This example shows how to create a 2D grid using meshgrid
and ndgrid
.
In MATLAB®, gridded data means data ordered in a grid. You can understand ordered data by thinking about how MATLAB stores data in matrices.
Define some data.
A = gallery('uniformdata',[3 5],0)
A = 3×5
0.9501 0.4860 0.4565 0.4447 0.9218
0.2311 0.8913 0.0185 0.6154 0.7382
0.6068 0.7621 0.8214 0.7919 0.1763
MATLAB stores the data in a matrix. You can think of A
as a set of places for the elements that are ordered by the indices of the matrix. The linear indices of A
are:
$$\left[\begin{array}{ccccc}1& 4& 7& 10& 13\\ 2& 5& 8& 11& 14\\ 3& 6& 9& 12& 15\end{array}\right]$$
Any element in the matrix can be retrieved by indexing, that is, by asking for the element at that place in the matrix. The ith element in A
is retrieved by A(i)
.
Retrieve the 7th element in A
.
A(7)
ans = 0.4565
For an mbyn matrix, you can find the column elements adjacent to the ith element by offsetting i by 1. To find the row elements adjacent to the ith element, offset i by m.
$$\begin{array}{ccc}& i1& \\ im& i& i+m\\ & i+1& \end{array}$$
Retrieve the column elements adjacent to A(7)
.
A(6),A(8)
ans = 0.7621
ans = 0.0185
MATLAB uses a similar idea for creating data grids. A grid is not just a set of points that meet certain geometric properties. Rather, a gridded data set relies on an ordered relationship among the points in the grid. The adjacency information readily available in the grid structure is very useful for many applications and particularly gridbased interpolation.
MATLAB provides two functions for creating grids:
meshgrid
creates 2D and 3D grids that are Cartesian axis aligned. To create a 2D grid, the syntax is [X,Y] = meshgrid(xgv, ygv)
where xgv
is a vector of length m
, and ygv
is a vector of length n
. meshgrid
replicates xgv
to form the n
bym
matrix X
, and it replicates ygv
to form another n
bym
matrix Y
. X
and Y
represent the coordinates of the grid points. The rows of X
are aligned with the horizontal X
axis, and the columns of Y
are aligned with the negative Y
axis.
ndgrid
creates ND grids that are array space aligned. In array space the axes are row, column, page, etc. The calling syntax is [X1, X2, X3,...,Xn] = ndgrid(x1gv, x2gv, x3gv,...,xngv)
where x1gv,x2gv,x3gv,...,xngv
are vectors that span the grid in each dimension. X1,X2,X3,...,Xn
are output arrays that can be used for evaluating functions of multiple variables and for multidimensional interpolation.
Use meshgrid
to create a 2D axis aligned grid from two vectors, xgv
and ygv
.
xgv = [1 2 3]; ygv = [1 2 3 4 5]; [X,Y] = meshgrid(xgv, ygv)
X = 5×3
1 2 3
1 2 3
1 2 3
1 2 3
1 2 3
Y = 5×3
1 1 1
2 2 2
3 3 3
4 4 4
5 5 5
Now use ndgrid
to create a 2D space aligned grid from the same two vectors, xgv
and ygv
.
[X1,X2] = ndgrid(xgv,ygv)
X1 = 3×5
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
X2 = 3×5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
Notice that ndgrid
's X1
is the transpose of meshgrid
's X
. The same is true for X2
and Y
.
For a given set of inputs, the meshgrid
and ndgrid
functions will produce grids with exactly the same coordinates. The only difference between their outputs is the format of the coordinate arrays. Plot both outputs and see that they are the same.
figure() [X1_ndgrid,X2_ndgrid] = ndgrid(1:3,1:5); Z = zeros(3,5); mesh(X1_ndgrid,X2_ndgrid,Z,'EdgeColor','black') axis equal; % Set the axis labeling and title h1 = gca; h1.XTick = [1 2 3]; h1.YTick = [1 2 3 4 5]; xlabel('ndgrid Output')
figure() [X_meshgrid,Y_meshgrid] = meshgrid(1:3, 1:5); mesh(X_meshgrid,Y_meshgrid,Z','EdgeColor','black') axis equal; % Set the axis labeling and title h2 = gca; h2.XTick = [1 2 3]; h2.YTick = [1 2 3 4 5]; xlabel('meshgrid Output')
Depending on how you intend to use your grid, you may prefer one format or the other. Some functions in MATLAB might require your data to have a meshgrid
format while others might require an ndgrid
format.
