Interpolation in MATLAB

Mathematics

one-dimensional interpolation

There is 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 Eval Uating the appropriate function at the desired interpolation points. Its more general form is

• Yi = Interp1 (x,y,xi, method )

yis a vector containing the values of a function, and are x a vector of the same length containing the points for which The values in was y given. is xi a vector containing the points at which to Interpolate. was an method optional strin G Specifying an interpolation method:

• Nearest neighbor interpolation (method = ‘nearest‘). This method sets the value of a 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 Relev Ant function at the points specified by xi . The default method for the interp1 function.
• Cubic spline interpolation (method = ‘spline‘). This method fits a different cubic function between all pair of existing data points, and uses the spline function to perf ORM Cubic spline interpolation at the data points.
• Cubic interpolation ( method = ‘pchip‘ or ‘cubic‘ ). These methods is 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 was outside the interval spanned x by, the specified interpolation method are used for Extrapola tion. Alternatively, yi = interp1(x,Y,xi,method,extrapval) replaces extrapolated values with extrapval . Are NaN often used for extrapval .

All methods work with nonuniformly spaced data.

Speed, Memory, and smoothness considerations

When choosing a interpolation method, keep in mind, some require more memory or longer computation time than other S. However, 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 is continuous, but the slope changes at the vertex points.
• Cubic spline interpolation has the longest relative execution time, although it r Equires less memory than cubic interpolation. It produces the smoothest results of all the interpolation methods. Obtain unexpected results, however, if your input data is non-uniform and some points are much closer together tha n 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 is 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)

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

MATLAB Function Reference

INTERP1

One-dimensional data interpolation (table lookup)

Syntax

• Yi = Interp1 (x,y,xi) Yi = interp1 (y,xi) Yi = interp1 (x,y,xi,method) Yi = Interp1 (x,y,xi,method, ' extrap ') Yi = Interp1 (x,y,xi , Method,extrapval)

Description

yi = interp1(x,Y,xi) Returns vector yi containing elements corresponding to the elements of and xi determined by interpolation within VEC Tors x and Y . The vector x specifies the points at which, the data is Y given. If Y is a matrix, then the interpolation are performed for each column of and is Y yi length(xi) -by- size(Y,2) .

yi = interp1(Y,xi) Assumes x = 1:N that, where is N Y the length Y of a for vector, or for size(Y,1) matrix Y .

yi = interp1(x,Y,xi,method) Interpolates using alternative methods:

‘nearest‘	Nearest neighbor interpolation
‘linear‘	Linear interpolation (default)
‘spline‘	Cubic spline interpolation
‘pchip‘	piecewise Cubic Hermite Interpolation
‘cubic‘	(Same as ' pchip‘ )
‘v5cubic‘	Cubic interpolation used in MATLAB 5

For ‘nearest‘ the, ‘linear‘ , and ‘v5cubic‘ methods, returns for any element of which is interp1(x,Y,xi,method) outside the NaN xi interval sp Anned by x . For all and methods, interp1 performs extrapolation for out of range values.

yi = interp1(x,Y,xi,method,‘extrap‘) Uses the specified method to perform extrapolation for out of range values.

yi = interp1(x,Y,xi,method,extrapval) Returns the scalar for out of extrapval range values. And is NaN 0 often used for extrapval .

The interp1 command interpolates between data points. It finds values at intermediate points, for a one-dimensional function that underlies the data. This function was shown below, along with the relationship between vectors,,, and x Y xi yi .

Interpolation is the same operation as table lookup. Described in table lookup terms, the table was [x,Y] and interp1 looks xi x up the elements of In, a ND, based upon their locations, returns values interpolated within the elements of yi Y .

Note interp1q is quicker than on interp1 non-uniformly spaced data because it does no input checking. interp1qfor-to-work properly, x must is a monotonically increasing column vector and Y must be a column vector or MA Trix with length(X) rows. The Type at the command line is more help interp1q information.

Examples

Example 1. Generate a coarse sine curve and interpolate over a finer abscissa.

• x = 0:10; y = sin (x); Xi = 0:.25:10; Yi = Interp1 (x,y,xi); Plot (x, y, ' o ', xi,yi)

• With ' spline ' method:

x = 0:10;

y = sin (x);

Xi = 0:.25:10;

Yi = interp1 (x,y,xi, 'spline');  

figure;plot (x, y, ' o ', xi,yi)

　　

Example 2. Here is the vectors representing the census years from 1900 to 1990 and the corresponding of the states population in mil Lions of people.

• t = 1900:10:1990;p = [75.995  91.972  105.711  123.203  131.669 ...     150.697  179.323  203.212  226.505  249.633];

The expression interp1(t,p,1975) interpolates within the census data to estimate the population in 1975. The result is

• Ans =    214.8585

Now interpolate within the data at every year from 1900 to, and plot the result.

• x = 1900:1:2000; y = Interp1 (t,p,x, ' spline '); Plot (t,p, ' O ', x, y)

Sometimes it is more convenient to think of interpolation in table lookup terms, where the data was stored in a s Ingle table. If A portion of the census data is stored with a single 5-by-2 table,

• tab =    1950    150.697    1960    179.323    1970    203.212 1980 226.505 1990 249.633

Then the population in 1975, obtained by table lookup within the matrix tab , is

• p = INTERP1 (tab (:, 1), Tab (:, 2), 1975) P =    214.8585

Algorithm

The interp1 command is a MATLAB m-file. The ' and nearest‘ ' linear‘ methods have straightforward implementations.

spline‘for the "method, interp1 calls a function that spline uses the functions ppval , mkpp and unmkpp . These routines form a small suite of functions for working with piecewise polynomials. splineuses them to perform the cubic spline interpolation. For access to more advanced features, see the spline reference page, the M-file Help for these functions, and the Spline T Oolbox.

pchip‘ ‘cubic‘ for the ' and methods, interp1 calls a function that pchip performs piecewise cubic interpolation within the Vect ORS x and y . This method preserves monotonicity and the shape of the data. See the pchip Reference page for more information.

See Also

interpft, interp2, interp3, interpn, pchip,spline

References

[1] de Boor, C., A PracticalGuide to Splines, Springer-verlag, 1978.

Interpolation in MATLAB