<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta http-equiv="X-UA-Compatible" content="IE=Edge" /> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <title>Wavelet Packets — PyWavelets Documentation</title> <link rel="stylesheet" href="../_static/nature.css" type="text/css" /> <link rel="stylesheet" href="../_static/pygments.css" type="text/css" /> <script type="text/javascript" id="documentation_options" data-url_root="../" src="../_static/documentation_options.js"></script> <script type="text/javascript" src="../_static/jquery.js"></script> <script type="text/javascript" src="../_static/underscore.js"></script> <script type="text/javascript" src="../_static/doctools.js"></script> <script type="text/javascript" src="../_static/language_data.js"></script> <script async="async" type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/latest.js?config=TeX-AMS-MML_HTMLorMML"></script> <link rel="search" type="application/opensearchdescription+xml" title="Search within PyWavelets Documentation" href="../_static/opensearch.xml"/> <link rel="shortcut icon" href="../_static/favicon.ico"/> <link rel="index" title="Index" href="../genindex.html" /> <link rel="search" title="Search" href="../search.html" /> <link rel="next" title="2D Wavelet Packets" href="wp2d.html" /> <link rel="prev" title="Multilevel DWT, IDWT and SWT" href="multilevel.html" /> <meta name="description" content="PyWavelets is a scientific Python module for Wavelet Transform calculations."/> <meta name="keywords" content="PyWavelets, wavelets, Python, wavelet transform, discrete wavelet transform, dwt, idwt, swt, wavelet packets, stationary wavelet transform, pywt"/> <meta name="author" content="Filip Wasilewski"/> <meta name="Distribution" content="Global"/> <meta name="Robots" content="INDEX,FOLLOW"/> <script type="text/javascript"> (function ($) { $(document).ready(function () { $("#toggle-edit-info").click(function (e) { e.preventDefault(); $("#edit-info").toggle(); }); }); })(jQuery); </script> <script type="text/javascript"> var _gaq = _gaq || []; _gaq.push(['_setAccount', 'UA-3396395-3']); _gaq.push(['_trackPageview']); (function () { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); </script> </head><body> <div class="related" role="navigation" aria-label="related navigation"> <h3>Navigation</h3> <ul> <li class="right" style="margin-right: 10px"> <a href="../genindex.html" title="General Index" accesskey="I">index</a></li> <li class="right" > <a href="wp2d.html" title="2D Wavelet Packets" accesskey="N">next</a> |</li> <li class="right" > <a href="multilevel.html" title="Multilevel DWT, IDWT and SWT" accesskey="P">previous</a> |</li> <li><a href="../index.html">Home »</a></li> <li class="nav-item nav-item-1"><a href="index.html" accesskey="U">Usage examples</a> »</li> </ul> </div> <div class="document"> <div class="documentwrapper"> <div class="bodywrapper"> <div class="body" role="main"> <div class="highlight-default notranslate" id="reg-wp"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">__future__</span> <span class="k">import</span> <span class="n">print_function</span> </pre></div> </div> <div class="section" id="wavelet-packets"> <h1>Wavelet Packets<a class="headerlink" href="#wavelet-packets" title="Permalink to this headline">¶</a></h1> <div class="section" id="import-pywt"> <h2>Import pywt<a class="headerlink" href="#import-pywt" title="Permalink to this headline">¶</a></h2> <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> </pre></div> </div> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="k">def</span> <span class="nf">format_array</span><span class="p">(</span><span class="n">a</span><span class="p">):</span> <span class="gp">... </span> <span class="sd">"""Consistent array representation across different systems"""</span> <span class="gp">... </span> <span class="kn">import</span> <span class="nn">numpy</span> <span class="gp">... </span> <span class="n">a</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o"><</span> <span class="mf">1e-5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span> <span class="gp">... </span> <span class="k">return</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array2string</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">precision</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">separator</span><span class="o">=</span><span class="s1">' '</span><span class="p">,</span> <span class="n">suppress_small</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span> </pre></div> </div> </div> <div class="section" id="create-wavelet-packet-structure"> <h2>Create Wavelet Packet structure<a class="headerlink" href="#create-wavelet-packet-structure" title="Permalink to this headline">¶</a></h2> <p>Ok, let’s create a sample <a class="reference internal" href="../ref/wavelet-packets.html#pywt.WaveletPacket" title="pywt.WaveletPacket"><code class="xref py py-class docutils literal notranslate"><span class="pre">WaveletPacket</span></code></a>:</p> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</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="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">]</span> <span class="gp">>>> </span><span class="n">wp</span> <span class="o">=</span> <span class="n">pywt</span><span class="o">.