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<a name="Multi_002ddimensional-Interpolation"></a>
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Previous: <a href="One_002ddimensional-Interpolation.html#One_002ddimensional-Interpolation" accesskey="p" rel="prev">One-dimensional Interpolation</a>, Up: <a href="Interpolation.html#Interpolation" 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#Concept-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Multi_002ddimensional-Interpolation-1"></a>
<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">Interpolation on Scattered Data</a>.
</p>
<a name="XREFinterp2"></a><dl>
<dt><a name="index-interp2"></a><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><a name="index-interp2-1"></a><em><var>zi</var> =</em> <strong>interp2</strong> <em>(<var>z</var>, <var>xi</var>, <var>yi</var>)</em></dt>
<dt><a name="index-interp2-2"></a><em><var>zi</var> =</em> <strong>interp2</strong> <em>(<var>z</var>, <var>n</var>)</em></dt>
<dt><a name="index-interp2-3"></a><em><var>zi</var> =</em> <strong>interp2</strong> <em>(<var>z</var>)</em></dt>
<dt><a name="index-interp2-4"></a><em><var>zi</var> =</em> <strong>interp2</strong> <em>(…, <var>method</var>)</em></dt>
<dt><a name="index-interp2-5"></a><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>
<a name="XREFinterp3"></a><dl>
<dt><a name="index-interp3"></a><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><a name="index-interp3-1"></a><em><var>vi</var> =</em> <strong>interp3</strong> <em>(<var>v</var>, <var>xi</var>, <var>yi</var>, <var>zi</var>)</em></dt>
<dt><a name="index-interp3-2"></a><em><var>vi</var> =</em> <strong>interp3</strong> <em>(<var>v</var>, <var>n</var>)</em></dt>
<dt><a name="index-interp3-3"></a><em><var>vi</var> =</em> <strong>interp3</strong> <em>(<var>v</var>)</em></dt>
<dt><a name="index-interp3-4"></a><em><var>vi</var> =</em> <strong>interp3</strong> <em>(…, <var>method</var>)</em></dt>
<dt><a name="index-interp3-5"></a><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>
<a name="XREFinterpn"></a><dl>
<dt><a name="index-interpn"></a><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><a name="index-interpn-1"></a><em><var>vi</var> =</em> <strong>interpn</strong> <em>(<var>v</var>, <var>y1</var>, <var>y2</var>, …)</em></dt>
<dt><a name="index-interpn-2"></a><em><var>vi</var> =</em> <strong>interpn</strong> <em>(<var>v</var>, <var>m</var>)</em></dt>
<dt><a name="index-interpn-3"></a><em><var>vi</var> =</em> <strong>interpn</strong> <em>(<var>v</var>)</em></dt>
<dt><a name="index-interpn-4"></a><em><var>vi</var> =</em> <strong>interpn</strong> <em>(…, <var>method</var>)</em></dt>
<dt><a name="index-interpn-5"></a><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. The parameters <var>y1</var>, etc.
respect a similar format to <var>x1</var>, etc., and they represent the points
at which the array <var>vi</var> is interpolated.
</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"><a name="fig_003ainterpn"></a>
<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>
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