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MATLAB Interpolation

Interpolation is the process of describing a function which “connects the dots” between specified (data) points. The most common interpolation technique is Linear Interpolation.

A more exotic interpolation scheme is to link the data points using third degree or cubic polynomials.

In MATLAB, we can interpolate our data using splines or Hermite interpolants on a fly.

MATLAB provides the following functions to help interpolation:

• interp1
• interp2
• interp3
• spline
• interpfit

Let us understand these functions of MATLAB one by one:

interp1

One-dimensional data interpolation.

Given y_i at x_i, finds y_j at x_j from y_j=f(xj ). Here f is a continuous function that is seen from interpolation. It is called one-dimensional interpolation because y depends on a single variable x.

Syntax

ynew=interp1(x,y,xnew,method)

where the method is an optional argument discussed after the description of interp2 and interp3.

interp2

Two-dimensional data interpolation.

Given z_i at (x_i, y_i), finds z_j at desired (x_j,y_j )from z=f(x,y).The function f is found from interpolation. It is called two-dimensional interpolation because z depends on two variables, x, and y.

Syntax

znew=interp2(x,y,z,xnew,ynew,method).

interp3

Three-dimensional analogue of interp1.

Given v_i at (x_i, y_i,z_i), finds v_j at desired(x_j,y_j,z_j).

Syntax

vnew=interp3(x,y,z,xnew,ynew,znew,method).

Also, there is an n-dimensional analog, interpn, if you ever need it. In each function, we have the option of specifying a method of interpolation. The choice for the technique is nearest, linear, cubic, or spline. The choice of the method dictates the smoothness of the interpolated data. The default method is linear.

To specify cubic interpolation instead of linear, for example, in interp1, use the syntax:

ynew=interp1(x,y,xnew,’ cubic’ ).

spline

One-dimensional interpolation that uses cubic splines to find y_j at desired x_j,given y_i at x_i. Cubic splines fit separate cubic polynomials between successive data points by matching the slopes as well as the curvature of each segment at the given data points.

Syntax

ynew=spline(x,y,xnew,method).

interpfit

Fast Fourier Transform (FFT)-based 1-D data interpolation. This is similar to interp1 except that the data is interpolated first by taking the Fourier Transform of the given data and then calculating the inverse transform using more data points. The interpolation is especially useful for periodic functions (i.e., if values of y are periodic).

Syntax

y = interpft(X,n)
y = interpft(X,n,dim)

Example

There are two simple steps contained in interpolation: providing a list (a vector) of points at which you wish to get interpolated data and executing the appropriate function (e.g., interp1) with the desired choice for the method of interpolation. We illustrate these steps through an example on the x and y data given in the following table:

x	0         0.785         1.570         2.356         3.141         3.927         4.712         5.497         6.283
y	0         0.707         1.000         0.707         0.000         -0.707       – 1.000        -0.707        -0.000

Step1: Generate a vector x_i containing desired points for interpolation.

% take equally spaced fifty points.
x_i=linspace(0, 2*pi, 50);

Step2: Generate data y_i at x_i.

% generate y_i at x_i with cubic interpolation.
y_i=interp1(x,y,x_i ,’ cubic’ );

Here ‘cubic’ is the choice for the interpolation scheme. The other schemes we could use are nearest, linear, and spline. The data generated by each scheme is shown in fig, along with the original input.

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