Mathematics

One-Dimensional Interpolation

There are two kinds of one-dimensional interpolation in MATLAB:

• Polynomial interpolation
• FFT-based interpolation

Polynomial Interpolation

The function interp1 performs one-dimensional interpolation, an important operation for data analysis and curve fitting. This function uses polynomial techniques, fitting the supplied data with polynomial functions between data points and evaluating the appropriate function at the desired interpolation points. Its most general form is

• yi = interp1(x,y,xi,method)
  

y is a vector containing the values of a function, and x is a vector of the same length containing the points for which the values in y are given. xi is a vector containing the points at which to interpolate. method is an optional string specifying an interpolation method:

• Nearest neighbor interpolation (method = 'nearest'). This method sets the value of an interpolated point to the value of the nearest existing data point.
• Linear interpolation (method = 'linear'). This method fits a different linear function between each pair of existing data points, and returns the value of the relevant function at the points specified by xi. This is the default method for the interp1 function.
• Cubic spline interpolation (method = 'spline'). This method fits a different cubic function between each pair of existing data points, and uses the spline function to perform cubic spline interpolation at the data points.
• Cubic interpolation (method = 'pchip' or 'cubic'). These methods are identical. They use the pchip function to perform piecewise cubic Hermite interpolation within the vectors x and y. These methods preserve monotonicity and the shape of the data.

If any element of xi is outside the interval spanned by x, the specified interpolation method is used for extrapolation. Alternatively, yi = interp1(x,Y,xi,method,extrapval) replaces extrapolated values with extrapval. NaN is often used for extrapval.

All methods work with nonuniformly spaced data.

Speed, Memory, and Smoothness Considerations

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.

• Nearest neighbor interpolation is the fastest method. However, it provides the worst results in terms of smoothness.
• Linear interpolation uses more memory than the nearest neighbor method, and requires slightly more execution time. Unlike nearest neighbor interpolation its results are continuous, but the slope changes at the vertex points.
• Cubic spline interpolation has the longest relative execution time, although it requires less memory than cubic interpolation. It produces the smoothest results of all the interpolation methods. You may obtain unexpected results, however, if your input data is non-uniform and some points are much closer together than others.
• Cubic interpolation requires more memory and execution time than either the nearest neighbor or linear methods. However, both the interpolated data and its derivative are continuous.

The relative performance of each method holds true even for interpolation of two-dimensional or multidimensional data. For a graphical comparison of interpolation methods, see the section Comparing Interpolation Methods.

FFT-Based Interpolation

The function interpft performs one-dimensional interpolation using an FFT-based method. This method calculates the Fourier transform of a vector that contains the values of a periodic function. It then calculates the inverse Fourier transform using more points. Its form is

• y = interpft(x,n)
  

x is a vector containing the values of a periodic function, sampled at equally spaced points. n is the number of equally spaced points to return.

Interpolation Function Summary

Two-Dimensional Interpolation