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<div class="section" id="dwt-and-idwt">
<span id="reg-dwt-idwt"></span><h1>DWT and IDWT<a class="headerlink" href="#dwt-and-idwt" title="Permalink to this headline">¶</a></h1>
<div class="section" id="discrete-wavelet-transform">
<h2>Discrete Wavelet Transform<a class="headerlink" href="#discrete-wavelet-transform" title="Permalink to this headline">¶</a></h2>
<p>Let’s do a <a class="reference internal" href="../ref/dwt-discrete-wavelet-transform.html#pywt.dwt" title="pywt.dwt"><code class="xref py py-func docutils literal notranslate"><span class="pre">Discrete</span> <span class="pre">Wavelet</span> <span class="pre">Transform</span></code></a> of a sample data <code class="docutils literal notranslate"><span class="pre">x</span></code>
using the <code class="docutils literal notranslate"><span class="pre">db2</span></code> wavelet. It’s simple..</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">pywt</span>
<span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">6</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">cA</span><span class="p">,</span> <span class="n">cD</span> <span class="o">=</span> <span class="n">pywt</span><span class="o">.</span><span class="n">dwt</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="s1">'db2'</span><span class="p">)</span>
</pre></div>
</div>
<p>And the approximation and details coefficients are in <code class="docutils literal notranslate"><span class="pre">cA</span></code> and <code class="docutils literal notranslate"><span class="pre">cD</span></code>
respectively:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">cA</span><span class="p">)</span>
<span class="go">[ 5.65685425 7.39923721 0.22414387 3.33677403 7.77817459]</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">cD</span><span class="p">)</span>
<span class="go">[-2.44948974 -1.60368225 -4.44140056 -0.41361256 1.22474487]</span>
</pre></div>
</div>
</div>
<div class="section" id="inverse-discrete-wavelet-transform">
<h2>Inverse Discrete Wavelet Transform<a class="headerlink" href="#inverse-discrete-wavelet-transform" title="Permalink to this headline">¶</a></h2>
<p>Now let’s do an opposite operation
- <a class="reference internal" href="../ref/idwt-inverse-discrete-wavelet-transform.html#pywt.idwt" title="pywt.idwt"><code class="xref py py-func docutils literal notranslate"><span class="pre">Inverse</span> <span class="pre">Discrete</span> <span class="pre">Wavelet</span> <span class="pre">Transform</span></code></a>:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">pywt</span><span class="o">.</span><span class="n">idwt</span><span class="p">(</span><span class="n">cA</span><span class="p">,</span> <span class="n">cD</span><span class="p">,</span> <span class="s1">'db2'</span><span class="p">))</span>
<span class="go">[ 3. 7. 1. 1. -2. 5. 4. 6.]</span>
</pre></div>
</div>
<p>Voilà! That’s it!</p>
</div>
<div class="section" id="more-examples">
<h2>More Examples<a class="headerlink" href="#more-examples" title="Permalink to this headline">¶</a></h2>
<p>Now let’s experiment with the <a class="reference internal" href="../ref/dwt-discrete-wavelet-transform.html#pywt.dwt" title="pywt.dwt"><code class="xref py py-func docutils literal notranslate"><span class="pre">dwt()</span></code></a> some more. For example let’s pass a
<a class="reference internal" href="../ref/wavelets.html#pywt.Wavelet" title="pywt.Wavelet"><code class="xref py py-class docutils literal notranslate"><span class="pre">Wavelet</span></code></a> object instead of the wavelet name and specify signal
extension mode (the default is <a class="reference internal" href="../ref/signal-extension-modes.html#modes-symmetric"><span class="std std-ref">symmetric</span></a>) for the
border effect handling:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">w</span> <span class="o">=</span> <span class="n">pywt</span><span class="o">.</span><span class="n">Wavelet</span><span class="p">(</span><span class="s1">'sym3'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">cA</span><span class="p">,</span> <span class="n">cD</span> <span class="o">=</span> <span class="n">pywt</span><span class="o">.</span><span class="n">dwt</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">wavelet</span><span class="o">=</span><span class="n">w</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s1">'constant'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">cA</span><span class="p">)</span>