Converting Between Grid Formats. To convert a 2D grid output from meshgrid
to
the ndgrid
format, transpose the coordinate matrices:
[X_meshgrid,Y_meshgrid] = meshgrid(1:3, 1:5); [X1_ndgrid,X2_ndgrid] = ndgrid(1:3,1:5); isequal(X_meshgrid',X1_ndgrid) ans = 1 isequal(Y_meshgrid',X2_ndgrid) ans = 1
permute
function. isequal(permute(X_meshgrid,[2 1]),X1_ndgrid) ans = 1
meshgrid
to ndgrid
,
transpose each page of the coordinate array. For a given array my_array
, permute(my_array,
[2 1 3])
interchanges the rows and columns, and the net
effect is the transposition of every page:[X_meshgrid,Y_meshgrid,Z_meshgrid] = meshgrid(1:3, 1:5, [1 2]); [X1_ndgrid,X2_ndgrid,X3_ndgrid] = ndgrid(1:3,1:5, [1 2]); isequal(permute(X_meshgrid,[2 1 3]),X1_ndgrid) ans = 1 isequal(permute(Y_meshgrid,[2 1 3]),X2_ndgrid) ans = 1 isequal(permute(Z_meshgrid,[2 1 3]),X3_ndgrid) ans = 1
Grid Vectors. The inputs that you pass to the grid functions are called grid
vectors. The grid vectors implicitly define the grid.
Consider two vectors, x1gv = (1:3)
and x2gv = (1:5)
. You can think of these vectors as a set of coordinates
in the x1
direction and a set of coordinates in
the x2
direction, like so:
Each arrow points to
a location. You can use these two vectors to define a set of grid
points, where one set of coordinates is given by x1gv
and
the other set of coordinates is given by x2gv
.
When the grid vectors are replicated they form two coordinate arrays
that make up the full grid:
Monotonic and Nonmonotonic Grids. Your input grid vectors may be monotonic or nonmonotonic.
Monotonic vectors contain values that either increase in that dimension
or decrease in that dimension. Conversely, nonmonotonic vectors contain
values that fluctuate. If the input grid vector is nonmonotonic, such
as [2 4 6 8 3 1]
, ndgrid
outputs
the following:
[X1,X2] = ndgrid([2 4 6 3 1]) X1 = 2 2 2 2 2 4 4 4 4 4 6 6 6 6 6 3 3 3 3 3 1 1 1 1 1 X2 = 2 4 6 3 1 2 4 6 3 1 2 4 6 3 1 2 4 6 3 1 2 4 6 3 1
Uniform and Nonuniform Grids. A uniform grid is one in which all neighboring
points in a given dimension have equal spacing. For example, [X1,
X2] = ndgrid([1 3 5 9],[11 13 15])
is a uniform with a spacing
of 2 units in each dimension.
It is not necessary for the spacing in a uniform grid to be
equal in all dimensions. For example, [X1, X2] = ndgrid([1
2 3 4],[11 13 15])
is considered uniform even though the
spacing in X1
and X2
are different.
A grid whose spacing varies within any dimension is a nonuniform grid.
For example, [X1, X2] = ndgrid([1 5 6 9],[11 13 15])
creates
a nonuniform grid because the spacing varies along the first dimension.



Uniform  Uniform  Nonuniform 
MATLAB allows you to represent a grid in one of three representations: full grid, compact grid, or default grid. The compact grid and default grid are used primarily for convenience and improved efficiency.
Full Grid. A full grid is one in which the points
are explicitly defined. The outputs of ndgrid
and meshgrid
define
a full grid.
