occurs, you can use normalize to rescale the data and improve the results. See Normalize Data with Differing Magnitudes for more information.

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.