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<span id="Multi_002ddimensional-Interpolation"></span><div class="header">
<p>
Previous: <a href="One_002ddimensional-Interpolation.html" accesskey="p" rel="prev">One-dimensional Interpolation</a>, Up: <a href="Interpolation.html" accesskey="u" rel="up">Interpolation</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
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<hr>
<span id="Multi_002ddimensional-Interpolation-1"></span><h3 class="section">29.2 Multi-dimensional Interpolation</h3>
<p>There are three multi-dimensional interpolation functions in Octave, with
similar capabilities. Methods using Delaunay tessellation are described
in <a href="Interpolation-on-Scattered-Data.html">Interpolation on Scattered Data</a>.
</p>
<span id="XREFinterp2"></span><dl>
<dt id="index-interp2">: <em><var>zi</var> =</em> <strong>interp2</strong> <em>(<var>x</var>, <var>y</var>, <var>z</var>, <var>xi</var>, <var>yi</var>)</em></dt>
<dt id="index-interp2-1">: <em><var>zi</var> =</em> <strong>interp2</strong> <em>(<var>z</var>, <var>xi</var>, <var>yi</var>)</em></dt>
<dt id="index-interp2-2">: <em><var>zi</var> =</em> <strong>interp2</strong> <em>(<var>z</var>, <var>n</var>)</em></dt>
<dt id="index-interp2-3">: <em><var>zi</var> =</em> <strong>interp2</strong> <em>(<var>z</var>)</em></dt>
<dt id="index-interp2-4">: <em><var>zi</var> =</em> <strong>interp2</strong> <em>(…, <var>method</var>)</em></dt>
<dt id="index-interp2-5">: <em><var>zi</var> =</em> <strong>interp2</strong> <em>(…, <var>method</var>, <var>extrap</var>)</em></dt>
<dd>
<p>Two-dimensional interpolation.
</p>
<p>Interpolate reference data <var>x</var>, <var>y</var>, <var>z</var> to determine <var>zi</var>
at the coordinates <var>xi</var>, <var>yi</var>. The reference data <var>x</var>, <var>y</var>
can be matrices, as returned by <code>meshgrid</code>, in which case the sizes of
<var>x</var>, <var>y</var>, and <var>z</var> must be equal. If <var>x</var>, <var>y</var> are
vectors describing a grid then <code>length (<var>x</var>) == columns (<var>z</var>)</code>
and <code>length (<var>y</var>) == rows (<var>z</var>)</code>. In either case the input
data must be strictly monotonic.
</p>
<p>If called without <var>x</var>, <var>y</var>, and just a single reference data matrix
<var>z</var>, the 2-D region
<code><var>x</var> = 1:columns (<var>z</var>), <var>y</var> = 1:rows (<var>z</var>)</code> is assumed.
This saves memory if the grid is regular and the distance between points is
not important.
</p>
<p>If called with a single reference data matrix <var>z</var> and a refinement
value <var>n</var>, then perform interpolation over a grid where each original
interval has been recursively subdivided <var>n</var> times. This results in
<code>2^<var>n</var>-1</code> additional points for every interval in the original
grid. If <var>n</var> is omitted a value of 1 is used. As an example, the
interval [0,1] with <code><var>n</var>==2</code> results in a refined interval with
points at [0, 1/4, 1/2, 3/4, 1].
</p>
<p>The interpolation <var>method</var> is one of:
</p>
<dl compact="compact">
<dt><code>"nearest"</code></dt>
<dd><p>Return the nearest neighbor.
</p>
</dd>
<dt><code>"linear"</code> (default)</dt>
<dd><p>Linear interpolation from nearest neighbors.
</p>
</dd>
<dt><code>"pchip"</code></dt>
<dd><p>Piecewise cubic Hermite interpolating polynomial—shape-preserving
interpolation with smooth first derivative.
</p>
</dd>
<dt><code>"cubic"</code></dt>
<dd><p>Cubic interpolation (same as <code>"pchip"</code>).
