1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
|
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
<!-- Created by GNU Texinfo 6.8, https://www.gnu.org/software/texinfo/ -->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<title>Interpolation on Scattered Data (GNU Octave (version 7.3.0))</title>
<meta name="description" content="Interpolation on Scattered Data (GNU Octave (version 7.3.0))">
<meta name="keywords" content="Interpolation on Scattered Data (GNU Octave (version 7.3.0))">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<meta name="viewport" content="width=device-width,initial-scale=1">
<link href="index.html" rel="start" title="Top">
<link href="Concept-Index.html" rel="index" title="Concept Index">
<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
<link href="Geometry.html" rel="up" title="Geometry">
<link href="Vector-Rotation-Matrices.html" rel="next" title="Vector Rotation Matrices">
<link href="Convex-Hull.html" rel="prev" title="Convex Hull">
<style type="text/css">
<!--
a.copiable-anchor {visibility: hidden; text-decoration: none; line-height: 0em}
a.summary-letter {text-decoration: none}
blockquote.indentedblock {margin-right: 0em}
div.display {margin-left: 3.2em}
div.example {margin-left: 3.2em}
kbd {font-style: oblique}
pre.display {font-family: inherit}
pre.format {font-family: inherit}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
span.nolinebreak {white-space: nowrap}
span.roman {font-family: initial; font-weight: normal}
span.sansserif {font-family: sans-serif; font-weight: normal}
span:hover a.copiable-anchor {visibility: visible}
ul.no-bullet {list-style: none}
-->
</style>
<link rel="stylesheet" type="text/css" href="octave.css">
</head>
<body lang="en">
<div class="section" id="Interpolation-on-Scattered-Data">
<div class="header">
<p>
Next: <a href="Vector-Rotation-Matrices.html" accesskey="n" rel="next">Vector Rotation Matrices</a>, Previous: <a href="Convex-Hull.html" accesskey="p" rel="prev">Convex Hull</a>, Up: <a href="Geometry.html" accesskey="u" rel="up">Geometry</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>
</div>
<hr>
<span id="Interpolation-on-Scattered-Data-1"></span><h3 class="section">30.4 Interpolation on Scattered Data</h3>
<p>An important use of the Delaunay tessellation is that it can be used to
interpolate from scattered data to an arbitrary set of points. To do
this the N-simplex of the known set of points is calculated with
<code>delaunay</code> or <code>delaunayn</code>. Then the simplices in to which the
desired points are found are identified. Finally the vertices of the simplices
are used to interpolate to the desired points. The functions that perform this
interpolation are <code>griddata</code>, <code>griddata3</code> and <code>griddatan</code>.
</p>
<span id="XREFgriddata"></span><dl class="def">
<dt id="index-griddata"><span class="category">: </span><span><em><var>zi</var> =</em> <strong>griddata</strong> <em>(<var>x</var>, <var>y</var>, <var>z</var>, <var>xi</var>, <var>yi</var>)</em><a href='#index-griddata' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-griddata-1"><span class="category">: </span><span><em><var>zi</var> =</em> <strong>griddata</strong> <em>(<var>x</var>, <var>y</var>, <var>z</var>, <var>xi</var>, <var>yi</var>, <var>method</var>)</em><a href='#index-griddata-1' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-griddata-2"><span class="category">: </span><span><em>[<var>xi</var>, <var>yi</var>, <var>zi</var>] =</em> <strong>griddata</strong> <em>(…)</em><a href='#index-griddata-2' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-griddata-3"><span class="category">: </span><span><em><var>vi</var> =</em> <strong>griddata</strong> <em>(<var>x</var>, <var>y</var>, <var>z</var>, <var>v</var>, <var>xi</var>, <var>yi</var>, <var>zi</var>)</em><a href='#index-griddata-3' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-griddata-4"><span class="category">: </span><span><em><var>vi</var> =</em> <strong>griddata</strong> <em>(<var>x</var>, <var>y</var>, <var>z</var>, <var>v</var>, <var>xi</var>, <var>yi</var>, <var>zi</var>, <var>method</var>)</em><a href='#index-griddata-4' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-griddata-5"><span class="category">: </span><span><em><var>vi</var> =</em> <strong>griddata</strong> <em>(<var>x</var>, <var>y</var>, <var>z</var>, <var>v</var>, <var>xi</var>, <var>yi</var>, <var>zi</var>, <var>method</var>, <var>options</var>)</em><a href='#index-griddata-5' class='copiable-anchor'> ¶</a></span></dt>
<dd>
<p>Interpolate irregular 2-D and 3-D source data at specified points.
</p>
<p>For 2-D interpolation, the inputs <var>x</var> and <var>y</var> define the points
where the function <code><var>z</var> = f (<var>x</var>, <var>y</var>)</code> is evaluated.
The inputs <var>x</var>, <var>y</var>, <var>z</var> are either vectors of the same length,
or the unequal vectors <var>x</var>, <var>y</var> are expanded to a 2-D grid with
<code>meshgrid</code> and <var>z</var> is a 2-D matrix matching the resulting size of
the X-Y grid.
