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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 title="pywt.dwt" class="reference external" href="../ref/dwt-discrete-wavelet-transform.html#pywt.dwt"><tt class="xref docutils literal"><span class="pre">Discrete</span> <span class="pre">Wavelet</span> <span class="pre">Transform</span></tt></a> of a sample data <em>x</em> using
the <tt class="docutils literal"><span class="pre">db2</span></tt> wavelet. It’s simple..</p>
<div class="highlight-python"><div class="highlight"><pre><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="s">'db2'</span><span class="p">)</span>
</pre></div>
</div>
<p>And the approximation and details coefficients are in <tt class="docutils literal"><span class="pre">cA</span></tt> and <tt class="docutils literal"><span class="pre">cD</span></tt>
respectively:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="k">print</span> <span class="n">cA</span>
<span class="go">[ 5.65685425 7.39923721 0.22414387 3.33677403 7.77817459]</span>
<span class="gp">>>> </span><span class="k">print</span> <span class="n">cD</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 title="pywt.idwt" class="reference external" href="../ref/idwt-inverse-discrete-wavelet-transform.html#pywt.idwt"><tt class="xref docutils literal"><span class="pre">Inverse</span> <span class="pre">Discrete</span> <span class="pre">Wavelet</span> <span class="pre">Transform</span></tt></a>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="k">print</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="s">'db2'</span><span class="p">)</span>
<span class="go">[ 3. 7. 1. 1. -2. 5. 4. 6.]</span>
</pre></div>
</div>
<p>Violla! 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 title="pywt.dwt" class="reference external" href="../ref/dwt-discrete-wavelet-transform.html#pywt.dwt"><tt class="xref docutils literal"><span class="pre">dwt()</span></tt></a> some more. For example let’s pass a
<a title="pywt.Wavelet" class="reference external" href="../ref/wavelets.html#pywt.Wavelet"><tt class="xref docutils literal"><span class="pre">Wavelet</span></tt></a> object instead of the wavelet name and specify signal extension
mode (the default is <a class="reference external" href="../ref/signal-extension-modes.html#modes-sym"><em>sym</em></a>) for the border effect handling:</p>
<div class="highlight-python"><div class="highlight"><pre><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="s">'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="s">'cpd'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="k">print</span> <span class="n">cA</span>
<span class="go">[ 4.38354585 3.80302657 7.31813271 -0.58565539 4.09727044 7.81994027]</span>
<span class="gp">>>> </span><span class="k">print</span> <span class="n">cD</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 lenght depends not only on the input
data length but also on the :class:Wavelet type (particularly on it’s
<tt class="xref docutils literal"><span class="pre">filters</span> <span class="pre">lenght</span></tt> that are used in the transformation).</p>
<p>To find out what will be the output data size use the <a title="pywt.dwt_coeff_len" class="reference external" href="../ref/dwt-discrete-wavelet-transform.html#pywt.dwt_coeff_len"><tt class="xref docutils literal"><span class="pre">dwt_coeff_len()</span></tt></a>
function:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </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="s">'sym'</span><span class="p">)</span>
<span class="go">6</span>
<span class="gp">>>> </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="s">'sym'</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 tradeoff that allows for the perfect
reconstruction...).</p>
<p>The third argument of the <a title="pywt.dwt_coeff_len" class="reference external" href="../ref/dwt-discrete-wavelet-transform.html#pywt.dwt_coeff_len"><tt class="xref docutils literal"><span class="pre">dwt_coeff_len()</span></tt></a> is the already mentioned signal
extension mode (please refer to the PyWavelets’ documentation for the
<a class="reference external" href="../ref/signal-extension-modes.html#modes"><em>modes</em></a> description). Currently there are six
<a class="reference external" href="../ref/signal-extension-modes.html#modes"><em>extension modes</em></a> available:</p>
<blockquote>
<div class="highlight-python"><div class="highlight"><pre><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">['zpd', 'cpd', 'sym', 'ppd', 'sp1', 'per']</span>
</pre></div>
</div>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </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="o">.</span><span class="n">dec_len</span><span class="p">,</span> <span class="n">mode</span><span class="p">)</span> <span class="k">for</span> <span class="n">mode</span> <span class="ow">in</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="p">]</span>