</span><span class="n">WaveletPacket</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">x</span><span class="p">,</span> <span class="n">wavelet</span><span class="o">=</span><span class="s1">'db1'</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s1">'symmetric'</span><span class="p">)</span> </pre></div> </div> <p>The input <code class="docutils literal notranslate"><span class="pre">data</span></code> and decomposition coefficients are stored in the <code class="xref py py-attr docutils literal notranslate"><span class="pre">WaveletPacket.data</span></code> attribute:</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">wp</span><span class="o">.</span><span class="n">data</span><span class="p">)</span> <span class="go">[1, 2, 3, 4, 5, 6, 7, 8]</span> </pre></div> </div> <p><a class="reference internal" href="../ref/wavelet-packets.html#pywt.Node" title="pywt.Node"><code class="xref py py-class docutils literal notranslate"><span class="pre">Nodes</span></code></a> are identified by <code class="xref py py-attr docutils literal notranslate"><span class="pre">paths</span></code>. For the root node the path is <code class="docutils literal notranslate"><span class="pre">''</span></code> and the decomposition level is <code class="docutils literal notranslate"><span class="pre">0</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="nb">repr</span><span class="p">(</span><span class="n">wp</span><span class="o">.</span><span class="n">path</span><span class="p">))</span> <span class="go">''</span> <span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">wp</span><span class="o">.</span><span class="n">level</span><span class="p">)</span> <span class="go">0</span> </pre></div> </div> <p>The <code class="docutils literal notranslate"><span class="pre">maxlevel</span></code>, if not given as param in the constructor, is automatically computed:</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">wp</span><span class="p">[</span><span class="s1">'ad'</span><span class="p">]</span><span class="o">.</span><span class="n">maxlevel</span><span class="p">)</span> <span class="go">3</span> </pre></div> </div> </div> <div class="section" id="traversing-wp-tree"> <h2>Traversing WP tree:<a class="headerlink" href="#traversing-wp-tree" title="Permalink to this headline">¶</a></h2> <div class="section" id="accessing-subnodes"> <h3>Accessing subnodes:<a class="headerlink" href="#accessing-subnodes" title="Permalink to this headline">¶</a></h3> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</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="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">]</span> <span class="gp">>>> </span><span class="n">wp</span> <span class="o">=</span> <span class="n">pywt</span><span class="o">.</span><span class="n">WaveletPacket</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">x</span><span class="p">,</span> <span class="n">wavelet</span><span class="o">=</span><span class="s1">'db1'</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s1">'symmetric'</span><span class="p">)</span> </pre></div> </div> <p>First check what is the maximum level of decomposition:</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">wp</span><span class="o">.</span><span class="n">maxlevel</span><span class="p">)</span> <span class="go">3</span> </pre></div> </div> <p>and try accessing subnodes of the WP tree:</p> <blockquote> <div><ul> <li><p class="first">1st level:</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">wp</span><span class="p">[</span><span class="s1">'a'</span><span class="p">]</span><span class="o">.</span><span class="n">data</span><span class="p">)</span> <span class="go">[ 2.12132034 4.94974747 7.77817459 10.60660172]</span> <span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">wp</span><span class="p">[</span><span class="s1">'a'</span><span class="p">]</span><span class="o">.</span><span class="n">path</span><span class="p">)</span> <span class="go">a</span> </pre></div> </div> </li> <li><p class="first">2nd level:</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">wp</span><span class="p">[</span><span class="s1">'aa'</span><span class="p">]</span><span class="o">.