</p>
<p>The interpolation points are (<var>xi</var>, <var>yi</var>). If, and only if,
<var>xi</var> is a row vector and <var>yi</var> is a column vector, then
<code>meshgrid</code> will be used to create a mesh of interpolation points.
</p>
<p>For 3-D interpolation, the inputs <var>x</var>, <var>y</var>, and <var>z</var> define the
points where the function <code><var>v</var> = f (<var>x</var>, <var>y</var>, <var>z</var>)</code>
is evaluated. The inputs <var>x</var>, <var>y</var>, <var>z</var> are either vectors of
the same length, or if they are of unequal length, then they are expanded to
a 3-D grid with <code>meshgrid</code>. The size of the input <var>v</var> must match
the size of the original data, either as a vector or a matrix.
</p>
<p>The optional input interpolation <var>method</var> can be <code>"nearest"</code>,
<code>"linear"</code>, or for 2-D data <code>"v4"</code>. When the method is
<code>"nearest"</code>, the output <var>vi</var> will be the closest point in the
original data (<var>x</var>, <var>y</var>, <var>z</var>) to the query point (<var>xi</var>,
<var>yi</var>, <var>zi</var>). When the method is <code>"linear"</code>, the output
<var>vi</var> will be a linear interpolation between the two closest points in
the original source data in each dimension. For 2-D cases only, the
<code>"v4"</code> method is also available which implements a biharmonic spline
interpolation. If <var>method</var> is omitted or empty, it defaults to
<code>"linear"</code>.
</p>
<p>For 3-D interpolation, the optional argument <var>options</var> is passed
directly to Qhull when computing the Delaunay triangulation used for
interpolation. For more information on the defaults and how to pass
different values, see <a href="Delaunay-Triangulation.html#XREFdelaunayn"><code>delaunayn</code></a>.
</p>
<p>Programming Notes: If the input is complex the real and imaginary parts
are interpolated separately. Interpolation is normally based on a
Delaunay triangulation. Any query values outside the convex hull of the
input points will return <code>NaN</code>. However, the <code>"v4"</code> method does
not use the triangulation and will return values outside the original data
(extrapolation).
</p>
<p><strong>See also:</strong> <a href="#XREFgriddata3">griddata3</a>, <a href="#XREFgriddatan">griddatan</a>, <a href="Delaunay-Triangulation.html#XREFdelaunay">delaunay</a>.
</p></dd></dl>
<span id="XREFgriddata3"></span><dl class="def">
<dt id="index-griddata3"><span class="category">: </span><span><em><var>vi</var> =</em> <strong>griddata3</strong> <em>(<var>x</var>, <var>y</var>, <var>z</var>, <var>v</var>, <var>xi</var>, <var>yi</var>, <var>zi</var>)</em><a href='#index-griddata3' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-griddata3-1"><span class="category">: </span><span><em><var>vi</var> =</em> <strong>griddata3</strong> <em>(<var>x</var>, <var>y</var>, <var>z</var>, <var>v</var>, <var>xi</var>, <var>yi</var>, <var>zi</var>, <var>method</var>)</em><a href='#index-griddata3-1' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-griddata3-2"><span class="category">: </span><span><em><var>vi</var> =</em> <strong>griddata3</strong> <em>(<var>x</var>, <var>y</var>, <var>z</var>, <var>v</var>, <var>xi</var>, <var>yi</var>, <var>zi</var>, <var>method</var>, <var>options</var>)</em><a href='#index-griddata3-2' class='copiable-anchor'> ¶</a></span></dt>
<dd>
<p>Interpolate irregular 3-D source data at specified points.
</p>
<p>The inputs <var>x</var>, <var>y</var>, and <var>z</var> define the points where the
function <code><var>v</var> = f (<var>x</var>, <var>y</var>, <var>z</var>)</code> is evaluated. The
inputs <var>x</var>, <var>y</var>, <var>z</var> are either vectors of the same length, or
if they are of unequal length, then they are expanded to a 3-D grid with
<code>meshgrid</code>. The size of the input <var>v</var> must match the size of the
original data, either as a vector or a matrix.
</p>
<p>The interpolation points are specified by <var>xi</var>, <var>yi</var>, <var>zi</var>.
</p>
<p>The optional input interpolation <var>method</var> can be <code>"nearest"</code> or
<code>"linear"</code>. When the method is <code>"nearest"</code>, the output <var>vi</var>
will be the closest point in the original data (<var>x</var>, <var>y</var>, <var>z</var>)
to the query point (<var>xi</var>, <var>yi</var>, <var>zi</var>). When the method is
<code>"linear"</code>, the output <var>vi</var> will be a linear interpolation between
the two closest points in the original source data in each dimension.
If <var>method</var> is omitted or empty, it defaults to <code>"linear"</code>.
</p>
<p>The optional argument <var>options</var> is passed directly to Qhull when
computing the Delaunay triangulation used for interpolation. See
<code>delaunayn</code> for information on the defaults and how to pass different
values.
