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<p>
<a name="Multi-dimensional-Interpolation"></a>
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Previous: <a rel="previous" accesskey="p" href="One_002ddimensional-Interpolation.html#One_002ddimensional-Interpolation">One-dimensional Interpolation</a>,
Up: <a rel="up" accesskey="u" href="Interpolation.html#Interpolation">Interpolation</a>
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<h3 class="section">28.2 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>.
<!-- ./general/interp2.m -->
<p><a name="doc_002dinterp2"></a>
<div class="defun">
— Function File: <var>zi</var> = <b>interp2</b> (<var>x, y, z, xi, yi</var>)<var><a name="index-interp2-2066"></a></var><br>
— Function File: <var>zi</var> = <b>interp2</b> (<var>Z, xi, yi</var>)<var><a name="index-interp2-2067"></a></var><br>
— Function File: <var>zi</var> = <b>interp2</b> (<var>Z, n</var>)<var><a name="index-interp2-2068"></a></var><br>
— Function File: <var>zi</var> = <b>interp2</b> (<var><small class="dots">...</small>, method</var>)<var><a name="index-interp2-2069"></a></var><br>
— Function File: <var>zi</var> = <b>interp2</b> (<var><small class="dots">...</small>, method, extrapval</var>)<var><a name="index-interp2-2070"></a></var><br>
<blockquote>
<p>Two-dimensional interpolation. <var>x</var>, <var>y</var> and <var>z</var> describe a
surface function. If <var>x</var> and <var>y</var> are vectors their length
must correspondent to the size of <var>z</var>. <var>x</var> and <var>y</var> must be
monotonic. If they are matrices they must have the <code>meshgrid</code>
format.
<dl>
<dt><code>interp2 (</code><var>x</var><code>, </code><var>y</var><code>, </code><var>Z</var><code>, </code><var>xi</var><code>, </code><var>yi</var><code>, ...)</code><dd>Returns a matrix corresponding to the points described by the
matrices <var>xi</var>, <var>yi</var>.
<p>If the last argument is a string, the interpolation method can
be specified. The method can be 'linear', 'nearest' or 'cubic'.
If it is omitted 'linear' interpolation is assumed.
<br><dt><code>interp2 (</code><var>z</var><code>, </code><var>xi</var><code>, </code><var>yi</var><code>)</code><dd>Assumes <var>x</var><code> = 1:rows (</code><var>z</var><code>)</code> and <var>y</var><code> =
1:columns (</code><var>z</var><code>)</code>
<br><dt><code>interp2 (</code><var>z</var><code>, </code><var>n</var><code>)</code><dd>Interleaves the matrix <var>z</var> n-times. If <var>n</var> is omitted a value
of <var>n</var><code> = 1</code> is assumed.
</dl>
<p>The variable <var>method</var> defines the method to use for the
interpolation. It can take one of the following values
<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 (not implemented yet).
<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>If a scalar value <var>extrapval</var> is defined as the final value, then
values outside the mesh as set to this value. Note that in this case
<var>method</var> must be defined as well. If <var>extrapval</var> is not
defined then NA is assumed.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dinterp1.html#doc_002dinterp1">interp1</a>.
</p></blockquote></div>
<!-- ./general/interp3.m -->
<p><a name="doc_002dinterp3"></a>
<div class="defun">
— Function File: <var>vi</var> = <b>interp3</b> (<var>x, y,z, v, xi, yi, zi</var>)<var><a name="index-interp3-2071"></a></var><br>
— Function File: <var>vi</var> = <b>interp3</b> (<var>v, xi, yi, zi</var>)<var><a name="index-interp3-2072"></a></var><br>
— Function File: <var>vi</var> = <b>interp3</b> (<var>v, m</var>)<var><a name="index-interp3-2073"></a></var><br>
— Function File: <var>vi</var> = <b>interp3</b> (<var>v</var>)<var><a name="index-interp3-2074"></a></var><br>
— Function File: <var>vi</var> = <b>interp3</b> (<var><small class="dots">...</small>, method</var>)<var><a name="index-interp3-2075"></a></var><br>
— Function File: <var>vi</var> = <b>interp3</b> (<var><small class="dots">...</small>, method, extrapval</var>)<var><a name="index-interp3-2076"></a></var><br>
<blockquote>
<p>Perform 3-dimensional interpolation. Each element of the 3-dimensional
array <var>v</var> represents a value at a location given by the parameters
<var>x</var>, <var>y</var>, and <var>z</var>. The parameters <var>x</var>, <var>x</var>, and
<var>z</var> are either 3-dimensional arrays of the same size as the array
<var>v</var> in the 'meshgrid' format or vectors. The parameters <var>xi</var>, etc.
respect a similar format to <var>x</var>, etc., and they represent the points
at which the array <var>vi</var> is interpolated.
