120 lines
12 KiB
HTML
120 lines
12 KiB
HTML
<html>
|
|
<head>
|
|
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
|
<title>Whittaker-Shannon interpolation</title>
|
|
<link rel="stylesheet" href="../math.css" type="text/css">
|
|
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
|
|
<link rel="home" href="../index.html" title="Math Toolkit 2.11.0">
|
|
<link rel="up" href="../interpolation.html" title="Chapter 12. Interpolation">
|
|
<link rel="prev" href="cardinal_quintic_b.html" title="Cardinal Quintic B-spline interpolation">
|
|
<link rel="next" href="barycentric.html" title="Barycentric Rational Interpolation">
|
|
</head>
|
|
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
|
<table cellpadding="2" width="100%"><tr>
|
|
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../boost.png"></td>
|
|
<td align="center"><a href="../../../../../index.html">Home</a></td>
|
|
<td align="center"><a href="../../../../../libs/libraries.htm">Libraries</a></td>
|
|
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
|
|
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
|
|
<td align="center"><a href="../../../../../more/index.htm">More</a></td>
|
|
</tr></table>
|
|
<hr>
|
|
<div class="spirit-nav">
|
|
<a accesskey="p" href="cardinal_quintic_b.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../interpolation.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="barycentric.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
|
|
</div>
|
|
<div class="section">
|
|
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
|
|
<a name="math_toolkit.whittaker_shannon"></a><a class="link" href="whittaker_shannon.html" title="Whittaker-Shannon interpolation">Whittaker-Shannon interpolation</a>
|
|
</h2></div></div></div>
|
|
<h4>
|
|
<a name="math_toolkit.whittaker_shannon.h0"></a>
|
|
<span class="phrase"><a name="math_toolkit.whittaker_shannon.synopsis"></a></span><a class="link" href="whittaker_shannon.html#math_toolkit.whittaker_shannon.synopsis">Synopsis</a>
|
|
</h4>
|
|
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">interpolators</span><span class="special">/</span><span class="identifier">whittaker_shannon</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
|
</pre>
|
|
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">interpolators</span> <span class="special">{</span>
|
|
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RandomAccessContainer</span><span class="special">></span>
|
|
<span class="keyword">class</span> <span class="identifier">whittaker_shannon</span>
|
|
<span class="special">{</span>
|
|
<span class="keyword">public</span><span class="special">:</span>
|
|
|
|
<span class="keyword">using</span> <span class="identifier">Real</span> <span class="special">=</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span>
|
|
|
|
<span class="identifier">whittaker_shannon</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span><span class="special">&&</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">left_endpoint</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">step_size</span><span class="special">);</span>
|
|
|
|
<span class="identifier">Real</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
|
|
|
|
<span class="identifier">Real</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
|
|
<span class="special">};</span>
|
|
|
|
<span class="special">}}}</span> <span class="comment">// namespaces</span>
|
|
</pre>
|
|
<h4>
|
|
<a name="math_toolkit.whittaker_shannon.h1"></a>
|
|
<span class="phrase"><a name="math_toolkit.whittaker_shannon.whittaker_shannon_interpolation"></a></span><a class="link" href="whittaker_shannon.html#math_toolkit.whittaker_shannon.whittaker_shannon_interpolation">Whittaker-Shannon
|
|
Interpolation</a>
|
|
</h4>
|
|
<p>
|
|
The Whittaker-Shannon interpolator takes equispaced data and interpolates between
|
|
them via a sum of sinc functions. This interpolation is stable and infinitely
|
|
smooth, but has linear complexity in the data, making it slow relative to compactly-supported
|
|
b-splines. In addition, we cannot pass an infinite amount of data into the
|
|
class, and must truncate the (perhaps) infinite sinc series to a finite number
|
|
of terms. Since the sinc function has slow <span class="emphasis"><em>1/x</em></span> decay,
|
|
the truncation of the series can incur large error. Hence this interpolator
|
|
works best when operating on samples of compactly-supported functions. Here
|
|
is an example of interpolating a smooth "bump function":