</p>
<p>Programming Notes: If the input is complex the real and imaginary parts
are interpolated separately. Interpolation is based on a Delaunay
triangulation and any query values outside the convex hull of the input
points will return <code>NaN</code>.
</p>
<p><strong>See also:</strong> <a href="#XREFgriddata">griddata</a>, <a href="#XREFgriddatan">griddatan</a>, <a href="Delaunay-Triangulation.html#XREFdelaunayn">delaunayn</a>.
</p></dd></dl>
<span id="XREFgriddatan"></span><dl class="def">
<dt id="index-griddatan"><span class="category">: </span><span><em><var>yi</var> =</em> <strong>griddatan</strong> <em>(<var>x</var>, <var>y</var>, <var>xi</var>)</em><a href='#index-griddatan' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-griddatan-1"><span class="category">: </span><span><em><var>yi</var> =</em> <strong>griddatan</strong> <em>(<var>x</var>, <var>y</var>, <var>xi</var>, <var>method</var>)</em><a href='#index-griddatan-1' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-griddatan-2"><span class="category">: </span><span><em><var>yi</var> =</em> <strong>griddatan</strong> <em>(<var>x</var>, <var>y</var>, <var>xi</var>, <var>method</var>, <var>options</var>)</em><a href='#index-griddatan-2' class='copiable-anchor'> ¶</a></span></dt>
<dd>
<p>Interpolate irregular source data <var>x</var>, <var>y</var> at points specified by
<var>xi</var>.
</p>
<p>The input <var>x</var> is an MxN matrix representing M points in an N-dimensional
space. The input <var>y</var> is a single-valued column vector (Mx1)
representing a function evaluated at the points <var>x</var>, i.e.,
<code><var>y</var> = fcn (<var>x</var>)</code>. The input <var>xi</var> is a list of points
for which the function output <var>yi</var> should be approximated through
interpolation. <var>xi</var> must have the same number of columns (<var>N</var>)
as <var>x</var> so that the dimensionality matches.
</p>
<p>The optional input interpolation <var>method</var> can be <code>"nearest"</code> or
<code>"linear"</code>. When the method is <code>"nearest"</code>, the output <var>yi</var>
will be the closest point in the original data <var>x</var> to the query point
<var>xi</var>. When the method is <code>"linear"</code>, the output <var>yi</var> will
be a linear interpolation between the two closest points in the original
source data. If <var>method</var> is omitted or empty, it defaults to
<code>"linear"</code>.
</p>
<p>The optional argument <var>options</var> is passed directly to Qhull when
computing the Delaunay triangulation used for interpolation. See
<code>delaunayn</code> for information on the defaults and how to pass different
values.
</p>
<p>Example
</p>
<div class="example">
<pre class="example">## Evaluate sombrero() function at irregular data points
x = 16*gallery ("uniformdata", [200,1], 1) - 8;
y = 16*gallery ("uniformdata", [200,1], 11) - 8;
z = sin (sqrt (x.^2 + y.^2)) ./ sqrt (x.^2 + y.^2);
## Create a regular grid and interpolate data
[xi, yi] = ndgrid (linspace (-8, 8, 50));
zi = griddatan ([x, y], z, [xi(:), yi(:)]);
zi = reshape (zi, size (xi));
## Plot results
clf ();
plot3 (x, y, z, "or");
hold on
surf (xi, yi, zi);
legend ("Original Data", "Interpolated Data");
</pre></div>
<p>Programming Notes: If the input is complex the real and imaginary parts
are interpolated separately. Interpolation is based on a Delaunay
triangulation and any query values outside the convex hull of the input
points will return <code>NaN</code>. For 2-D and 3-D data additional
interpolation methods are available by using the <code>griddata</code> function.
</p>
<p><strong>See also:</strong> <a href="#XREFgriddata">griddata</a>, <a href="#XREFgriddata3">griddata3</a>, <a href="Delaunay-Triangulation.html#XREFdelaunayn">delaunayn</a>.
</p></dd></dl>
<p>An example of the use of the <code>griddata</code> function is
</p>
<div class="example">
<pre class="example">rand ("state", 1);
x = 2*rand (1000,1) - 1;
y = 2*rand (size (x)) - 1;
z = sin (2*(x.^2+y.^2));
[xx,yy] = meshgrid (linspace (-1,1,32));
zz = griddata (x, y, z, xx, yy);
mesh (xx, yy, zz);
</pre></div>
<p>that interpolates from a random scattering of points, to a uniform grid.
The output of the above can be seen in <a href="#fig_003agriddata">Figure 30.6</a>.
</p>
<div class="float"><span id="fig_003agriddata"></span>
<div align="center"><img src="griddata.png" alt="griddata">
</div>
<div class="float-caption"><p><strong>Figure 30.6: </strong>Interpolation from a scattered data to a regular grid</p></div></div>
</div>
<hr>
<div class="header">
<p>
Next: <a href="Vector-Rotation-Matrices.html">Vector Rotation Matrices</a>, Previous: <a href="Convex-Hull.html">Convex Hull</a>, Up: <a href="Geometry.html">Geometry</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>
</div>
</body>
</html>
|
