Next: Multi-dimensional Interpolation, Up: Interpolation
Octave supports several methods for one-dimensional interpolation, most of which are described in this section. Polynomial Interpolation and Interpolation on Scattered Data describes further methods.
One-dimensional interpolation. Interpolate y, defined at the points x, at the points xi. The sample points x must be strictly monotonic. If y is an array, treat the columns of y seperately.
Method is one of:
- 'nearest'
- Return the nearest neighbour.
- 'linear'
- Linear interpolation from nearest neighbours
- 'pchip'
- Piece-wise cubic hermite interpolating polynomial
- 'cubic'
- Cubic interpolation from four nearest neighbours
- 'spline'
- Cubic spline interpolation–smooth first and second derivatives throughout the curve
Appending '*' to the start of the above method forces
interp1
to assume that x is uniformly spaced, and only x(1)
and x(2)
are referenced. This is usually faster, and is never slower. The default method is 'linear'.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 supplied and
interp1
returns the piece-wise polynomial that can later be used withppval
to evaluate the interpolation. There is an equivalence, such thatppval (interp1 (
x,
y,
method, 'pp'),
xi) == interp1 (
x,
y,
xi,
method, 'extrap')
.An example of the use of
interp1
isxf=[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 methods. The '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 discontinuous, then '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');
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 an array, then operate along each column of the array seperately. If dim is specified, then interpolate along the dimension dim.
interpft
assumes that the interpolated function is periodic, and so assumption are made about the end points of the inetrpolation.See also: interp1.
There are two significant limitations on Fourier interpolation. Firstly,
the function signal is assumed to be periodic, and so no 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 interpft
is
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.
In additional the support function spline
and lookup
that
underlie the interp1
function can be called directly.
Returns the cubic spline interpolation of y at the point x. Called with two arguments the piece-wise polynomial pp that may later be used with
ppval
to evaluate the polynomial at specific points.The variable x must be a vector of length n, and y can be either a vector or array. In the case where y is a vector, it can 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 y is n+ 2
, then the first and last values of the vector y are the first derivative of the cubic spline at the end-points.If y is an array, then the size of y must have the form
[
s1,
s2, ...,
sk,
n]
or[
s1,
s2, ...,
sk,
n+ 2]
. The array is then reshaped internally to a matrix where to leading dimension is given by s1*
s2* ... *
sk and each row this matrix is then treated seperately. Note that this is exactly the opposite treatment thaninterp1
and is done for compatiability.Called with a third input argument,
spline
evaluates the piece-wise spline at the points xi. There is an equivalence betweenppval (spline (
x,
y),
xi)
andspline (
x,
y,
xi)
.See also: ppval, mkpp, unmkpp.
The lookup
is used by other interpolation function to identify
the points of the original data that are closest to the current point
of interest.
Lookup values in a sorted table. Usually used as a prelude to interpolation.
If table is strictly increasing and
idx = lookup (table, y)
, thentable(idx(i)) <= y(i) < table(idx(i+1))
for ally(i)
within the table. Ify(i)
is before the table, thenidx(i)
is 0. Ify(i)
is after the table thenidx(i)
istable(n)
.If the table is strictly decreasing, then the tests are reversed. There are no guarantees for tables which are non-monotonic or are not strictly monotonic.
To get an index value which lies within an interval of the table, use:
idx = lookup (table(2:length(table)-1), y) + 1This expression puts values before the table into the first interval, and values after the table into the last interval.