Octave supports several methods for one-dimensional interpolation, most of which are described in this section. Polynomial Interpolation and Interpolation on Scattered Data describe additional methods.
One-dimensional interpolation. Interpolates to determine the value of yi at the points, xi. If not specified, x is taken to be the indices of y. If y is a matrix or an N-dimensional array, the interpolation is performed on each column of y.
Method is one of:
Return the nearest neighbor.
Linear interpolation from nearest neighbors
Piecewise cubic Hermite interpolating polynomial
Cubic interpolation (same as
Cubic spline interpolation—smooth first and second derivatives throughout the curve
Appending ’*’ to the start of the above method forces
to assume that x is uniformly spaced, and only
x(2) are referenced. This is usually faster,
and is never slower. The default method is
If extrap is the string
"extrap", then extrapolate values
beyond the endpoints. If extrap is a number, replace values beyond
the endpoints with that number. If extrap is missing, assume NA.
If the string argument
"pp" is specified, then xi should not be
interp1 returns the piecewise polynomial that
can later be used with
ppval to evaluate the interpolation.
There is an equivalence, such that
ppval (interp1 (x,
y, method, .
"pp"), xi) == interp1 (x, y,
Duplicate points in x specify a discontinuous interpolant. There
may be at most 2 consecutive points with the same value.
If x is increasing, the default discontinuous interpolant is
right-continuous. If x is decreasing, the default discontinuous
interpolant is left-continuous.
The continuity condition of the interpolant may be specified by using
"-right", to select a left-continuous
or right-continuous interpolant, respectively.
Discontinuous interpolation is only allowed for
"linear" methods; in all other cases, the x-values must be
An example of the use of
xf = [0:0.05:10]; yf = sin (2*pi*xf/5); xp = [0:10]; yp = sin (2*pi*xp/5); lin = interp1 (xp, yp, xf); spl = interp1 (xp, yp, xf, "spline"); cub = interp1 (xp, yp, xf, "cubic"); near = interp1 (xp, yp, xf, "nearest"); plot (xf, yf, "r", xf, lin, "g", xf, spl, "b", xf, cub, "c", xf, near, "m", xp, yp, "r*"); legend ("original", "linear", "spline", "cubic", "nearest");
See also: interpft.
There are some important differences between the various interpolation
"spline" method enforces that both the first and second
derivatives of the interpolated values have a continuous derivative,
whereas the other methods do not. This means that the results of the
"spline" method are generally smoother. If the function to be
interpolated is in fact smooth, then
"spline" will give excellent
results. However, if the function to be evaluated is in some manner
"pchip" interpolation might give better results.
This can be demonstrated by the code
t = -2:2; dt = 1; ti =-2:0.025:2; dti = 0.025; y = sign (t); ys = interp1 (t,y,ti,"spline"); yp = interp1 (t,y,ti,"pchip"); ddys = diff (diff (ys)./dti) ./ dti; ddyp = diff (diff (yp)./dti) ./ dti; figure (1); plot (ti,ys,"r-", ti,yp,"g-"); legend ("spline", "pchip", 4); figure (2); plot (ti,ddys,"r+", ti,ddyp,"g*"); legend ("spline", "pchip");
The result of which can be seen in Figure 29.1 and Figure 29.2.
Fourier interpolation, is a resampling technique where a signal is converted to the frequency domain, padded with zeros and then reconverted to the time domain.
Fourier interpolation. If x is a vector, then x is resampled with n points. The data in x is assumed to be equispaced. If x is a matrix or an N-dimensional array, the interpolation is performed on each column of x. If dim is specified, then interpolate along the dimension dim.
interpft assumes that the interpolated function is periodic,
and so assumptions are made about the endpoints of the interpolation.
See also: interp1.
There are two significant limitations on Fourier interpolation. Firstly,
the function signal is assumed to be periodic, and so non-periodic
signals will be poorly represented at the edges. Secondly, both the
signal and its interpolation are required to be sampled at equispaced
points. An example of the use of
t = 0 : 0.3 : pi; dt = t(2)-t(1); n = length (t); k = 100; ti = t(1) + [0 : k-1]*dt*n/k; y = sin (4*t + 0.3) .* cos (3*t - 0.1); yp = sin (4*ti + 0.3) .* cos (3*ti - 0.1); plot (ti, yp, "g", ti, interp1 (t, y, ti, "spline"), "b", ... ti, interpft (y, k), "c", t, y, "r+"); legend ("sin(4t+0.3)cos(3t-0.1)", "spline", "interpft", "data");
which demonstrates the poor behavior of Fourier interpolation for non-periodic functions, as can be seen in Figure 29.3.
In addition, the support functions
interp1 function can be called directly.
Return the cubic spline interpolant of points x and y.
When called with two arguments, return the piecewise polynomial pp
that may be used with
ppval to evaluate the polynomial at specific
points. When called with a third input argument,
the spline at the points xi. The third calling form
(x, y, xi) is equivalent to
(x, y), xi).
The variable x must be a vector of length n. y can be
either a vector or array. If y is a vector it must have a length of
either n or
n + 2. If the length of y is
n, then the "not-a-knot" end condition is used. If the length of
n + 2, then the first and last values of the
vector y are the values of the first derivative of the cubic spline
at the endpoints.
If y is an array, then the size of y must have the form
[s1, s2, …, sk, n]
[s1, s2, …, sk, n + 2].
The array is reshaped internally to a matrix where the leading
dimension is given by
s1 * s2 * … * sk
and each row of this matrix is then treated separately. Note that this
is exactly opposite to
interp1 but is done for MATLAB
See also: pchip, ppval, mkpp, unmkpp.