</span><span class="n">data</span><span class="p">)</span> <span class="go">[ 5. 13.]</span> <span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">wp</span><span class="p">[</span><span class="s1">'aa'</span><span class="p">]</span><span class="o">.</span><span class="n">path</span><span class="p">)</span> <span class="go">aa</span> </pre></div> </div> </li> <li><p class="first">3rd level:</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">wp</span><span class="p">[</span><span class="s1">'aaa'</span><span class="p">]</span><span class="o">.</span><span class="n">data</span><span class="p">)</span> <span class="go">[ 12.72792206]</span> <span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">wp</span><span class="p">[</span><span class="s1">'aaa'</span><span class="p">]</span><span class="o">.</span><span class="n">path</span><span class="p">)</span> <span class="go">aaa</span> </pre></div> </div> <p>Ups, we have reached the maximum level of decomposition and got an <code class="xref py py-exc docutils literal notranslate"><span class="pre">IndexError</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">wp</span><span class="p">[</span><span class="s1">'aaaa'</span><span class="p">]</span><span class="o">.</span><span class="n">data</span><span class="p">)</span> <span class="gt">Traceback (most recent call last):</span> <span class="c">...</span> <span class="gr">IndexError</span>: <span class="n">Path length is out of range.</span> </pre></div> </div> </li> </ul> </div></blockquote> <p>Now try some invalid path:</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">wp</span><span class="p">[</span><span class="s1">'ac'</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">Subnode name must be in ['a', 'd'], not 'c'.</span> </pre></div> </div> <p>which just yielded a <code class="xref py py-exc docutils literal notranslate"><span class="pre">ValueError</span></code>.</p> </div> <div class="section" id="accessing-node-s-attributes"> <h3>Accessing Node’s attributes:<a class="headerlink" href="#accessing-node-s-attributes" title="Permalink to this headline">¶</a></h3> <p><a class="reference internal" href="../ref/wavelet-packets.html#pywt.WaveletPacket" title="pywt.WaveletPacket"><code class="xref py py-class docutils literal notranslate"><span class="pre">WaveletPacket</span></code></a> object is a tree data structure, which evaluates to a set of <a class="reference internal" href="../ref/wavelet-packets.html#pywt.Node" title="pywt.Node"><code class="xref py py-class docutils literal notranslate"><span class="pre">Node</span></code></a> objects. <a class="reference internal" href="../ref/wavelet-packets.html#pywt.WaveletPacket" title="pywt.WaveletPacket"><code class="xref py py-class docutils literal notranslate"><span class="pre">WaveletPacket</span></code></a> is just a special subclass of the <a class="reference internal" href="../ref/wavelet-packets.html#pywt.Node" title="pywt.Node"><code class="xref py py-class docutils literal notranslate"><span class="pre">Node</span></code></a> class (which in turn inherits from the <a class="reference internal" href="../ref/wavelet-packets.html#pywt.BaseNode" title="pywt.BaseNode"><code class="xref py py-class docutils literal notranslate"><span class="pre">BaseNode</span></code></a>).</p> <p>Tree nodes can be accessed using the <code class="docutils literal notranslate"><span class="pre">obj[x]</span></code> (<code class="xref py py-meth docutils literal notranslate"><span class="pre">Node.__getitem__()</span></code>) operator. Each tree node has a set of attributes: <code class="xref py py-attr docutils literal notranslate"><span class="pre">data</span></code>, <code class="xref py py-attr docutils literal notranslate"><span class="pre">path</span></code>, <code class="xref py py-attr docutils literal notranslate"><span class="pre">node_name</span></code>, <code class="xref py py-attr docutils literal notranslate"><span class="pre">parent</span></code>, <code class="xref py py-attr docutils literal notranslate"><span class="pre">level</span></code>, <code class="xref py py-attr docutils literal notranslate"><span class="pre">maxlevel</span></code> and <code class="xref py py-attr docutils literal notranslate"><span class="pre">mode</span></code>.</p> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</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="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">]</span> <span class="gp">>>> </span><span class="n">wp</span> <span class="o">=</span> <span class="n">pywt</span><span class="o">.