Compact Grid. The explicit definition of every point in a grid is expensive
in terms of memory. The compact grid representation
is a way to dispense with the memory overhead of a full grid. The
compact grid representation stores just the grid vectors instead of
the full grid. (For information on how the griddedInterpolant
class
uses the compact grid representation, see Interpolation with the griddedInterpolant Class.)
Default Grid. For many cases, when you are analyzing data, you care about both the distances between points in the grid and their value. For example, a data set might represent rainfall at specific points in a geographic area. In this case, you would care about the value at each grid point and the distance between a point and its neighbor. In other cases, you might care only about the value for a given point and not about the relative distances. For example, you could be working with input data from an MRI scan where the distance between the points is completely uniform. In this case, you would care about the values of the points, but you can be certain of a completely uniform grid. In this case, the default grid representation is useful. The default grid representation stores the value at a grid point explicitly and creates grid point coordinates implicitly.
This example shows how you can use the default grid instead of the full grid to produce a plot.
Create a grid and a function that you want to plot.
[X,Y] = meshgrid(11:15,11:16); Z = X.^2 + Y.^2;
Use the full grid that you created with meshgrid
by using X
and Y
.
figure
surf(X,Y,Z)
title('Full Grid')
In contrast, have MATLAB create a default grid instead of using the full grid.
figure
surf(Z)
title('Default Grid')
Notice the difference in the scaling of the axes. When you plot the full grid the X
 and Y
axis have ranges of 11 to 15 and 10 to 16 respectively. When you plot the default grid the X
 and Y
axis have ranges of 1 to m and 1 to n, where Z
is mbyn.
In some cases, you may need to approximate a grid for your data. An approximate grid can be idealized by a standard MATLAB grid by choosing an appropriate set of grid vectors. For example, a grid can have points that lie along curved lines. A data set like this might occur if your data is longitude and latitude based:
In this case, although the input data cannot be gridded directly, you can approximate straight grid lines at appropriate intervals:
You can also use the default grid.
Degenerate Grid. A degenerate grid is a special case of
grid where one or more dimensions of the grid are singletons. A singleton
dimension can be internal, as in 7:7
in this example:
[X1,X2,X3] = ndgrid(1:2:10,7:7,1:3:15);
[X1,X2,X3] = ndgrid(1:2:10,1:3:15,7:7);
If you use indexing to
extract the desired data, the resultant grid is degenerate in the X3
dimension:
[X1,X2,X3] = ndgrid(1:3); X1_slice = X1(:,:,2) X1_slice = 1 1 1 2 2 2 3 3 3 X2_slice = X2(:,:,2) X2_slice = 1 2 3 1 2 3 1 2 3 X3_slice = X3(:,:,2) X3_slice = 2 2 2 2 2 2 2 2 2
The concept of data gridding is very important for understanding the ways in which MATLAB does gridbased interpolation.
In gridbased interpolation, the data to be interpolated is represented by an ordered grid. For example, an arrangement of temperature measurements over a rectangular flat surface at 1cm intervals from toptobottom vertically and lefttoright horizontally is considered 2D gridded data. Gridbased interpolation provides an efficient way to approximate the temperature at any location between the grid points.
Gridbased interpolation provides significant savings in computational overhead because the gridded structure allows MATLAB to locate a query point and its adjacent neighbors very quickly. To understand how this works, consider the following points on a 1D grid:
The lines connecting
adjacent points represent the cells of the
grid. The first cell occurs between x = 1
and x
= 3
, the second occurs between x = 3
and x
= 5
, and so on. Each number represents a coordinate in the
grid. If you want to query the grid at x = 6
, you
would have to use interpolation because 6 is not explicitly defined
in the grid. Since this grid has a uniform spacing of 2, you can narrow
down the location of the query point with a single integer division
(6/2 = 3
). This tells you that the point is in
the 3rd cell of the grid. Locating a cell in a 2D grid involves performing
this operation once in each dimension. This operation is called a fast
lookup, and MATLAB uses this technique only when
the data is arranged in a uniform grid.
This fast lookup efficiently locates the cell containing a query
point Xq
. A binary search proceeds as follows:
Locate the center grid point.