<span class="go">[ 4.38354585 3.80302657 7.31813271 -0.58565539 4.09727044 7.81994027]</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">cD</span><span class="p">)</span>
<span class="go">[-1.33068221 -2.78795192 -3.16825651 -0.67715519 -0.09722957 -0.07045258]</span>
</pre></div>
</div>
<p>Note that the output coefficients arrays length depends not only on the input
data length but also on the :class:Wavelet type (particularly on its
<code class="xref py py-attr docutils literal notranslate"><span class="pre">filters</span> <span class="pre">length</span></code> that are used in the transformation).</p>
<p>To find out what will be the output data size use the <a class="reference internal" href="../ref/dwt-discrete-wavelet-transform.html#pywt.dwt_coeff_len" title="pywt.dwt_coeff_len"><code class="xref py py-func docutils literal notranslate"><span class="pre">dwt_coeff_len()</span></code></a>
function:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="c1"># int() is for normalizing Python integers and long integers for documentation tests</span>
<span class="gp">>>> </span><span class="nb">int</span><span class="p">(</span><span class="n">pywt</span><span class="o">.</span><span class="n">dwt_coeff_len</span><span class="p">(</span><span class="n">data_len</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">filter_len</span><span class="o">=</span><span class="n">w</span><span class="o">.</span><span class="n">dec_len</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s1">'symmetric'</span><span class="p">))</span>
<span class="go">6</span>
<span class="gp">>>> </span><span class="nb">int</span><span class="p">(</span><span class="n">pywt</span><span class="o">.</span><span class="n">dwt_coeff_len</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">w</span><span class="p">,</span> <span class="s1">'symmetric'</span><span class="p">))</span>
<span class="go">6</span>
<span class="gp">>>> </span><span class="nb">len</span><span class="p">(</span><span class="n">cA</span><span class="p">)</span>
<span class="go">6</span>
</pre></div>
</div>
<p>Looks fine. (And if you expected that the output length would be a half of the
input data length, well, that’s the trade-off that allows for the perfect
reconstruction…).</p>
<p>The third argument of the <a class="reference internal" href="../ref/dwt-discrete-wavelet-transform.html#pywt.dwt_coeff_len" title="pywt.dwt_coeff_len"><code class="xref py py-func docutils literal notranslate"><span class="pre">dwt_coeff_len()</span></code></a> is the already mentioned signal
extension mode (please refer to the PyWavelets’ documentation for the
<a class="reference internal" href="../ref/signal-extension-modes.html#modes"><span class="std std-ref">modes</span></a> description). Currently there are six
<a class="reference internal" href="../ref/signal-extension-modes.html#modes"><span class="std std-ref">extension modes</span></a> available:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">pywt</span><span class="o">.</span><span class="n">Modes</span><span class="o">.</span><span class="n">modes</span>
<span class="go">['zero', 'constant', 'symmetric', 'periodic', 'smooth', 'periodization', 'reflect']</span>
</pre></div>
</div>
<p>As you see in the above example, the <a class="reference internal" href="../ref/signal-extension-modes.html#modes-periodization"><span class="std std-ref">periodization</span></a>
(periodization) mode is slightly different from the others. It’s aim when
doing the <a class="reference internal" href="../ref/dwt-discrete-wavelet-transform.html#pywt.dwt" title="pywt.dwt"><code class="xref py py-func docutils literal notranslate"><span class="pre">DWT</span></code></a> transform is to output coefficients arrays that
are half of the length of the input data.</p>
<p>Knowing that, you should never mix the periodization mode with other modes when
doing <a class="reference internal" href="../ref/dwt-discrete-wavelet-transform.html#pywt.dwt" title="pywt.dwt"><code class="xref py py-func docutils literal notranslate"><span class="pre">DWT</span></code></a> and <a class="reference internal" href="../ref/idwt-inverse-discrete-wavelet-transform.html#pywt.idwt" title="pywt.idwt"><code class="xref py py-func docutils literal notranslate"><span class="pre">IDWT</span></code></a>. Otherwise, it will produce
<strong>invalid results</strong>:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span>
