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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.wavelet_transforms"></a><a class="link" href="wavelet_transforms.html" title="Wavelet Transforms">Wavelet Transforms</a>
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<h4>
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<a name="math_toolkit.wavelet_transforms.h0"></a>
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<span class="phrase"><a name="math_toolkit.wavelet_transforms.synopsis"></a></span><a class="link" href="wavelet_transforms.html#math_toolkit.wavelet_transforms.synopsis">Synopsis</a>
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</h4>
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<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">quadrature</span><span class="special">/</span><span class="identifier">wavelet_transforms</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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<span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">quadrature</span> <span class="special">{</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">p</span><span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">daubechies_wavelet_transform</span>
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<span class="special">{</span>
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<span class="keyword">public</span><span class="special">:</span>
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<span class="identifier">daubechies_wavelet_transform</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">grid_refinements</span> <span class="special">=</span> <span class="special">-</span><span class="number">1</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">tol</span> <span class="special">=</span> <span class="number">100</span><span class="special">*</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">(),</span>
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<span class="keyword">int</span> <span class="identifier">max_refinements</span> <span class="special">=</span> <span class="number">12</span><span class="special">)</span> <span class="special">{}</span>
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<span class="identifier">daubechies_wavelet_transform</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</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">daubechies_wavelet</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">p</span><span class="special">></span> <span class="identifier">wavelet</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">tol</span> <span class="special">=</span> <span class="number">100</span><span class="special">*</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">(),</span>
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<span class="keyword">int</span> <span class="identifier">max_refinements</span> <span class="special">=</span> <span class="number">12</span><span class="special">);</span>
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<span class="keyword">auto</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">s</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">t</span><span class="special">)-></span><span class="keyword">decltype</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">declval</span><span class="special"><</span><span class="identifier">F</span><span class="special">>()(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">declval</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>()))</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="special">};</span>
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<span class="special">}</span>
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</pre>
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<p>
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The wavelet transform of a function <span class="emphasis"><em>f</em></span> with respect to
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a wavelet ψ is
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</p>
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<p>
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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/wavelet_transform_definition.svg" width="333" height="48"></object></span>
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</p>
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<p>
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For compactly supported Daubechies wavelets, the bounds can always be taken
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as finite, and we have
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</p>
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<p>
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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/daubechies_wavelet_transform_definition.svg" width="328" height="44"></object></span>
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</p>
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<p>
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which also defines the <span class="emphasis"><em>s</em></span>=0 case.
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</p>
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<p>
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The code provided by Boost merely forwards a lambda to the trapezoidal quadrature
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routine, which converges quickly due to the Euler-Maclaurin summation formula.
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However, the convergence is not as rapid as for infinitely differentiable functions,
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so the default tolerances are modified.
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</p>
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<p>
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A basic usage is
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</p>
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<pre class="programlisting"><span class="keyword">auto</span> <span class="identifier">psi</span> <span class="special">=</span> <span class="identifier">daubechies_wavelet</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="number">8</span><span class="special">>();</span>
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<span class="keyword">auto</span> <span class="identifier">f</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>
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<span class="keyword">return</span> <span class="identifier">sin</span><span class="special">(</span><span class="number">1</span><span class="special">/</span><span class="identifier">x</span><span class="special">);</span>
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<span class="special">};</span>
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<span class="keyword">auto</span> <span class="identifier">Wf</span> <span class="special">=</span> <span class="identifier">daubechies_wavelet_transform</span><span class="special">(</span><span class="identifier">f</span><span class="special">,</span> <span class="identifier">psi</span><span class="special">);</span>
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<span class="keyword">double</span> <span class="identifier">w</span> <span class="special">=</span> <span class="identifier">Wf</span><span class="special">(</span><span class="number">0.8</span><span class="special">,</span> <span class="number">7.2</span><span class="special">);</span>
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</pre>
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<p>
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An image from this function is shown below.
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../graphs/scalogram_sin1t_light.png"></span>
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</p>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
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Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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>)
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