</p>
</dd>
<dt><code>"spline"</code></dt>
<dd><p>Cubic spline interpolation—smooth first and second derivatives
throughout the curve.
</p></dd>
</dl>
<p><var>extrap</var> is a scalar number. It replaces values beyond the endpoints
with <var>extrap</var>. Note that if <var>extrap</var> is used, <var>method</var> must
be specified as well. If <var>extrap</var> is omitted and the <var>method</var> is
<code>"spline"</code>, then the extrapolated values of the <code>"spline"</code> are
used. Otherwise the default <var>extrap</var> value for any other <var>method</var>
is <code>"NA"</code>.
</p>
<p><strong>See also:</strong> <a href="One_002ddimensional-Interpolation.html#XREFinterp1">interp1</a>, <a href="#XREFinterp3">interp3</a>, <a href="#XREFinterpn">interpn</a>, <a href="Three_002dDimensional-Plots.html#XREFmeshgrid">meshgrid</a>.
</p></dd></dl>
<span id="XREFinterp3"></span><dl>
<dt id="index-interp3">: <em><var>vi</var> =</em> <strong>interp3</strong> <em>(<var>x</var>, <var>y</var>, <var>z</var>, <var>v</var>, <var>xi</var>, <var>yi</var>, <var>zi</var>)</em></dt>
<dt id="index-interp3-1">: <em><var>vi</var> =</em> <strong>interp3</strong> <em>(<var>v</var>, <var>xi</var>, <var>yi</var>, <var>zi</var>)</em></dt>
<dt id="index-interp3-2">: <em><var>vi</var> =</em> <strong>interp3</strong> <em>(<var>v</var>, <var>n</var>)</em></dt>
<dt id="index-interp3-3">: <em><var>vi</var> =</em> <strong>interp3</strong> <em>(<var>v</var>)</em></dt>
<dt id="index-interp3-4">: <em><var>vi</var> =</em> <strong>interp3</strong> <em>(…, <var>method</var>)</em></dt>
<dt id="index-interp3-5">: <em><var>vi</var> =</em> <strong>interp3</strong> <em>(…, <var>method</var>, <var>extrapval</var>)</em></dt>
<dd>
<p>Three-dimensional interpolation.
</p>
<p>Interpolate reference data <var>x</var>, <var>y</var>, <var>z</var>, <var>v</var> to determine
<var>vi</var> at the coordinates <var>xi</var>, <var>yi</var>, <var>zi</var>. The reference
data <var>x</var>, <var>y</var>, <var>z</var> can be matrices, as returned by
<code>meshgrid</code>, in which case the sizes of <var>x</var>, <var>y</var>, <var>z</var>, and
<var>v</var> must be equal. If <var>x</var>, <var>y</var>, <var>z</var> are vectors describing
a cubic grid then <code>length (<var>x</var>) == columns (<var>v</var>)</code>,
<code>length (<var>y</var>) == rows (<var>v</var>)</code>, and
<code>length (<var>z</var>) == size (<var>v</var>, 3)</code>. In either case the input
data must be strictly monotonic.
</p>
<p>If called without <var>x</var>, <var>y</var>, <var>z</var>, and just a single reference
data matrix <var>v</var>, the 3-D region
<code><var>x</var> = 1:columns (<var>v</var>), <var>y</var> = 1:rows (<var>v</var>),
<var>z</var> = 1:size (<var>v</var>, 3)</code> is assumed.
This saves memory if the grid is regular and the distance between points is
not important.
</p>
<p>If called with a single reference data matrix <var>v</var> and a refinement
value <var>n</var>, then perform interpolation over a 3-D grid where each
original interval has been recursively subdivided <var>n</var> times. This
results in <code>2^<var>n</var>-1</code> additional points for every interval in the
original grid. If <var>n</var> is omitted a value of 1 is used. As an
example, the interval [0,1] with <code><var>n</var>==2</code> results in a refined
interval with points at [0, 1/4, 1/2, 3/4, 1].