</span><span class="n">WaveletPacket</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">x</span><span class="p">,</span> <span class="n">wavelet</span><span class="o">=</span><span class="s1">'db1'</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s1">'symmetric'</span><span class="p">)</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">wp</span><span class="p">[</span><span class="s1">'ad'</span><span class="p">]</span><span class="o">.</span><span class="n">data</span><span class="p">)</span> <span class="go">[-2. -2.]</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">wp</span><span class="p">[</span><span class="s1">'ad'</span><span class="p">]</span><span class="o">.</span><span class="n">path</span><span class="p">)</span> <span class="go">ad</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">wp</span><span class="p">[</span><span class="s1">'ad'</span><span class="p">]</span><span class="o">.</span><span class="n">node_name</span><span class="p">)</span> <span class="go">d</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">wp</span><span class="p">[</span><span class="s1">'ad'</span><span class="p">]</span><span class="o">.</span><span class="n">parent</span><span class="o">.</span><span class="n">path</span><span class="p">)</span> <span class="go">a</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">wp</span><span class="p">[</span><span class="s1">'ad'</span><span class="p">]</span><span class="o">.</span><span class="n">level</span><span class="p">)</span> <span class="go">2</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">wp</span><span class="p">[</span><span class="s1">'ad'</span><span class="p">]</span><span class="o">.</span><span class="n">maxlevel</span><span class="p">)</span> <span class="go">3</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">wp</span><span class="p">[</span><span class="s1">'ad'</span><span class="p">]</span><span class="o">.</span><span class="n">mode</span><span class="p">)</span> <span class="go">symmetric</span> </pre></div> </div> </div> <div class="section" id="collecting-nodes"> <h3>Collecting nodes<a class="headerlink" href="#collecting-nodes" title="Permalink to this headline">¶</a></h3> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</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="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">]</span> <span class="gp">>>> </span><span class="n">wp</span> <span class="o">=</span> <span class="n">pywt</span><span class="o">.</span><span class="n">WaveletPacket</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">x</span><span class="p">,</span> <span class="n">wavelet</span><span class="o">=</span><span class="s1">'db1'</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s1">'symmetric'</span><span class="p">)</span> </pre></div> </div> <p>We can get all nodes on the particular level either in <code class="docutils literal notranslate"><span class="pre">natural</span></code> order:</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">node</span><span class="o">.</span><span class="n">path</span> <span class="k">for</span> <span class="n">node</span> <span class="ow">in</span> <span class="n">wp</span><span class="o">.</span><span class="n">get_level</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="s1">'natural'</span><span class="p">)])</span> <span class="go">['aaa', 'aad', 'ada', 'add', 'daa', 'dad', 'dda', 'ddd']</span> </pre></div> </div> <p>or sorted based on the band frequency (<code class="docutils literal notranslate"><span class="pre">freq</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">node</span><span class="o">.</span><span class="n">path</span> <span class="k">for</span> <span class="n">node</span> <span class="ow">in</span> <span class="n">wp</span><span class="o">.</span><span class="n">get_level</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="s1">'freq'</span><span class="p">)])</span> <span class="go">['aaa', 'aad', 'add', 'ada', 'dda', 'ddd', 'dad', 'daa']</span> </pre></div> </div> <p>Note that <a class="reference internal" href="../ref/wavelet-packets.html#pywt.WaveletPacket.get_level" title="pywt.WaveletPacket.get_level"><code class="xref py py-meth docutils literal notranslate"><span class="pre">WaveletPacket.get_level()</span></code></a> also performs automatic decomposition until it reaches the specified <code class="docutils literal notranslate"><span class="pre">level</span></code>.</p> </div> </div> <div class="section" id="reconstructing-data-from-wavelet-packets"> <h2>Reconstructing data from Wavelet Packets:<a class="headerlink" href="#reconstructing-data-from-wavelet-packets" title="Permalink to this headline">¶</a></h2> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</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="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">]</span> <span class="gp">>>> </span><span class="n">wp</span> <span class="o">=</span> <span class="n">pywt</span><span class="o">.