Compare Xq
to the point at the
center of the grid.
If Xq
is less than the point found
at the center, eliminate all of the grid points greater than central
point from the search. Similarly, if Xq
is greater
than the one found at the center, we eliminate all of the grid points
that are less than the central point. Note that by doing this, we
have reduced the number of points we must search by half.
Find the center of the remaining grid points and repeat
from Step 2 until you are left with one grid point on either side
of your query. These two points mark the boundary of the cell that
contains Xq
.
To illustrate the power of binary searches, consider the following example. Before the advent of electronic credit card authorizations, the only protection merchants had against fraudulent credit card purchases was to compare the account number on each customer's credit card against a list of "bad" account numbers. These lists were bound booklets with tens of thousands of card numbers arranged in ascending order. How many comparisons would be required to search through a list of 10,000 account numbers for one sale? It turns out that for any list of n
ordered items, the maximum number of comparisons will be no more than the number of times you can divide the list in half, or log2(n)
. Therefore, the credit card search will take no more than log2(10e3)
or about 13 comparisons. That's a pretty impressive if you consider how many comparisons it would take to perform a sequential search.
By contrast, consider a problem with a scattered data set.
x = rand(20,1); y = rand(20,1); scatter(x,y)
To find the points in close proximity to a query point would require many more operations. If your data can be approximated as a grid, gridbased interpolation will provide substantial savings in computation and memory usage.
If your data is scattered, you can use the tools detailed in Interpolating Scattered Data.
The interpolation methods available in MATLAB create interpolating functions that pass though the sample data points. If you were to query the interpolation function at a sample location, you would get back the value at that sample data point. By contrast, curve and surface fitting algorithms do not necessarily pass through the sample data points.
Gridbased interpolation offers several different methods for interpolation. When choosing an interpolation method, keep in mind that some require more memory or longer computation time than others. However, you may need to trade off these resources to achieve the desired smoothness in the result. The following table provides an overview of the benefits, tradeoffs, and requirements for each method.
Method  Description  Continuity  Memory Usage and Performance  Requirements 

Nearest Neighbor  The interpolated value at a query point is the value at the nearest sample grid point.  Discontinuous 


Next Neighbor  The interpolated value at a query point is the value at the next sample grid point.  Discontinuous  Same memory requirements and computation time as nearest neighbor. 

Previous Neighbor  The interpolated value at a query point is the value at the previous sample grid point.  Discontinuous  Same memory requirements and computation time as nearest neighbor. 

Linear  The interpolated value at a query point is based on linear interpolation of the values at neighboring grid points in each respective dimension.  C0 


Pchip  The interpolated value at a query point is based on a shapepreserving piecewise cubic interpolation of the values at neighboring grid points.  C1 


Cubic  The interpolated value at a query point is based on cubic interpolation of the values at neighboring grid points in each respective dimension.  C1 


Modified Akima  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.  C2 


This figure has some visual comparisons of the different interpolation methods for 1D data.
Comparison of Interpolation Methods
MATLAB provides support for gridbased interpolation in several ways:
The interp
family of functions: interp1
, interp2
, interp3
,
and interpn
.
The griddedInterpolant
class.
Both the interp
family of functions
and griddedInterpolant
support ND gridbased interpolation.
However, there are memory and performance benefits to using the griddedInterpolant
class
over the interp
functions. Moreover, the griddedInterpolant
class
provides a single consistent interface for working with gridded data
in any number of dimensions.
This example shows how the interp1
function can be used to interpolate a set of sample values using the 'pchip'
method.
The function interp1
performs onedimensional interpolation. Its most general form is:
Vq = interp1(X,V,Xq,method)
where X
is a vector of coordinates and V
is a vector containing the values at those coordinates. Xq
is a vector containing the query points at which to interpolate, and method
optionally specifies any of four interpolation methods: 'nearest'
, 'linear'
, 'pchip'
, or 'spline'
.
Create a set of 1D grid points X
and corresponding sample values V
.
