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<h3 class="section">29.1 One-dimensional Interpolation</h3>
<p>Octave supports several methods for one-dimensional interpolation, most
of which are described in this section. <a href="Polynomial-Interpolation.html#Polynomial-Interpolation">Polynomial Interpolation</a>
and <a href="Interpolation-on-Scattered-Data.html#Interpolation-on-Scattered-Data">Interpolation on Scattered Data</a> describe further methods.
<!-- interp1 scripts/general/interp1.m -->
<p><a name="doc_002dinterp1"></a>
<div class="defun">
— Function File: <var>yi</var> = <b>interp1</b> (<var>x, y, xi</var>)<var><a name="index-interp1-2747"></a></var><br>
— Function File: <var>yi</var> = <b>interp1</b> (<var>y, xi</var>)<var><a name="index-interp1-2748"></a></var><br>
— Function File: <var>yi</var> = <b>interp1</b> (<var><small class="dots">...</small>, method</var>)<var><a name="index-interp1-2749"></a></var><br>
— Function File: <var>yi</var> = <b>interp1</b> (<var><small class="dots">...</small>, extrap</var>)<var><a name="index-interp1-2750"></a></var><br>
— Function File: <var>pp</var> = <b>interp1</b> (<var><small class="dots">...</small>, 'pp'</var>)<var><a name="index-interp1-2751"></a></var><br>
<blockquote>
<p>One-dimensional interpolation. Interpolate <var>y</var>, defined at the
points <var>x</var>, at the points <var>xi</var>. The sample points <var>x</var>
must be monotonic. If not specified, <var>x</var> is taken to be the
indices of <var>y</var>. If <var>y</var> is an array, treat the columns
of <var>y</var> separately.
<p>Method is one of:
<dl>
<dt>'nearest'<dd>Return the nearest neighbor.
<br><dt>'linear'<dd>Linear interpolation from nearest neighbors
<br><dt>'pchip'<dd>Piecewise cubic Hermite interpolating polynomial
<br><dt>'cubic'<dd>Cubic interpolation (same as <code>pchip</code>)
<br><dt>'spline'<dd>Cubic spline interpolation—smooth first and second derivatives
throughout the curve
</dl>
<p>Appending '*' to the start of the above method forces <code>interp1</code>
to assume that <var>x</var> is uniformly spaced, and only <var>x</var><code>
(1)</code> and <var>x</var><code> (2)</code> are referenced. This is usually faster,
and is never slower. The default method is 'linear'.
<p>If <var>extrap</var> is the string 'extrap', then extrapolate values beyond
the endpoints. If <var>extrap</var> is a number, replace values beyond the
endpoints with that number. If <var>extrap</var> is missing, assume NA.
<p>If the string argument 'pp' is specified, then <var>xi</var> should not be
supplied and <code>interp1</code> returns the piecewise polynomial that
can later be used with <code>ppval</code> to evaluate the interpolation.
There is an equivalence, such that <code>ppval (interp1 (</code><var>x</var><code>,
</code><var>y</var><code>, </code><var>method</var><code>, 'pp'), </code><var>xi</var><code>) == interp1 (</code><var>x</var><code>, </code><var>y</var><code>,
</code><var>xi</var><code>, </code><var>method</var><code>, 'extrap')</code>.
<p>Duplicate points in <var>x</var> specify a discontinuous interpolant. There
should be at most 2 consecutive points with the same value.
The discontinuous interpolant is right-continuous if <var>x</var> is increasing,
left-continuous if it is decreasing.
Discontinuities are (currently) only allowed for "nearest" and "linear"
methods; in all other cases, <var>x</var> must be strictly monotonic.
<p>An example of the use of <code>interp1</code> is
<pre class="example"> 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");
</pre>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dinterpft.html#doc_002dinterpft">interpft</a>.
</p></blockquote></div>
<p>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.
<p>This can be demonstrated by the code
<pre class="example"> 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');
</pre>
<p class="noindent">The result of which can be seen in <a href="fig_003ainterpderiv1.html#fig_003ainterpderiv1">fig:interpderiv1</a> and
<a href="fig_003ainterpderiv2.html#fig_003ainterpderiv2">fig:interpderiv2</a>.
<div class="float">
<a name="fig_003ainterpderiv1"></a><div align="center"><img src="interpderiv1.png" alt="interpderiv1.png"></div>
<p><strong class="float-caption">Figure 29.1: Comparison of 'pchip' and 'spline' interpolation methods for a
step function</strong></p></div>
<div class="float">
<a name="fig_003ainterpderiv2"></a><div align="center"><img src="interpderiv2.png" alt="interpderiv2.png"></div>
<p><strong class="float-caption">Figure 29.2: Comparison of the second derivative of the 'pchip' and 'spline'
interpolation methods for a step function</strong></p></div>
<p>A simplified version of <code>interp1</code> that performs only linear
interpolation is available in <code>interp1q</code>. This argument is slightly
faster than <code>interp1</code> as to performs little error checking.
<!-- interp1q scripts/general/interp1q.m -->
<p><a name="doc_002dinterp1q"></a>
<div class="defun">
— Function File: <var>yi</var> = <b>interp1q</b> (<var>x, y, xi</var>)<var><a name="index-interp1q-2752"></a></var><br>
<blockquote><p>One-dimensional linear interpolation without error checking.