</span><span class="n">WaveletPacket</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">x</span><span class="p">,</span> <span class="n">wavelet</span><span class="o">=</span><span class="s1">'db1'</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s1">'symmetric'</span><span class="p">)</span> </pre></div> </div> <p>Now create a new <a class="reference internal" href="../ref/wavelet-packets.html#pywt.WaveletPacket" title="pywt.WaveletPacket"><code class="xref py py-class docutils literal notranslate"><span class="pre">Wavelet</span> <span class="pre">Packet</span></code></a> and set its nodes with some data.</p> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">new_wp</span> <span class="o">=</span> <span class="n">pywt</span><span class="o">.</span><span class="n">WaveletPacket</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">wavelet</span><span class="o">=</span><span class="s1">'db1'</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s1">'symmetric'</span><span class="p">)</span> </pre></div> </div> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">new_wp</span><span class="p">[</span><span class="s1">'aa'</span><span class="p">]</span> <span class="o">=</span> <span class="n">wp</span><span class="p">[</span><span class="s1">'aa'</span><span class="p">]</span><span class="o">.</span><span class="n">data</span> <span class="gp">>>> </span><span class="n">new_wp</span><span class="p">[</span><span class="s1">'ad'</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mf">2.</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.</span><span class="p">]</span> </pre></div> </div> <p>For convenience, <code class="xref py py-attr docutils literal notranslate"><span class="pre">Node.data</span></code> gets automatically extracted from the <a class="reference internal" href="../ref/wavelet-packets.html#pywt.Node" title="pywt.Node"><code class="xref py py-class docutils literal notranslate"><span class="pre">Node</span></code></a> object:</p> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">new_wp</span><span class="p">[</span><span class="s1">'d'</span><span class="p">]</span> <span class="o">=</span> <span class="n">wp</span><span class="p">[</span><span class="s1">'d'</span><span class="p">]</span> </pre></div> </div> <p>And reconstruct the data from the <code class="docutils literal notranslate"><span class="pre">aa</span></code>, <code class="docutils literal notranslate"><span class="pre">ad</span></code> and <code class="docutils literal notranslate"><span class="pre">d</span></code> packets.</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">new_wp</span><span class="o">.</span><span class="n">reconstruct</span><span class="p">(</span><span class="n">update</span><span class="o">=</span><span class="kc">False</span><span class="p">))</span> <span class="go">[ 1. 2. 3. 4. 5. 6. 7. 8.]</span> </pre></div> </div> <p>If the <code class="docutils literal notranslate"><span class="pre">update</span></code> param in the reconstruct method is set to <code class="docutils literal notranslate"><span class="pre">False</span></code>, the node’s <code class="xref py py-attr docutils literal notranslate"><span class="pre">data</span></code> will not be updated.</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">new_wp</span><span class="o">.</span><span class="n">data</span><span class="p">)</span> <span class="go">None</span> </pre></div> </div> <p>Otherwise, the <code class="xref py py-attr docutils literal notranslate"><span class="pre">data</span></code> attribute will be set to the reconstructed value.</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">new_wp</span><span class="o">.</span><span class="n">reconstruct</span><span class="p">(</span><span class="n">update</span><span class="o">=</span><span class="kc">True</span><span class="p">))</span> <span class="go">[ 1. 2. 3. 4. 5. 6. 7. 8.]</span> <span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">new_wp</span><span class="o">.</span><span class="n">data</span><span class="p">)</span> <span class="go">[ 1. 2. 3. 4. 5. 6. 7. 8.]</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">n</span><span class="o">.</span><span class="n">path</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">new_wp</span><span class="o">.</span><span class="n">get_leaf_nodes</span><span class="p">(</span><span class="kc">False</span><span class="p">)])</span> <span class="go">['aa', 'ad', 'd']</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">n</span><span class="o">.</span><span class="n">path</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">new_wp</span><span class="o">.