</p>
<p>The interpolation <var>method</var> is one of:
</p>
<dl compact="compact">
<dt><code>"nearest"</code></dt>
<dd><p>Return the nearest neighbor.
</p>
</dd>
<dt><code>"linear"</code> (default)</dt>
<dd><p>Linear interpolation from nearest neighbors.
</p>
</dd>
<dt><code>"cubic"</code></dt>
<dd><p>Piecewise cubic Hermite interpolating polynomial—shape-preserving
interpolation with smooth first derivative (not implemented yet).
</p>
</dd>
<dt><code>"spline"</code></dt>
<dd><p>Cubic spline interpolation—smooth first and second derivatives
throughout the curve.
</p></dd>
</dl>
<p><var>extrapval</var> is a scalar number. It replaces values beyond the endpoints
with <var>extrapval</var>. Note that if <var>extrapval</var> is used, <var>method</var>
must be specified as well. If <var>extrapval</var> is omitted and the
<var>method</var> is <code>"spline"</code>, then the extrapolated values of the
<code>"spline"</code> are used. Otherwise the default <var>extrapval</var> value for
any other <var>method</var> is <code>"NA"</code>.
</p>
<p><strong>See also:</strong> <a href="One_002ddimensional-Interpolation.html#XREFinterp1">interp1</a>, <a href="#XREFinterp2">interp2</a>, <a href="#XREFinterpn">interpn</a>, <a href="Three_002dDimensional-Plots.html#XREFmeshgrid">meshgrid</a>.
</p></dd></dl>
<span id="XREFinterpn"></span><dl>
<dt id="index-interpn">: <em><var>vi</var> =</em> <strong>interpn</strong> <em>(<var>x1</var>, <var>x2</var>, …, <var>v</var>, <var>y1</var>, <var>y2</var>, …)</em></dt>
<dt id="index-interpn-1">: <em><var>vi</var> =</em> <strong>interpn</strong> <em>(<var>v</var>, <var>y1</var>, <var>y2</var>, …)</em></dt>
<dt id="index-interpn-2">: <em><var>vi</var> =</em> <strong>interpn</strong> <em>(<var>v</var>, <var>m</var>)</em></dt>
<dt id="index-interpn-3">: <em><var>vi</var> =</em> <strong>interpn</strong> <em>(<var>v</var>)</em></dt>
<dt id="index-interpn-4">: <em><var>vi</var> =</em> <strong>interpn</strong> <em>(…, <var>method</var>)</em></dt>
<dt id="index-interpn-5">: <em><var>vi</var> =</em> <strong>interpn</strong> <em>(…, <var>method</var>, <var>extrapval</var>)</em></dt>
<dd>
<p>Perform <var>n</var>-dimensional interpolation, where <var>n</var> is at least two.
</p>
<p>Each element of the <var>n</var>-dimensional array <var>v</var> represents a value
at a location given by the parameters <var>x1</var>, <var>x2</var>, …, <var>xn</var>.
The parameters <var>x1</var>, <var>x2</var>, …, <var>xn</var> are either
<var>n</var>-dimensional arrays of the same size as the array <var>v</var> in
the <code>"ndgrid"</code> format or vectors.
</p>
<p>The parameters <var>y1</var>, <var>y2</var>, …, <var>yn</var> represent the points at
which the array <var>vi</var> is interpolated. They can be vectors of the same
length and orientation in which case they are interpreted as coordinates of
scattered points. If they are vectors of differing orientation or length,
they are used to form a grid in <code>"ndgrid"</code> format. They can also be
<var>n</var>-dimensional arrays of equal size.
</p>
<p>If <var>x1</var>, …, <var>xn</var> are omitted, they are assumed to be
<code>x1 = 1 : size (<var>v</var>, 1)</code>, etc. If <var>m</var> is specified, then
the interpolation adds a point half way between each of the interpolation
points. This process is performed <var>m</var> times. If only <var>v</var> is
specified, then <var>m</var> is assumed to be <code>1</code>.