X = [1 2 3 4 5]; V = [12 16 31 10 6];
Interpolate over finer intervals with 0.1 spacing.
Xq = (1:0.1:5);
Vq = interp1(X,V,Xq,'pchip');
Plot the samples and interpolated values.
plot(X,V,'o'); hold on plot(Xq,Vq,''); legend('samples','pchip'); hold off
This example shows how to use the 'extrap'
option to interpolate beyond the domain of your sample points.
Define the sample points and values.
X = [1 2 3 4 5]; V = [12 16 31 10 6];
Specify the query points, Xq
, that extend beyond the domain of X
.
Xq = (0:0.1:6); Vq = interp1(X,V,Xq,'pchip','extrap');
Plot the results.
figure plot(X,V,'o'); hold on plot(Xq,Vq,''); legend('samples','pchip'); hold off
In an alternative approach, you can introduce additional points to gain more control over the behavior in the extrapolated regions. For example, you can constrain the curve to remain flat in the extrapolated region by extending the domain with repeated values.
X = [0 1 2 3 4 5 6]; V = [12 12 16 31 10 6 6];
Specify the query points, Xq
, that extend further beyond the domain of X
.
Xq = (1:0.1:7);
Interpolate using 'pchip'
. You can omit the 'extrap'
option because it is the default with the 'pchip'
, 'makima'
, and 'spline'
methods.
Vq = interp1(X,V,Xq,'pchip');
Plot the results.
figure plot(X,V,'o'); hold on plot(Xq,Vq,''); legend('samples','pchip'); hold off
This example shows how the interp2
function can be used to interpolate the coarsely sampled peaks
function over a finer grid.
The interp2
and interp3
functions perform two and threedimensional interpolation respectively, and they interpolate grids in the meshgrid
format. The calling syntax for interp2
has the general form:
Vq = interp2(X,Y,V,Xq,Yq,method)
where X
and Y
are arrays of coordinates that define a grid in meshgrid
format, and V
is an array containing the values at the grid points. Xq
and Yq
are arrays containing the coordinates of the query points at which to interpolate. method
optionally specifies any of four interpolation methods: 'nearest'
, 'linear'
, 'cubic'
, or 'spline'
.
The grid points that comprise X
and Y
must be monotonically increasing and should conform to the meshgrid
format.
Create a coarse grid and corresponding sample values.
[X,Y] = meshgrid(3:1:3); V = peaks(X,Y);
Plot the sample values.
surf(X,Y,V)
title('Sample Grid');
Generate a finer grid for interpolation.
[Xq,Yq] = meshgrid(3:0.25:3);
Use interp2
to interpolate at the query points.
Vq = interp2(X,Y,V,Xq,Yq,'linear');
Plot the results.
surf(Xq,Yq,Vq);
title('Refined Grid');
This example shows how to use interp3
to interpolate a 3D function at a single query point and compare it to the value generated by an analytic expression.
interp3
works in the same way as interp2
except that it takes two additional arguments: one for the third dimension in the sample grid and the other for the third dimension in the query points,
Vq = interp3(X,Y,Z,V,Xq,Yq,Zq,method)
.
As is the case with interp2
, the grid points you supply to interp3
must be monotonically increasing and should conform to the meshgrid
format.
Define a function that generates values for X, Y, and Z input.
generatedvalues = @(X,Y,Z)(X.^2 + Y.^3 + Z.^4);
Create the sample data.
[X,Y,Z] = meshgrid((5:.25:5)); V = generatedvalues(X,Y,Z);
Interpolate at a specific query point.
Vq = interp3(X,Y,Z,V,2.35,1.76,0.23,'cubic')
Vq = 10.9765
Compare Vq
to the value generated by the analytic expression.
V_actual = generatedvalues(2.35,1.76,0.23)
V_actual = 10.9771
This example shows how the interpn
function can be used to interpolate a coarsely sampled function over a finer grid using the 'cubic'
method.
