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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.daubechies"></a><a class="link" href="daubechies.html" title="Daubechies Wavelets and Scaling Functions">Daubechies Wavelets and Scaling
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Functions</a>
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</h2></div></div></div>
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<h5>
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<a name="math_toolkit.daubechies.h0"></a>
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<span class="phrase"><a name="math_toolkit.daubechies.synopsis"></a></span><a class="link" href="daubechies.html#math_toolkit.daubechies.synopsis">Synopsis</a>
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</h5>
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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">special_functions</span><span class="special">/</span><span class="identifier">daubechies_scaling</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>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</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_scaling</span> <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_scaling</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>
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<span class="keyword">inline</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>
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<span class="keyword">inline</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>
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<span class="keyword">inline</span> <span class="identifier">Real</span> <span class="identifier">double_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>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">></span> <span class="identifier">support</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="identifier">int64_t</span> <span class="identifier">bytes</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="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">order</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">Real</span><span class="special">></span> <span class="identifier">dyadic_grid</span><span class="special">(</span><span class="identifier">int64_t</span> <span class="identifier">j_max</span><span class="special">);</span>
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<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">special_functions</span><span class="special">/</span><span class="identifier">daubechies_wavelet</span><span class="special">.</span><span class="identifier">hpp</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">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</span> <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</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>
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<span class="keyword">inline</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>
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<span class="keyword">inline</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>
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<span class="keyword">inline</span> <span class="identifier">Real</span> <span class="identifier">double_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>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">></span> <span class="identifier">support</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="identifier">int64_t</span> <span class="identifier">bytes</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> <span class="comment">// namespaces</span>
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</pre>
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<p>
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Daubechies wavelets and scaling functions are a family of compactly supported
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functions indexed by an integer <span class="emphasis"><em>p</em></span> which have <span class="emphasis"><em>p</em></span>
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vanishing moments and an associated filter of length <span class="emphasis"><em>2p</em></span>.
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They are used in signal denoising, Galerkin methods for PDEs, and compression.
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</p>
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<p>
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The canonical reference on these functions is Daubechies' monograph <span class="emphasis"><em>Ten
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Lectures on Wavelets</em></span>, whose notational conventions we attempt to
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follow here.
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</p>
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<p>
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A basic usage is as follows:
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</p>
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<pre class="programlisting"><span class="keyword">auto</span> <span class="identifier">phi</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_scaling</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">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">phi</span><span class="special">(</span><span class="number">0.38</span><span class="special">);</span>
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<span class="keyword">double</span> <span class="identifier">dydx</span> <span class="special">=</span> <span class="identifier">phi</span><span class="special">.</span><span class="identifier">prime</span><span class="special">(</span><span class="number">0.38</span><span class="special">);</span>
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<span class="keyword">auto</span> <span class="identifier">psi</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="keyword">double</span><span class="special">,</span> <span class="number">8</span><span class="special">>();</span>
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<span class="identifier">y</span> <span class="special">=</span> <span class="identifier">psi</span><span class="special">(</span><span class="number">0.38</span><span class="special">);</span>
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</pre>
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<p>
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Note that the constructor call is expensive, as it must assemble a <span class="emphasis"><em>dyadic
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grid</em></span>--values of <sub><span class="emphasis"><em>p</em></span></sub>φ at dyadic rationals,
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i.e., numbers of the form n/2<sup><span class="emphasis"><em>j</em></span></sup>. You should only instantiate
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this class once in the duration of a program. The class is pimpl'd and all
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its member functions are threadsafe, so it can be copied cheaply and shared
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between threads. The default number of grid refinements is chosen so that the
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relative error is controlled to ~2-3 ULPs away from the right-hand side of
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the support, where superexponential growth of the condition number of function
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evaluation makes this impossible. However, controlling relative error of Daubechies
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wavelets and scaling functions is much more difficult than controlling absolute
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error, and the memory consumption is much higher in relative mode. The memory
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consumption of the class can be queried via
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</p>
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<pre class="programlisting"><span class="identifier">int64_t</span> <span class="identifier">mem</span> <span class="special">=</span> <span class="identifier">phi</span><span class="special">.</span><span class="identifier">bytes</span><span class="special">();</span>
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</pre>
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<p>
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and if this is deemed unacceptably large, the user may choose to control absolute
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error via calling the constructor with the <code class="computeroutput"><span class="identifier">grid_refinements</span></code>
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parameter set to -2, so
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</p>
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<pre class="programlisting"><span class="keyword">auto</span> <span class="identifier">phi</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_scaling</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="number">8</span><span class="special">>(-</span><span class="number">2</span><span class="special">);</span>
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</pre>
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<p>
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gives a scaling function which keeps the absolute error bounded by roughly
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the double precision unit roundoff.