<span class="go">[6, 6, 6, 6, 6, 4]</span>
</pre></div>
</div>
</blockquote>
<p>As you see in the above example, the <a class="reference external" href="../ref/signal-extension-modes.html#modes-per"><em>per</em></a> (periodization) mode
is slightly different from the others. It’s aim when doing the <a title="pywt.dwt" class="reference external" href="../ref/dwt-discrete-wavelet-transform.html#pywt.dwt"><tt class="xref docutils literal"><span class="pre">DWT</span></tt></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 title="pywt.dwt" class="reference external" href="../ref/dwt-discrete-wavelet-transform.html#pywt.dwt"><tt class="xref docutils literal"><span class="pre">DWT</span></tt></a> and <a title="pywt.idwt" class="reference external" href="../ref/idwt-inverse-discrete-wavelet-transform.html#pywt.idwt"><tt class="xref docutils literal"><span class="pre">IDWT</span></tt></a>. Otherwise, it will produce
<strong>invalid results</strong>:</p>
<div class="highlight-python"><div class="highlight"><pre><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="s">'per'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="k">print</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="s">'sym3'</span><span class="p">,</span> <span class="s">'sym'</span><span class="p">)</span> <span class="c"># invalid mode</span>
<span class="go">[ 1. 1. -2. 5.]</span>
<span class="gp">>>> </span><span class="k">print</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="s">'sym3'</span><span class="p">,</span> <span class="s">'per'</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 <tt class="xref docutils literal"><span class="pre">None</span></tt> instead of coefficients data to <a title="pywt.idwt" class="reference external" href="../ref/idwt-inverse-discrete-wavelet-transform.html#pywt.idwt"><tt class="xref docutils literal"><span class="pre">idwt()</span></tt></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 <tt class="xref docutils literal"><span class="pre">None</span></tt> as one of the coefficient arrays
parameters is similar to passing a <em>zero-filled</em> array. The results are simply
the same:</p>
<blockquote>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="k">print</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="bp">None</span><span class="p">,</span> <span class="s">'db2'</span><span class="p">,</span> <span class="s">'sym'</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-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="k">print</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="s">'db2'</span><span class="p">,</span> <span class="s">'sym'</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-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="k">print</span> <span class="n">pywt</span><span class="o">.</span><span class="n">idwt</span><span class="p">(</span><span class="bp">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="s">'db2'</span><span class="p">,</span> <span class="s">'sym'</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-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="k">print</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="s">'db2'</span><span class="p">,</span> <span class="s">'sym'</span><span class="p">)</span>
<span class="go">[ 0.57769726 -0.93125065 1.67303261 -0.96592583 -0.12940952 -0.22414387]</span>
</pre></div>
</div>
</blockquote>
<p>Remember that only one argument at a time can be <tt class="xref docutils literal"><span class="pre">None</span></tt>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="k">print</span> <span class="n">pywt</span><span class="o">.</span><span class="n">idwt</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="s">'db2'</span><span class="p">,</span> <span class="s">'sym'</span><span class="p">)</span>
<span class="gt">Traceback (most recent call last):</span>
<span class="c">...</span>
<span class="nc">ValueError</span>: <span class="n-Identifier">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 title="pywt.idwt" class="reference external" href="../ref/idwt-inverse-discrete-wavelet-transform.html#pywt.idwt"><tt class="xref docutils literal"><span class="pre">idwt</span></tt></a><a class="headerlink" href="#coefficients-data-size-in-idwt" title="Permalink to this headline">¶</a></h3>
<p>When doing the <a title="pywt.idwt" class="reference external" href="../ref/idwt-inverse-discrete-wavelet-transform.html#pywt.idwt"><tt class="xref docutils literal"><span class="pre">IDWT</span></tt></a> transform, usually the coefficient arrays