<span class="go">[3, 7, 1, 1, -2, 5, 4, 6]</span>
<span class="gp">>>> </span><span class="n">cA</span><span class="p">,</span> <span class="n">cD</span> <span class="o">=</span> <span class="n">pywt</span><span class="o">.</span><span class="n">dwt</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">wavelet</span><span class="o">=</span><span class="n">w</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s1">'periodization'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">pywt</span><span class="o">.</span><span class="n">idwt</span><span class="p">(</span><span class="n">cA</span><span class="p">,</span> <span class="n">cD</span><span class="p">,</span> <span class="s1">'sym3'</span><span class="p">,</span> <span class="s1">'symmetric'</span><span class="p">))</span> <span class="c1"># invalid mode</span>
<span class="go">[ 1. 1. -2. 5.]</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">pywt</span><span class="o">.</span><span class="n">idwt</span><span class="p">(</span><span class="n">cA</span><span class="p">,</span> <span class="n">cD</span><span class="p">,</span> <span class="s1">'sym3'</span><span class="p">,</span> <span class="s1">'periodization'</span><span class="p">))</span>
<span class="go">[ 3. 7. 1. 1. -2. 5. 4. 6.]</span>
</pre></div>
</div>
</div>
<div class="section" id="tips-tricks">
<h2>Tips & tricks<a class="headerlink" href="#tips-tricks" title="Permalink to this headline">¶</a></h2>
<div class="section" id="passing-none-instead-of-coefficients-data-to-idwt">
<h3>Passing <code class="docutils literal notranslate"><span class="pre">None</span></code> instead of coefficients data to <a class="reference internal" href="../ref/idwt-inverse-discrete-wavelet-transform.html#pywt.idwt" title="pywt.idwt"><code class="xref py py-func docutils literal notranslate"><span class="pre">idwt()</span></code></a><a class="headerlink" href="#passing-none-instead-of-coefficients-data-to-idwt" title="Permalink to this headline">¶</a></h3>
<p>Now some tips & tricks. Passing <code class="docutils literal notranslate"><span class="pre">None</span></code> as one of the coefficient arrays
parameters is similar to passing a <em>zero-filled</em> array. The results are simply
the same:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">pywt</span><span class="o">.</span><span class="n">idwt</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="kc">None</span><span class="p">,</span> <span class="s1">'db2'</span><span class="p">,</span> <span class="s1">'symmetric'</span><span class="p">))</span>
<span class="go">[ 1.19006969 1.54362308 0.44828774 -0.25881905 0.48296291 0.8365163 ]</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">pywt</span><span class="o">.</span><span class="n">idwt</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="s1">'db2'</span><span class="p">,</span> <span class="s1">'symmetric'</span><span class="p">))</span>
<span class="go">[ 1.19006969 1.54362308 0.44828774 -0.25881905 0.48296291 0.8365163 ]</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">pywt</span><span class="o">.</span><span class="n">idwt</span><span class="p">(</span><span class="kc">None</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="s1">'db2'</span><span class="p">,</span> <span class="s1">'symmetric'</span><span class="p">))</span>
<span class="go">[ 0.57769726 -0.93125065 1.67303261 -0.96592583 -0.12940952 -0.22414387]</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">pywt</span><span class="o">.</span><span class="n">idwt</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="s1">'db2'</span><span class="p">,</span> <span class="s1">'symmetric'</span><span class="p">))</span>
<span class="go">[ 0.57769726 -0.93125065 1.67303261 -0.96592583 -0.12940952 -0.22414387]</span>
</pre></div>
</div>
<p>Remember that only one argument at a time can be <code class="docutils literal notranslate"><span class="pre">None</span></code>:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">pywt</span><span class="o">.</span><span class="n">idwt</span><span class="p">(</span><span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="s1">'db2'</span><span class="p">,</span> <span class="s1">'symmetric'</span><span class="p">))</span>
<span class="gt">Traceback (most recent call last):</span>
<span class="c">...</span>
<span class="gr">ValueError</span>: <span class="n">At least one coefficient parameter must be specified.</span>
</pre></div>
</div>
</div>