The function interpn
performs ndimensional interpolation on grids that are in the ndgrid
format. Its most general form is:
Vq = interpn(X1,X2,X3,...Xn,V,Y1,Y2,Y3,...,Yn,method)
where X1,X2,X3,...,Xn
are arrays of coordinates that define a grid in ndgrid
format, and V
is an array containing the values at the grid points. Y1,Y2,Y3,...,Yn
are arrays containing the coordinates of the query points at which to interpolate. method
optionally specifies any of four interpolation methods: 'nearest'
, 'linear'
, 'cubic'
, or 'spline'
.
The grid points that comprise X1,X2,X3,...Xn
must be monotonically increasing and should conform to the ndgrid
format.
Create a set of grid points and corresponding sample values.
[X1,X2] = ndgrid((5:1:5)); R = sqrt(X1.^2 + X2.^2)+ eps; V = sin(R)./(R);
Plot the sample values.
mesh(X1,X2,V)
title('Sample Grid');
Create a finer grid for interpolation.
[Y1,Y2] = ndgrid((5:.5:5));
Interpolate over the finer grid and plot the results.
Vq = interpn(X1,X2,V,Y1,Y2,'cubic'); mesh(Y1,Y2,Vq) title('Refined Grid');
interpn
has an alternate syntax: Vq = interpn(V,ntimes,method)
that allows you to interpolate over a grid that is an integer number of times finer than the sample grid. In the previous code, Y1
and Y2
queried the interpolant over a grid that contained one extra point between each of the samples. The following code demonstrates how you can achieve the same result with ntimes=1
.
Interpolate over a finer grid using ntimes=1
.
Vq = interpn(V,1,'cubic');
Plot the result.
mesh(Vq)
title('Refined Grid with NTIMES');
Notice that the plot is scaled differently than in the previous example. This is because we called mesh
and passed the values only. The mesh
function used a default grid based on the dimensions of Vq
. The output values are the same in both cases.
You can also supply a nonuniform grid of query points. This can be useful if you are interested in querying the interpolant at a higher resolution in one region of the grid. The following code shows how this can be done.
Interpolate over a biased grid.
[Y1, Y2] = ndgrid([5 4 3 2.5 2 1.5 1.25 1 0.75 0.5 0.20 0]);
Vq = interpn(X1,X2,V,Y1,Y2,'cubic');
Plot the result.
mesh(Y1,Y2,Vq);
title('Biased Grid');
Like the interpn
function, the griddedInterpolant
class provides
a single interface for gridbased interpolation in n dimensions.
However griddedInterpolant
offers the following additional
benefits:
It offers substantial performance improvements for repeated queries to the interpolant.
It offers additional performance improvements and savings in memory consumption because it stores the sample points as a compact grid.
griddedInterpolant
accepts sample data that
conforms to the ndgrid
format. If you wish to create
a griddedInterpolant
with meshgrid
data,
you will need to convert the data to the ndgrid
format.
See Converting meshgrid Data to the ndgrid Format for
a demonstration of how to convert 2D and 3D meshgrids
.
The griddedInterpolant
class supports the interpolation
methods that are also supported by interpn
:
nearest
, linear
,
pchip
, cubic
,
makima
, and spline
. However griddedInterpolant
offers greater performance with
less overhead.
An interpolant is a function that performs
interpolation. You create the interpolant by calling the griddedInterpolant
constructor
and passing the sample data: the grid and corresponding sample values.
You can also specify the interpolation method if you wish to override
the default “linear” method. The calling syntax has
the following forms:
For 1D interpolation, you can pass x
,
a set of points, and v
, a vector of the same length
containing the corresponding values.
F = griddedInterpolant(x,v)
For higher dimensions, you can supply a full grid. X1,X2,...,Xn
specify
the grid as a set of n nD arrays.
These arrays conform to the ndgrid
format and are
the same size as the sample array V
.
F = griddedInterpolant(X1,X2,...,Xn,V)
If you know that the distances between adjacent sample
points are uniform, you can let griddedInterpolant
create
a default grid by passing only the sample points V
.
F = griddedInterpolant(V)
You can also specify the coordinates of your sample
data as a compact grid. The compact grid is represented by a set of
vectors. These vectors are then packaged into a cell array by enclosing
them in curly braces; for example,{x1g,x2g,...,xng}
where
vectors x1g,x2g,...,xng
define the grid coordinates
in each dimension.