</span><span class="n">get_leaf_nodes</span><span class="p">(</span><span class="kc">True</span><span class="p">)])</span> <span class="go">['aaa', 'aad', 'ada', 'add', 'daa', 'dad', 'dda', 'ddd']</span> </pre></div> </div> </div> <div class="section" id="removing-nodes-from-wavelet-packet-tree"> <h2>Removing nodes from Wavelet Packet tree:<a class="headerlink" href="#removing-nodes-from-wavelet-packet-tree" title="Permalink to this headline">¶</a></h2> <p>Let’s create a sample data:</p> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</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="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">]</span> <span class="gp">>>> </span><span class="n">wp</span> <span class="o">=</span> <span class="n">pywt</span><span class="o">.</span><span class="n">WaveletPacket</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">x</span><span class="p">,</span> <span class="n">wavelet</span><span class="o">=</span><span class="s1">'db1'</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s1">'symmetric'</span><span class="p">)</span> </pre></div> </div> <p>First, start with a tree decomposition at level 2. Leaf nodes in the tree are:</p> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">dummy</span> <span class="o">=</span> <span class="n">wp</span><span class="o">.</span><span class="n">get_level</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="gp">>>> </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">wp</span><span class="o">.</span><span class="n">get_leaf_nodes</span><span class="p">(</span><span class="kc">False</span><span class="p">):</span> <span class="gp">... </span> <span class="nb">print</span><span class="p">(</span><span class="n">n</span><span class="o">.</span><span class="n">path</span><span class="p">,</span> <span class="n">format_array</span><span class="p">(</span><span class="n">n</span><span class="o">.</span><span class="n">data</span><span class="p">))</span> <span class="go">aa [ 5. 13.]</span> <span class="go">ad [-2. -2.]</span> <span class="go">da [-1. -1.]</span> <span class="go">dd [ 0. 0.]</span> </pre></div> </div> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">node</span> <span class="o">=</span> <span class="n">wp</span><span class="p">[</span><span class="s1">'ad'</span><span class="p">]</span> <span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">node</span><span class="p">)</span> <span class="go">ad: [-2. -2.]</span> </pre></div> </div> <p>To remove a node from the WP tree, use Python’s <code class="docutils literal notranslate"><span class="pre">del</span> <span class="pre">obj[x]</span></code> (<code class="xref py py-class docutils literal notranslate"><span class="pre">Node.__delitem__</span></code>):</p> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="k">del</span> <span class="n">wp</span><span class="p">[</span><span class="s1">'ad'</span><span class="p">]</span> </pre></div> </div> <p>The leaf nodes that left in the tree are:</p> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">wp</span><span class="o">.</span><span class="n">get_leaf_nodes</span><span class="p">():</span> <span class="gp">... </span> <span class="nb">print</span><span class="p">(</span><span class="n">n</span><span class="o">.</span><span class="n">path</span><span class="p">,</span> <span class="n">format_array</span><span class="p">(</span><span class="n">n</span><span class="o">.</span><span class="n">data</span><span class="p">))</span> <span class="go">aa [ 5. 13.]</span> <span class="go">da [-1. -1.]</span> <span class="go">dd [ 0. 0.]</span> </pre></div> </div> <p>And the reconstruction is:</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">wp</span><span class="o">.</span><span class="n">reconstruct</span><span class="p">())</span> <span class="go">[ 2. 3. 2. 3. 6. 7. 6. 7.]</span> </pre></div> </div> <p>Now restore the deleted node value.</p> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">wp</span><span class="p">[</span><span class="s1">'ad'</span><span class="p">]</span><span class="o">.</span><span class="n">data</span> <span class="o">=</span> <span class="n">node</span><span class="o">.</span><span class="n">data</span> </pre></div> </div> <p>Printing leaf nodes and tree reconstruction confirms the original state of the tree:</p> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">wp</span><span class="o">.</span><span class="n">get_leaf_nodes</span><span class="p">(</span><span class="kc">False</span><span class="p">):</span> <span class="gp">... </span> <span class="nb">print</span><span class="p">(</span><span class="n">n</span><span class="o">.</span><span class="n">path</span><span class="p">,</span> <span class="n">format_array</span><span class="p">(</span><span class="n">n</span><span class="o">.