<div class="section" id="coefficients-data-size-in-idwt">
<h3>Coefficients data size in <a class="reference internal" href="../ref/idwt-inverse-discrete-wavelet-transform.html#pywt.idwt" title="pywt.idwt"><code class="xref py py-attr docutils literal notranslate"><span class="pre">idwt</span></code></a><a class="headerlink" href="#coefficients-data-size-in-idwt" title="Permalink to this headline">¶</a></h3>
<p>When doing the <a class="reference internal" href="../ref/idwt-inverse-discrete-wavelet-transform.html#pywt.idwt" title="pywt.idwt"><code class="xref py py-func docutils literal notranslate"><span class="pre">IDWT</span></code></a> transform, usually the coefficient arrays
must have the same size.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">pywt</span><span class="o">.</span><span class="n">idwt</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="s1">'db2'</span><span class="p">,</span> <span class="s1">'symmetric'</span><span class="p">))</span>
<span class="gt">Traceback (most recent call last):</span>
<span class="c">...</span>
<span class="gr">ValueError</span>: <span class="n">Coefficients arrays must have the same size.</span>
</pre></div>
</div>
<p>Not every coefficient array can be used in <a class="reference internal" href="../ref/idwt-inverse-discrete-wavelet-transform.html#pywt.idwt" title="pywt.idwt"><code class="xref py py-func docutils literal notranslate"><span class="pre">IDWT</span></code></a>. In the
following example the <a class="reference internal" href="../ref/idwt-inverse-discrete-wavelet-transform.html#pywt.idwt" title="pywt.idwt"><code class="xref py py-func docutils literal notranslate"><span class="pre">idwt()</span></code></a> will fail because the input arrays are
invalid - they couldn’t be created as a result of <a class="reference internal" href="../ref/dwt-discrete-wavelet-transform.html#pywt.dwt" title="pywt.dwt"><code class="xref py py-func docutils literal notranslate"><span class="pre">DWT</span></code></a>, because
the minimal output length for dwt using <code class="docutils literal notranslate"><span class="pre">db4</span></code> wavelet and the <a class="reference internal" href="../ref/signal-extension-modes.html#modes-symmetric"><span class="std std-ref">symmetric</span></a> mode is <code class="docutils literal notranslate"><span class="pre">4</span></code>, not <code class="docutils literal notranslate"><span class="pre">3</span></code>:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">pywt</span><span class="o">.</span><span class="n">idwt</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">],</span> <span class="s1">'db4'</span><span class="p">,</span> <span class="s1">'symmetric'</span><span class="p">)</span>
<span class="gt">Traceback (most recent call last):</span>
<span class="c">...</span>
<span class="gr">ValueError</span>: <span class="n">Invalid coefficient arrays length for specified wavelet. Wavelet and mode must be the same as used for decomposition.</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="nb">int</span><span class="p">(</span><span class="n">pywt</span><span class="o">.</span><span class="n">dwt_coeff_len</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">pywt</span><span class="o">.</span><span class="n">Wavelet</span><span class="p">(</span><span class="s1">'db4'</span><span class="p">)</span><span class="o">.</span><span class="n">dec_len</span><span class="p">,</span> <span class="s1">'symmetric'</span><span class="p">))</span>
<span class="go">4</span>
</pre></div>
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<li><a class="reference internal" href="#">DWT and IDWT</a><ul>
<li><a class="reference internal" href="#discrete-wavelet-transform">Discrete Wavelet Transform</a></li>
<li><a class="reference internal" href="#inverse-discrete-wavelet-transform">Inverse Discrete Wavelet Transform</a></li>
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<li><a class="reference internal" href="#passing-none-instead-of-coefficients-data-to-idwt">Passing <code class="docutils literal notranslate"><span class="pre">None</span></code> instead of coefficients data to <code class="docutils literal notranslate"><span class="pre">idwt()</span></code></a></li>
<li><a class="reference internal" href="#coefficients-data-size-in-idwt">Coefficients data size in <code class="docutils literal notranslate"><span class="pre">idwt</span></code></a></li>
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