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Next: <a rel="next" accesskey="n" href="Multi_002ddimensional-Interpolation.html#Multi_002ddimensional-Interpolation">Multi-dimensional Interpolation</a>,
Up: <a rel="up" accesskey="u" href="Interpolation.html#Interpolation">Interpolation</a>
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<h3 class="section">28.1 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.
<!-- ./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-interp1-2056"></a></var><br>
— Function File: <var>yi</var> = <b>interp1</b> (<var><small class="dots">...</small>, method</var>)<var><a name="index-interp1-2057"></a></var><br>
— Function File: <var>yi</var> = <b>interp1</b> (<var><small class="dots">...</small>, extrap</var>)<var><a name="index-interp1-2058"></a></var><br>
— Function File: <var>pp</var> = <b>interp1</b> (<var><small class="dots">...</small>, 'pp'</var>)<var><a name="index-interp1-2059"></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 strictly monotonic. 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>Piece-wise cubic hermite interpolating polynomial
<br><dt>'cubic'<dd>Cubic interpolation from four nearest neighbors
<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 piece-wise 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>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>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.
<!-- ./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-2060"></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.
<!-- ./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-2061"></a></var><br>
— Function File: <b>interpft</b> (<var>x, n, dim</var>)<var><a name="index-interpft-2062"></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 end points 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.
<p>In additional the support function <code>spline</code> and <code>lookup</code> that
underlie the <code>interp1</code> function can be called directly.
<!-- ./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-2063"></a></var><br>
— Function File: <var>yi</var> = <b>spline</b> (<var>x, y, xi</var>)<var><a name="index-spline-2064"></a></var><br>
<blockquote>
<p>Return the cubic spline interpolant of <var>y</var> at points <var>x</var>.
If called with two arguments, <code>spline</code> returns the piece-wise
polynomial <var>pp</var> that may later be used with <code>ppval</code> to
evaluate the polynomial at specific points.
If called with a third input argument, <code>spline</code> evaluates the
spline at the points <var>xi</var>. There is an equivalence
between <code>ppval (spline (</code><var>x</var><code>, </code><var>y</var><code>), </code><var>xi</var><code>)</code> and
<code>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>, and <var>y</var>
can be either a vector or array. In the case where <var>y</var> is a
vector, it can 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 end-points.
<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 then 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 the opposite treatment than <code>interp1</code> and is done
for compatibility.
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<p class="noindent"><strong>See also:</strong> <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>
<p>The <code>lookup</code> function is used by other interpolation functions to identify
the points of the original data that are closest to the current point
of interest.
<!-- ./DLD-FUNCTIONS/lookup.cc -->
<p><a name="doc_002dlookup"></a>
<div class="defun">
— Loadable Function: <var>idx</var> = <b>lookup</b> (<var>table, y, opt</var>)<var><a name="index-lookup-2065"></a></var><br>
<blockquote><p>Lookup values in a sorted table. Usually used as a prelude to
interpolation.
<p>If table is strictly increasing and <code>idx = lookup (table, y)</code>, then
<code>table(idx(i)) <= y(i) < table(idx(i+1))</code> for all <code>y(i)</code>
within the table. If <code>y(i) < table (1)</code> then
<code>idx(i)</code> is 0. If <code>y(i) >= table(end)</code> then
<code>idx(i)</code> is <code>table(n)</code>.
<p>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.
<p>The algorithm used by lookup is standard binary search, with optimizations
to speed up the case of partially ordered arrays (dense downsampling).
In particular, looking up a single entry is of logarithmic complexity
(unless a conversion occurs due to non-numeric or unequal types).
<p><var>table</var> and <var>y</var> can also be cell arrays of strings
(or <var>y</var> can be a single string). In this case, string lookup
is performed using lexicographical comparison.
<p>If <var>opts</var> is specified, it shall be a string with letters indicating
additional options.
For numeric lookup, 'l' in <var>opts</var> indicates that
the leftmost subinterval shall be extended to infinity (i.e., all indices
at least 1), and 'r' indicates that the rightmost subinterval shall be
extended to infinity (i.e., all indices at most n-1).
<p>For string lookup, 'i' indicates case-insensitive comparison.
</p></blockquote></div>
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