</span><span class="n">data</span><span class="p">))</span> <span class="go">aa [ 5. 13.]</span> <span class="go">ad [-2. -2.]</span> <span class="go">da [-1. -1.]</span> <span class="go">dd [ 0. 0.]</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">wp</span><span class="o">.</span><span class="n">reconstruct</span><span class="p">())</span> <span class="go">[ 1. 2. 3. 4. 5. 6. 7. 8.]</span> </pre></div> </div> </div> <div class="section" id="lazy-evaluation"> <h2>Lazy evaluation:<a class="headerlink" href="#lazy-evaluation" title="Permalink to this headline">¶</a></h2> <div class="admonition note"> <p class="first admonition-title">Note</p> <p class="last">This section is for demonstration of pywt internals purposes only. Do not rely on the attribute access to nodes as presented in this example.</p> </div> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</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="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">]</span> <span class="gp">>>> </span><span class="n">wp</span> <span class="o">=</span> <span class="n">pywt</span><span class="o">.</span><span class="n">WaveletPacket</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">x</span><span class="p">,</span> <span class="n">wavelet</span><span class="o">=</span><span class="s1">'db1'</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s1">'symmetric'</span><span class="p">)</span> </pre></div> </div> <ol class="arabic"> <li><p class="first">At first the wp’s attribute <code class="docutils literal notranslate"><span class="pre">a</span></code> is None</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">wp</span><span class="o">.</span><span class="n">a</span><span class="p">)</span> <span class="go">None</span> </pre></div> </div> <p><strong>Remember that you should not rely on the attribute access.</strong></p> </li> <li><p class="first">At first attempt to access the node it is computed via decomposition of its parent node (the wp object itself).</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">wp</span><span class="p">[</span><span class="s1">'a'</span><span class="p">])</span> <span class="go">a: [ 2.12132034 4.94974747 7.77817459 10.60660172]</span> </pre></div> </div> </li> <li><p class="first">Now the <code class="docutils literal notranslate"><span class="pre">wp.a</span></code> is set to the newly created node:</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">wp</span><span class="o">.</span><span class="n">a</span><span class="p">)</span> <span class="go">a: [ 2.12132034 4.94974747 7.77817459 10.60660172]</span> </pre></div> </div> <p>And so is <code class="docutils literal notranslate"><span class="pre">wp.d</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">wp</span><span class="o">.</span><span class="n">d</span><span class="p">)</span> <span class="go">d: [-0.70710678 -0.70710678 -0.70710678 -0.70710678]</span> </pre></div> </div> </li> </ol> </div> </div> </div> </div> </div> <div class="sphinxsidebar" role="navigation" aria-label="main navigation"> <div class="sphinxsidebarwrapper"> <h3><a href="../index.html">Table of Contents</a></h3> <ul> <li><a class="reference internal" href="#">Wavelet Packets</a><ul> <li><a class="reference internal" href="#import-pywt">Import pywt</a></li> <li><a class="reference internal" href="#create-wavelet-packet-structure">Create Wavelet Packet structure</a></li> <li><a class="reference internal" href="#traversing-wp-tree">Traversing WP tree:</a><ul> <li><a class="reference internal" href="#accessing-subnodes">Accessing subnodes:</a></li> <li><a class="reference internal" href="#accessing-node-s-attributes">Accessing Node’s attributes:</a></li> <li><a class="reference internal" href="#collecting-nodes">Collecting nodes</a></li> </ul> </li> <li><a class="reference internal" href="#reconstructing-data-from-wavelet-packets">Reconstructing data from Wavelet Packets:</a></li> <li><a class="reference internal" href="#removing-nodes-from-wavelet-packet-tree">Removing nodes from Wavelet Packet tree:</a></li> <li><a class="reference internal" href="#lazy-evaluation">Lazy evaluation:</a></li> </ul> </li> </ul> <h4>Previous topic</h4> <p class="topless"><a href="multilevel.html" title="previous chapter">Multilevel DWT, IDWT and SWT</a></p> <h4>Next topic</h4> <p class="topless"><a href="wp2d.html" title="next chapter">2D Wavelet Packets</a></p><div> <h3>Quick links</h3> <ul> <li><a href="https://github.com/PyWavelets/pywt"><img src="../_static/github.png" height="16" width="16" alt="" /> Fork on Github</a></li> <li><a href="http://groups.google.com/group/pywavelets"><img src="../_static/comments.png" height="16" 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