F = griddedInterpolant({x1g,x2g,...,xng},V)
You can also specify the interpolation method as a final input argument with any of the calling syntaxes. This example specifies nearest neighbor interpolation.
F = griddedInterpolant({x1g,x2g,x3g},V,'nearest')
The griddedInterpolant
, F
,
is evaluated in the same way as you would call a function. Your query
points may be scattered or gridded, and you can pass them to F
in
any of the following ways:
You can specify an mbyn matrix Xq
,
which contains m scattered points in n dimensions.
The interpolated values Vq
are returned as an mby1 vector.
Vq = F(Xq)
You can also specify the query points as a series
of n column vectors x1q,x2q,...,xnq
of
length m. These vectors represent m points
in n dimensions. The interpolated values Vq
are
returned as an mby1 vector.
Vq = F(x1q,x2q,...,xnq)
You can specify the query points as a series of n ndimensional
arrays representing a full grid. The arrays X1q,X2q,...,Xnq
are
all the same size and conform to the ndgrid
format.
The interpolated values Vq
will also be the same
size.
Vq = F(X1q,X2q,...,Xnq)
You can also specify the query points as a compact
grid. x1gq,x2gq,...,xngq
are vectors that define
the grid points in each dimension.
Vq = F({x1gq,x2gq,...,xngq})
For example, in 2D:
Vq = F({(0:0.2:10),(5:0.5:5)});
The griddedInterpolant
class accepts ndgrid
formatted
sample data. If you want to create a griddedInterpolant
with meshgrid
data,
you should convert it to the ndgrid
format first.
The following example outlines the steps for converting 2D meshgrid
data
to the ndgrid
format. We begin by creating the meshgrid
and
corresponding sample values:
[X,Y] = meshgrid(2:.1:2,1:.1:1); V=0.75*Y.^33*Y2*X.^2;
To convert X
, Y
, and V
to ndgrid
format,
follow these steps:
Transpose each array in the grid as well as the sample data.
X=X'; Y=Y'; V=V';
Now create the interpolant.
F = griddedInterpolant(X,Y,V);
To convert a 3D meshgrid
, use the permute
function.
Again, start by creating the meshgrid
and corresponding
sample
values:
[X,Y,Z] = meshgrid(5:5,3:3,10:10); V = X.^3 + Y.^2 + Z;
To convert X
, Y
, Z
,
and V
to ndgrid
format, follow
these steps:
Use the permute
function to interchange
the rows and columns of each array. The net effect will be the transpose
of every page.
P = [2 1 3]; X = permute(X,P); Y = permute(Y,P); Z = permute(Z,P); V = permute(V,P);
Now create the interpolant.
F = griddedInterpolant(X,Y,Z,V);
This example shows how to create and plot a 1D interpolant using griddedInterpolant
with a cubic interpolation method.
Create a coarse grid and sample values.
X = [1 2 3 4 5]; V = [12 6 15 9 6];
Construct the griddedInterpolant
using a cubic interpolation method.
F = griddedInterpolant(X,V,'cubic')
F = griddedInterpolant with properties: GridVectors: {[1 2 3 4 5]} Values: [12 6 15 9 6] Method: 'cubic' ExtrapolationMethod: 'cubic'
The GridVectors
property contains the compact grid specifying the coordinates of the sample values V
. The Method
property specifies the interpolation method. Notice that we specified 'cubic'
when we created F
. If you omit the Method
argument, the default interpolation method, linear
, will be assigned to F
.
You can access any of the properties of F
in the same way you would access the fields in a struct
.
F.GridVectors; % Displays the grid vectors as a cell array F.Values; % Displays the sample values F.Method; % Displays the interpolation method
Interpolate over finer intervals with 0.1 spacing.
Xq = (1:0.1:5); Vq = F(Xq);
Plot the result.
plot(X,V,'o'); hold on plot(Xq,Vq,''); legend('samples','Cubic Interpolation');
This example shows how to create and plot a 2D interpolant using griddedInterpolant
.