<p>If <var>x</var>, <var>y</var>, <var>z</var> are omitted, they are assumed to be
<code>x = 1 : size (</code><var>v</var><code>, 2)</code>, <code>y = 1 : size (</code><var>v</var><code>, 1)</code> and
<code>z = 1 : size (</code><var>v</var><code>, 3)</code>. 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>Method is one of:
<dl>
<dt>'nearest'<dd>Return the nearest neighbor.
<br><dt>'linear'<dd>Linear interpolation from nearest neighbors.
<br><dt>'cubic'<dd>Cubic interpolation from four nearest neighbors (not implemented yet).
<br><dt>'spline'<dd>Cubic spline interpolation–smooth first and second derivatives
throughout the curve.
</dl>
<p>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.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dinterp1.html#doc_002dinterp1">interp1</a>, <a href="doc_002dinterp2.html#doc_002dinterp2">interp2</a>, <a href="doc_002dspline.html#doc_002dspline">spline</a>, <a href="doc_002dmeshgrid.html#doc_002dmeshgrid">meshgrid</a>.
</p></blockquote></div>
<!-- ./general/interpn.m -->
<p><a name="doc_002dinterpn"></a>
<div class="defun">
— Function File: <var>vi</var> = <b>interpn</b> (<var>x1, x2, <small class="dots">...</small>, v, y1, y2, <small class="dots">...</small></var>)<var><a name="index-interpn-2077"></a></var><br>
— Function File: <var>vi</var> = <b>interpn</b> (<var>v, y1, y2, <small class="dots">...</small></var>)<var><a name="index-interpn-2078"></a></var><br>
— Function File: <var>vi</var> = <b>interpn</b> (<var>v, m</var>)<var><a name="index-interpn-2079"></a></var><br>
— Function File: <var>vi</var> = <b>interpn</b> (<var>v</var>)<var><a name="index-interpn-2080"></a></var><br>
— Function File: <var>vi</var> = <b>interpn</b> (<var><small class="dots">...</small>, method</var>)<var><a name="index-interpn-2081"></a></var><br>
— Function File: <var>vi</var> = <b>interpn</b> (<var><small class="dots">...</small>, method, extrapval</var>)<var><a name="index-interpn-2082"></a></var><br>
<blockquote>
<p>Perform <var>n</var>-dimensional interpolation, where <var>n</var> is at least two.
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>, <small class="dots">...</small>, <var>xn</var>.
The parameters <var>x1</var>, <var>x2</var>, <small class="dots">...</small>, <var>xn</var> are either
<var>n</var>-dimensional arrays of the same size as the array <var>v</var> in
the 'ndgrid' 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>If <var>x1</var>, <small class="dots">...</small>, <var>xn</var> are omitted, they are assumed to be
<code>x1 = 1 : size (</code><var>v</var><code>, 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>Method is one of:
<dl>
<dt>'nearest'<dd>Return the nearest neighbor.
<br><dt>'linear'<dd>Linear interpolation from nearest neighbors.
<br><dt>'cubic'<dd>Cubic interpolation from four nearest neighbors (not implemented yet).
<br><dt>'spline'<dd>Cubic spline interpolation–smooth first and second derivatives
throughout the curve.
</dl>
<p>The default method is 'linear'.
<p>If <var>extrapval</var> is the scalar value, use it to replace the values
beyond the endpoints with that number. If <var>extrapval</var> is missing,
assume NA.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dinterp1.html#doc_002dinterp1">interp1</a>, <a href="doc_002dinterp2.html#doc_002dinterp2">interp2</a>, <a href="doc_002dspline.html#doc_002dspline">spline</a>, <a href="doc_002dndgrid.html#doc_002dndgrid">ndgrid</a>.
</p></blockquote></div>
<p>A significant difference between <code>interpn</code> and the other two
multidimensional 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
<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>
<p class="noindent">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.
<p>In additional the support function <code>bicubic</code> that underlies the
cubic interpolation of <code>interp2</code> function can be called directly.
<!-- ./general/bicubic.m -->
<p><a name="doc_002dbicubic"></a>
<div class="defun">
— Function File: <var>zi</var> = <b>bicubic</b> (<var>x, y, z, xi, yi, extrapval</var>)<var><a name="index-bicubic-2083"></a></var><br>
<blockquote>
<p>Return a matrix <var>zi</var> corresponding to the bicubic
interpolations at <var>xi</var> and <var>yi</var> of the data supplied
as <var>x</var>, <var>y</var> and <var>z</var>. Points outside the grid are set
to <var>extrapval</var>.
<p>See <a href="http://wiki.woodpecker.org.cn/moin/Octave/Bicubic">http://wiki.woodpecker.org.cn/moin/Octave/Bicubic</a>
for further information.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dinterp2.html#doc_002dinterp2">interp2</a>.
</p></blockquote></div>
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