</p>
<p>The interpolation <var>method</var> is one of:
</p>
<dl compact="compact">
<dt><code>"nearest"</code></dt>
<dd><p>Return the nearest neighbor.
</p>
</dd>
<dt><code>"linear"</code> (default)</dt>
<dd><p>Linear interpolation from nearest neighbors.
</p>
</dd>
<dt><code>"pchip"</code></dt>
<dd><p>Piecewise cubic Hermite interpolating polynomial—shape-preserving
interpolation with smooth first derivative (not implemented yet).
</p>
</dd>
<dt><code>"cubic"</code></dt>
<dd><p>Cubic interpolation (same as <code>"pchip"</code> [not implemented yet]).
</p>
</dd>
<dt><code>"spline"</code></dt>
<dd><p>Cubic spline interpolation—smooth first and second derivatives
throughout the curve.
</p></dd>
</dl>
<p>The default method is <code>"linear"</code>.
</p>
<p><var>extrapval</var> is a scalar number. It replaces values beyond the endpoints
with <var>extrapval</var>. Note that if <var>extrapval</var> is used, <var>method</var>
must be specified as well. If <var>extrapval</var> is omitted and the
<var>method</var> is <code>"spline"</code>, then the extrapolated values of the
<code>"spline"</code> are used. Otherwise the default <var>extrapval</var> value for
any other <var>method</var> is <code>"NA"</code>.
</p>
<p><strong>See also:</strong> <a href="One_002ddimensional-Interpolation.html#XREFinterp1">interp1</a>, <a href="#XREFinterp2">interp2</a>, <a href="#XREFinterp3">interp3</a>, <a href="One_002ddimensional-Interpolation.html#XREFspline">spline</a>, <a href="Three_002dDimensional-Plots.html#XREFndgrid">ndgrid</a>.
</p></dd></dl>
<p>A significant difference between <code>interpn</code> and the other two
multi-dimensional interpolation functions is the fashion in which the
dimensions are treated. For <code>interp2</code> and <code>interp3</code>, the y-axis is
considered to be the columns of the matrix, whereas the x-axis corresponds to
the rows of the array. As Octave indexes arrays in column major order, the
first dimension of any array is the columns, and so <code>interpn</code> effectively
reverses the ’x’ and ’y’ dimensions. Consider the example,
</p>
<div class="example">
<pre class="example">x = y = z = -1:1;
f = @(x,y,z) x.^2 - y - z.^2;
[xx, yy, zz] = meshgrid (x, y, z);
v = f (xx,yy,zz);
xi = yi = zi = -1:0.1:1;
[xxi, yyi, zzi] = meshgrid (xi, yi, zi);
vi = interp3 (x, y, z, v, xxi, yyi, zzi, "spline");
[xxi, yyi, zzi] = ndgrid (xi, yi, zi);
vi2 = interpn (x, y, z, v, xxi, yyi, zzi, "spline");
mesh (zi, yi, squeeze (vi2(1,:,:)));
</pre></div>
<p>where <code>vi</code> and <code>vi2</code> are identical. The reversal of the
dimensions is treated in the <code>meshgrid</code> and <code>ndgrid</code> functions
respectively.
The result of this code can be seen in <a href="#fig_003ainterpn">Figure 29.4</a>.
</p>
<div class="float"><span id="fig_003ainterpn"></span>
<div align="center"><img src="interpn.png" alt="interpn">
</div>
<div class="float-caption"><p><strong>Figure 29.4: </strong>Demonstration of the use of <code>interpn</code></p></div></div>
<hr>
<div class="header">
<p>
Previous: <a href="One_002ddimensional-Interpolation.html" accesskey="p" rel="prev">One-dimensional Interpolation</a>, Up: <a href="Interpolation.html" accesskey="u" rel="up">Interpolation</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
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