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</p>
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<p>
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If context precludes the ability to reuse the class throughout the program,
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it makes sense to reduce the accuracy even further. This can be done by specifying
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the grid refinements, for example,
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</p>
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<pre class="programlisting"><span class="keyword">auto</span> <span class="identifier">phi</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_scaling</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="number">8</span><span class="special">>(</span><span class="number">12</span><span class="special">);</span>
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</pre>
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<p>
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creates a Daubechies scaling function interpolated from a dyadic grid computed
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down to depth <span class="emphasis"><em>j</em></span> = 12. The call to the constructor is exponential
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time in the number of grid refinements, and the call operator, <code class="computeroutput"><span class="special">.</span><span class="identifier">prime</span></code>, and
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<code class="computeroutput"><span class="special">.</span><span class="identifier">double_prime</span></code>
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are constant time.
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</p>
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<p>
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Note that the only reason that this is a class, rather than a free function
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is that the dyadic grids would make the Boost source download extremely large.
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Hence, it may make sense to precompute the dyadic grid and dump it in a <code class="computeroutput"><span class="special">.</span><span class="identifier">cpp</span></code> file;
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this can be achieved via
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</p>
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<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">float128</span><span class="special">;</span>
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<span class="keyword">int</span> <span class="identifier">grid_refinements</span> <span class="special">=</span> <span class="number">12</span><span class="special">;</span>
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<span class="keyword">constexpr</span> <span class="keyword">const</span> <span class="identifier">derivative</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span>
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<span class="keyword">constexpr</span> <span class="keyword">const</span> <span class="identifier">p</span> <span class="special">=</span> <span class="number">8</span><span class="special">;</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">float128</span><span class="special">></span> <span class="identifier">v</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">dyadic_grid</span><span class="special"><</span><span class="identifier">float128</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">derivative</span><span class="special">>(</span><span class="identifier">grid_refinements</span><span class="special">);</span>
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</pre>
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<p>
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Note that quad precision is the most accurate precision provided, for both
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the dyadic grid and for the scaling function. 1ULP accuracy can only be achieved
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for float and double precision, in well-conditioned regions.
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</p>
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<p>
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Derivatives are only available if the wavelet and scaling function has sufficient
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smoothness. The compiler will gladly inform you of your error if you try to
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call <code class="computeroutput"><span class="special">.</span><span class="identifier">prime</span></code>
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on <sub>2</sub>φ, which is not differentiable, but be aware that smoothness increases
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with the number of vanishing moments.
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</p>
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<p>
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The axioms of a multiresolution analysis ensure that integer shifts of the
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scaling functions are elements of the multiresolution analysis; a side effect
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is that the supports of the (unshifted) wavelet and scaling functions are arbitrary.
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For this reason, we have provided <code class="computeroutput"><span class="special">.</span><span class="identifier">support</span><span class="special">()</span></code>
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so that you can check our conventions:
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</p>
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<pre class="programlisting"><span class="keyword">auto</span> <span class="special">[</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">phi</span><span class="special">.</span><span class="identifier">support</span><span class="special">();</span>
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</pre>
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<p>
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For definiteness though, for the scaling function, the support is always [0,
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<span class="emphasis"><em>2p</em></span> - 1], and the support of the wavelet is [ -<span class="emphasis"><em>p</em></span>
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+ 1, <span class="emphasis"><em>p</em></span>].
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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_2_scaling.svg"></object></span> The 2 vanishing
|
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moment scaling function.
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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_8_scaling.svg"></object></span> The 8 vanishing
|
||
moment scaling function.
|
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</p>
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<h4>
|
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<a name="math_toolkit.daubechies.h1"></a>
|
||
<span class="phrase"><a name="math_toolkit.daubechies.references"></a></span><a class="link" href="daubechies.html#math_toolkit.daubechies.references">References</a>
|
||
</h4>
|
||
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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||
Daubechies, Ingrid. <span class="emphasis"><em>Ten Lectures on Wavelets.</em></span> Vol.
|
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61. Siam, 1992.
|
||
</li>
|
||
<li class="listitem">
|
||
Mallat, Stephane. <span class="emphasis"><em>A Wavelet Tour of Signal Processing: the sparse
|
||
way</em></span> Academic press, 2008.
|
||
</li>
|
||
<li class="listitem">
|
||
Thompson, Nicholas, Maddock, John et al. <span class="emphasis"><em>Towards 1ULP Evaluation
|
||
of Daubechies Wavelets</em></span> https://arxiv.org/ftp/arxiv/papers/2005/2005.05424.pdf
|
||
</li>
|
||
</ul></div>
|
||
</div>
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||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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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,
|
||
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>)
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