must have the same size.</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="k">print</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="s">'db2'</span><span class="p">,</span> <span class="s">'sym'</span><span class="p">)</span>
<span class="gt">Traceback (most recent call last):</span>
<span class="c">...</span>
<span class="nc">ValueError</span>: <span class="n-Identifier">Coefficients arrays must have the same size.</span>
</pre></div>
</div>
<p>But for some applications like multilevel DWT and IDWT it is sometimes conveniet
to allow for a small departure from this behaviour. When the <em>correct_size</em> flag
is set, the approximation coefficients array can be larger from the details
coefficient array by one element:</p>
<blockquote>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="k">print</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="s">'db2'</span><span class="p">,</span> <span class="s">'sym'</span><span class="p">,</span> <span class="n">correct_size</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
<span class="go">[ 1.76776695 0.61237244 3.18198052 0.61237244 4.59619408 0.61237244]</span>
</pre></div>
</div>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="k">print</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="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="s">'db2'</span><span class="p">,</span> <span class="s">'sym'</span><span class="p">,</span> <span class="n">correct_size</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
<span class="gt">Traceback (most recent call last):</span>
<span class="c">...</span>
<span class="nc">ValueError</span>: <span class="n-Identifier">Coefficients arrays must satisfy (0 <= len(cA) - len(cD) <= 1).</span>
</pre></div>
</div>
</blockquote>
<p>Not every coefficient array can be used in <a title="pywt.idwt" class="reference external" href="../ref/idwt-inverse-discrete-wavelet-transform.html#pywt.idwt"><tt class="xref docutils literal"><span class="pre">IDWT</span></tt></a>. In the following
example the <a title="pywt.idwt" class="reference external" href="../ref/idwt-inverse-discrete-wavelet-transform.html#pywt.idwt"><tt class="xref docutils literal"><span class="pre">idwt()</span></tt></a> will fail because the input arrays are invalid - they
couldn’t be created as a result of <a title="pywt.dwt" class="reference external" href="../ref/dwt-discrete-wavelet-transform.html#pywt.dwt"><tt class="xref docutils literal"><span class="pre">DWT</span></tt></a>, beacuse the minimal output
length for dwt using <tt class="docutils literal"><span class="pre">db4</span></tt> wavelet and the <a class="reference external" href="../ref/signal-extension-modes.html#modes-sym"><em>sym</em></a> mode is
<tt class="docutils literal"><span class="pre">4</span></tt>, not <tt class="docutils literal"><span class="pre">3</span></tt>:</p>
<blockquote>
<div class="highlight-python"><div class="highlight"><pre><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="s">'db4'</span><span class="p">,</span> <span class="s">'sym'</span><span class="p">)</span>
<span class="gt">Traceback (most recent call last):</span>
<span class="c">...</span>
<span class="nc">ValueError</span>: <span class="n-Identifier">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-python"><div class="highlight"><pre><span class="gp">>>> </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="s">'db4'</span><span class="p">)</span><span class="o">.</span><span class="n">dec_len</span><span class="p">,</span> <span class="s">'sym'</span><span class="p">)</span>
<span class="go">4</span>
</pre></div>
</div>
</blockquote>
</div>
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<h3><a href="../index.html">Table Of Contents</a></h3>
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<li><a class="reference external" href="#">DWT and IDWT</a><ul>
<li><a class="reference external" href="#discrete-wavelet-transform">Discrete Wavelet Transform</a></li>
<li><a class="reference external" href="#inverse-discrete-wavelet-transform">Inverse Discrete Wavelet Transform</a></li>
<li><a class="reference external" href="#more-examples">More Examples</a></li>
<li><a class="reference external" href="#tips-tricks">Tips & tricks</a><ul>
<li><a class="reference external" href="#passing-none-instead-of-coefficients-data-to-idwt">Passing <tt class="xref docutils literal"><span class="pre">None</span></tt> instead of coefficients data to <tt class="docutils literal"><span class="pre">idwt()</span></tt></a></li>
<li><a class="reference external" href="#coefficients-data-size-in-idwt">Coefficients data size in <tt class="docutils literal"><span class="pre">idwt</span></tt></a></li>
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