In two dimensions and higher, you can specify the sample coordinates as an ndgrid
, a compact grid, or a default grid. In this example, we'll supply an ndgrid
.
Create a coarse grid and sample values.
[X,Y] = ndgrid(1:.3:1,2:.3:2); V = 0.75*Y.^3  3*Y  2*X.^2;
Construct the griddedInterpolant
.
F = griddedInterpolant(X,Y,V,'spline');
Interpolate over finer intervals with 0.1 spacing.
[Xq,Yq] = ndgrid(1:.1:1,2:.1:2); Vq = F(Xq,Yq);
Plot the result.
figure()
surf(X,Y,V);
view(65,60)
title('Sample Data');
figure()
surf(Xq,Yq,Vq);
view(65,60)
title('Refined with Spline');
This example shows how to create a 3D interpolant and evaluate over a slice plane so you can plot the values on that plane.
Create a full grid and sample values.
[X,Y,Z] = ndgrid((5:2:5)); V = X.^3 + Y.^2 + Z.^2;
Construct the griddedInterpolant
.
F = griddedInterpolant(X,Y,Z,V,'cubic');
Since we already created the full grid to generate our sample values, we had nothing to lose in passing it to griddedInterpolant
. In practice however, it's common to load the sample data into MATLAB from disk. The compact grid can very beneficial in such cases because it allows you to specify the entire grid in a form that is much more economical in terms of memory. If we had loaded V
into MATLAB instead of calculating it from a full grid, we could have created a compact grid to conserve memory in our workspace. For example,
gv = {(5:2:5),(5:2:5),(5:2:5)};
F = griddedInterpolant(gv,V,'cubic');
Now interpolate over a plane at Z = 2
with 0.25 spacing.
[Xq,Yq,Zq] = ndgrid((5:.25:5),(5:.25:5),2:2); Vq = F(Xq,Yq,Zq);
Plot the result.
surf(Xq,Yq,Vq);
title('Refined with Linear Interpolation');
This example creates a 4D interpolant and evaluates it at a single point.
Create a coarse grid and sample values.
[X1,X2,X3,X4] = ndgrid((5:2:5)); V = X1.^3 + X2.^2 + X3.^2 +X4;
Construct the
griddedInterpolant
.
F = griddedInterpolant(X1,X2,X3,X4,V,'linear');
Query the griddedInterpolant
at a single
point.
Vq = F([3.2 2.1 4.7 1.3])
MATLAB outputs the following:
ans = 10.1000
.
Depending on the arrangement of your query points, you may prefer one evaluation syntax over the others. For example, create the following interpolant:
[X,Y] = ndgrid(1:.25:1,2:.25:2); V = 0.75*Y.^33*Y2*X.^2; F = griddedInterpolant(X,Y,V);
Query F
using a full grid to give the values at the grid
points:
[Xq,Yq] = ndgrid(1:.1:0,2:.1:0); Vq = F(Xq,Yq);
You can also interpolate over the same grid using the compact grid format:
gvq = {1:.1:0,2:.1:0}; Vq = F(gvq);
Or you can query a single point:
Vq = F(.315,.738)
which returns:
Vq = 2.1308
or a random set of scattered points:
rng('default') Vq = F(rand(3,2))
which returns:
Vq = 3.4919 3.3557 0.3515
You can also examine the Values
in
F
:
F.Values(1,3)
which returns:
ans = 0.0313
Or you can replace the Values
array:
F.Values = 2*V;
You can edit the properties in F
onthefly.
For example, you can replace the interpolation method as follows:
F.Method = 'cubic';
You can also replace the GridVectors
in F
. First,
examine
GridVectors
:
gv = F.GridVectors; gv{1}
gv
is a cell array, and gv{1}
displays the first grid
vector:
ans = 1.0000 0.7500 0.5000 0.2500 0 0.2500 0.5000 0.7500 1.0000
Now replace the GridVectors
in F
by creating a new cell
array
new_gv
:
new_gv = {(0:0.3:1),(0:0.3:1)}; F.GridVectors = new_gv;