Interpolates <var>y</var>, defined at the points <var>x</var>, at the points
<var>xi</var>. The sample points <var>x</var> must be a strictly monotonically
increasing column vector. If <var>y</var> is an array, treat the columns
of <var>y</var> separately. If <var>y</var> is a vector, it must be a column
vector of the same length as <var>x</var>.
<p>Values of <var>xi</var> beyond the endpoints of the interpolation result
in NA being returned.
<p>Note that the error checking is only a significant portion of the
execution time of this <code>interp1</code> if the size of the input arguments
is relatively small. Therefore, the benefit of using <code>interp1q</code>
is relatively small.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dinterp1.html#doc_002dinterp1">interp1</a>.
</p></blockquote></div>
<p>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.
<!-- interpft scripts/general/interpft.m -->
<p><a name="doc_002dinterpft"></a>
<div class="defun">
— Function File: <b>interpft</b> (<var>x, n</var>)<var><a name="index-interpft-2753"></a></var><br>
— Function File: <b>interpft</b> (<var>x, n, dim</var>)<var><a name="index-interpft-2754"></a></var><br>
<blockquote>
<p>Fourier interpolation. If <var>x</var> is a vector, then <var>x</var> is
resampled with <var>n</var> points. The data in <var>x</var> is assumed to be
equispaced. If <var>x</var> is an array, then operate along each column of
the array separately. If <var>dim</var> is specified, then interpolate
along the dimension <var>dim</var>.
<p><code>interpft</code> assumes that the interpolated function is periodic,
and so assumptions are made about the endpoints of the interpolation.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dinterp1.html#doc_002dinterp1">interp1</a>.
</p></blockquote></div>
<p>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 <code>interpft</code> is
<pre class="example"> 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');
</pre>
<p class="noindent">which demonstrates the poor behavior of Fourier interpolation for non-periodic
functions, as can be seen in <a href="fig_003ainterpft.html#fig_003ainterpft">fig:interpft</a>.
<div class="float">
<a name="fig_003ainterpft"></a><div align="center"><img src="interpft.png" alt="interpft.png"></div>
<p><strong class="float-caption">Figure 29.3: Comparison of <code>interp1</code> and <code>interpft</code> for non-periodic data</strong></p></div>
<p>In additional the support function <code>spline</code> and <code>lookup</code> that
underlie the <code>interp1</code> function can be called directly.
<a href="Finding-Elements-and-Checking-Conditions.html#Finding-Elements-and-Checking-Conditions">Finding Elements and Checking Conditions</a>
<!-- spline scripts/polynomial/spline.m -->
<p><a name="doc_002dspline"></a>
<div class="defun">
— Function File: <var>pp</var> = <b>spline</b> (<var>x, y</var>)<var><a name="index-spline-2755"></a></var><br>
— Function File: <var>yi</var> = <b>spline</b> (<var>x, y, xi</var>)<var><a name="index-spline-2756"></a></var><br>
<blockquote><p>Return the cubic spline interpolant of points <var>x</var> and <var>y</var>.
<p>When called with two arguments, return the piecewise polynomial <var>pp</var>
that may be used with <code>ppval</code> to evaluate the polynomial at specific
points. When called with a third input argument, <code>spline</code> evaluates
the spline at the points <var>xi</var>. The third calling form <code>spline
(</code><var>x</var><code>, </code><var>y</var><code>, </code><var>xi</var><code>)</code> is equivalent to <code>ppval (spline
(</code><var>x</var><code>, </code><var>y</var><code>), </code><var>xi</var><code>)</code>.
<p>The variable <var>x</var> must be a vector of length <var>n</var>. <var>y</var> can be
either a vector or array. If <var>y</var> is a vector it must have a length of
either <var>n</var> or <var>n</var><code> + 2</code>. If the length of <var>y</var> is
<var>n</var>, then the "not-a-knot" end condition is used. If the length of
<var>y</var> is <var>n</var><code> + 2</code>, then the first and last values of the
vector <var>y</var> are the values of the first derivative of the cubic spline
at the endpoints.
<p>If <var>y</var> is an array, then the size of <var>y</var> must have the form
<code>[</code><var>s1</var><code>, </code><var>s2</var><code>, ..., </code><var>sk</var><code>, </code><var>n</var><code>]</code>
or
<code>[</code><var>s1</var><code>, </code><var>s2</var><code>, ..., </code><var>sk</var><code>, </code><var>n</var><code> + 2]</code>.
The array is reshaped internally to a matrix where the leading
dimension is given by
<var>s1</var><code> * </code><var>s2</var><code> * ... * </code><var>sk</var>
and each row of this matrix is then treated separately. Note that this
is exactly opposite to <code>interp1</code> but is done for <span class="sc">matlab</span>
compatibility.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dpchip.html#doc_002dpchip">pchip</a>, <a href="doc_002dppval.html#doc_002dppval">ppval</a>, <a href="doc_002dmkpp.html#doc_002dmkpp">mkpp</a>, <a href="doc_002dunmkpp.html#doc_002dunmkpp">unmkpp</a>.
</p></blockquote></div>
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