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">auto</span> <span class="identifier">bump</span> <span class="special">=</span> <span class="special">[](</span><span class="keyword">double</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">{</span> <span class="keyword">if</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">abs</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">>=</span> <span class="number">1</span><span class="special">)</span> <span class="special">{</span> <span class="keyword">return</span> <span class="number">0.0</span><span class="special">;</span> <span class="special">}</span> <span class="keyword">return</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exp</span><span class="special">(-</span><span class="number">1.0</span><span class="special">/(</span><span class="number">1.0</span><span class="special">-</span><span class="identifier">x</span><span class="special">*</span><span class="identifier">x</span><span class="special">));</span> <span class="special">};</span>
|
|
|
|
<span class="keyword">double</span> <span class="identifier">t0</span> <span class="special">=</span> <span class="special">-</span><span class="number">1</span><span class="special">;</span>
|
|
<span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">2049</span><span class="special">;</span>
|
|
<span class="keyword">double</span> <span class="identifier">h</span> <span class="special">=</span> <span class="number">2.0</span><span class="special">/(</span><span class="identifier">n</span><span class="special">-</span><span class="number">1.0</span><span class="special">);</span>
|
|
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">v</span><span class="special">(</span><span class="identifier">n</span><span class="special">);</span>
|
|
<span class="keyword">for</span><span class="special">(</span><span class="identifier">size_t</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special"><</span> <span class="identifier">n</span><span class="special">;</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span> <span class="special">{</span>
|
|
<span class="keyword">double</span> <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">t0</span> <span class="special">+</span> <span class="identifier">i</span><span class="special">*</span><span class="identifier">h</span><span class="special">;</span>
|
|
<span class="identifier">v</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">bump</span><span class="special">(</span><span class="identifier">t</span><span class="special">);</span>
|
|
<span class="special">}</span>
|
|
|
|
<span class="keyword">auto</span> <span class="identifier">ws</span> <span class="special">=</span> <span class="identifier">whittaker_shannon</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">v</span><span class="special">),</span> <span class="identifier">t0</span><span class="special">,</span> <span class="identifier">h</span><span class="special">);</span>
|
|
|
|
<span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">ws</span><span class="special">(</span><span class="number">0.3</span><span class="special">);</span>
|
|
</pre>
|
|
<p>
|
|
The derivative of the interpolant can also be evaluated, but the accuracy is
|
|
not as high:
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">yp</span> <span class="special">=</span> <span class="identifier">ws</span><span class="special">.</span><span class="identifier">prime</span><span class="special">(</span><span class="number">0.3</span><span class="special">);</span>
|
|
</pre>
|
|
<h4>
|
|
<a name="math_toolkit.whittaker_shannon.h2"></a>
|
|
<span class="phrase"><a name="math_toolkit.whittaker_shannon.complexity_and_performance"></a></span><a class="link" href="whittaker_shannon.html#math_toolkit.whittaker_shannon.complexity_and_performance">Complexity
|
|
and Performance</a>
|
|
</h4>
|
|
<p>
|
|
The call to the constructor requires 𝑶(1) operations, simply moving data into
|
|
the class. Each call to the interpolant is 𝑶(<span class="emphasis"><em>n</em></span>), where
|
|
<span class="emphasis"><em>n</em></span> is the number of points to interpolate.
|
|
</p>
|
|
</div>
|
|
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
|
<td align="left"></td>
|
|
<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
|
|
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
|
|
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
|
|
Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
|
|
Daryle Walker and Xiaogang Zhang<p>
|
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
|
</p>
|
|
</div></td>
|
|
</tr></table>
|
|
<hr>
|
|
<div class="spirit-nav">
|
|
<a accesskey="p" href="cardinal_quintic_b.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../interpolation.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="barycentric.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
|
|
</div>
|
|
</body>
|
|
</html>
|