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<title>PyWavelets - Discrete Wavelet Transform in Python</title>
<meta name="author" content="Filip Wasilewski" />
<meta name="date" content="2006-12-02 15:31" />
<meta content="pywavelets wavelets discrete wavelet transform Python module dwt idwt swt wavelet packets" name="keywords" />
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<div class="document" id="pywavelets-discrete-wavelet-transform-in-python">
<h1 class="title">PyWavelets - Discrete Wavelet Transform in Python</h1>
<h2 class="subtitle" id="user-guide">User Guide</h2>
<table class="docinfo" frame="void" rules="none">
<col class="docinfo-name" />
<col class="docinfo-content" />
<tbody valign="top">
<tr><th class="docinfo-name">Author:</th>
<td>Filip Wasilewski</td></tr>
<tr><th class="docinfo-name">Contact:</th>
<td><a class="first last reference" href="mailto:filip.wasilewski@gmail.com">filip.wasilewski@gmail.com</a></td></tr>
<tr><th class="docinfo-name">Version:</th>
<td>0.1.6</td></tr>
<tr><th class="docinfo-name">Status:</th>
<td>alpha</td></tr>
<tr><th class="docinfo-name">Date:</th>
<td>2006-12-02 15:31</td></tr>
<tr class="field"><th class="docinfo-name">License:</th><td class="field-body"><a class="reference" href="COPYING.txt">MIT</a></td>
</tr>
</tbody>
</table>
<div class="abstract topic">
<p class="topic-title first">Abstract</p>
<p><cite>PyWavelets</cite> is a <a class="reference" href="http://python.org/">Python</a> module for computing forward and inverse
1D and 2D Discrete Wavelet Transform, Stationary Wavelet Transform and Wavelet Packets
decomposition and reconstruction.
This document is a User Guide to <cite>PyWavelets</cite>.</p>
</div>
<div class="contents local topic">
<p class="topic-title first"><a id="table-of-contents" name="table-of-contents">Table of Contents</a></p>
<ul class="auto-toc simple">
<li><a class="reference" href="#introduction" id="id2" name="id2">1. Introduction</a><ul class="auto-toc">
<li><a class="reference" href="#requirements" id="id3" name="id3">1.1. Requirements</a></li>
<li><a class="reference" href="#download" id="id4" name="id4">1.2. Download</a></li>
<li><a class="reference" href="#install" id="id5" name="id5">1.3. Install</a></li>
<li><a class="reference" href="#license" id="id6" name="id6">1.4. License</a></li>
<li><a class="reference" href="#contact" id="id7" name="id7">1.5. Contact</a></li>
</ul>
</li>
<li><a class="reference" href="#wavelets" id="id8" name="id8">2. Wavelets</a><ul class="auto-toc">
<li><a class="reference" href="#wavelet-families" id="id9" name="id9">2.1. Wavelet <tt class="docutils literal"><span class="pre">families</span></tt></a></li>
<li><a class="reference" href="#built-in-wavelets-wavelist" id="id10" name="id10">2.2. Built-in wavelets - <tt class="docutils literal"><span class="pre">wavelist</span></tt></a></li>
<li><a class="reference" href="#wavelet-object" id="id11" name="id11">2.3. <tt class="docutils literal"><span class="pre">Wavelet</span></tt> object</a></li>
<li><a class="reference" href="#id1" id="id12" name="id12">2.4. Using custom wavelets</a></li>
</ul>
</li>
<li><a class="reference" href="#discrete-wavelet-transform-dwt" id="id13" name="id13">3. Discrete Wavelet Transform (DWT)</a><ul class="auto-toc">
<li><a class="reference" href="#single-level-dwt" id="id14" name="id14">3.1. Single level <tt class="docutils literal"><span class="pre">dwt</span></tt></a></li>
<li><a class="reference" href="#multilevel-decomposition-using-wavedec" id="id15" name="id15">3.2. Multilevel decomposition using <tt class="docutils literal"><span class="pre">wavedec</span></tt></a></li>
<li><a class="reference" href="#maximum-decomposition-level-dwt-max-level" id="id16" name="id16">3.3. Maximum decomposition level - <tt class="docutils literal"><span class="pre">dwt_max_level</span></tt></a></li>
<li><a class="reference" href="#result-coefficients-length-dwt-coeff-len" id="id17" name="id17">3.4. Result coefficients length - <tt class="docutils literal"><span class="pre">dwt_coeff_len</span></tt></a></li>
<li><a class="reference" href="#signal-extension-modes-modes" id="id18" name="id18">3.5. Signal extension modes - <tt class="docutils literal"><span class="pre">MODES</span></tt></a></li>
</ul>
</li>
<li><a class="reference" href="#inverse-discrete-wavelet-transform-idwt" id="id19" name="id19">4. Inverse Discrete Wavelet Transform (IDWT)</a><ul class="auto-toc">
<li><a class="reference" href="#single-level-idwt" id="id20" name="id20">4.1. Single level <tt class="docutils literal"><span class="pre">idwt</span></tt></a></li>
<li><a class="reference" href="#multilevel-reconstruction-using-waverec" id="id21" name="id21">4.2. Multilevel reconstruction using <tt class="docutils literal"><span class="pre">waverec</span></tt></a></li>
<li><a class="reference" href="#direct-reconstruction-with-upcoef" id="id22" name="id22">4.3. Direct reconstruction with <tt class="docutils literal"><span class="pre">upcoef</span></tt></a></li>
</ul>
</li>
<li><a class="reference" href="#d-dwt-and-idwt" id="id23" name="id23">5. 2D DWT and IDWT</a><ul class="auto-toc">
<li><a class="reference" href="#single-level-dwt2" id="id24" name="id24">5.1. Single level <tt class="docutils literal"><span class="pre">dwt2</span></tt></a></li>
<li><a class="reference" href="#single-level-idwt2" id="id25" name="id25">5.2. Single level <tt class="docutils literal"><span class="pre">idwt2</span></tt></a></li>
<li><a class="reference" href="#d-multilevel-decomposition-using-wavedec2" id="id26" name="id26">5.3. 2D multilevel decomposition using <tt class="docutils literal"><span class="pre">wavedec2</span></tt></a></li>
<li><a class="reference" href="#d-multilevel-reconstruction-using-waverec2" id="id27" name="id27">5.4. 2D multilevel reconstruction using <tt class="docutils literal"><span class="pre">waverec2</span></tt></a></li>
</ul>
</li>
<li><a class="reference" href="#wavelet-packets" id="id28" name="id28">6. Wavelet Packets</a><ul class="auto-toc">
<li><a class="reference" href="#wavelet-packet" id="id29" name="id29">6.1. Wavelet Packet</a></li>
<li><a class="reference" href="#node" id="id30" name="id30">6.2. Node</a></li>
</ul>
</li>
<li><a class="reference" href="#stationary-wavelet-transform" id="id31" name="id31">7. Stationary Wavelet Transform</a><ul class="auto-toc">
<li><a class="reference" href="#multilevel-swt" id="id32" name="id32">7.1. Multilevel <tt class="docutils literal"><span class="pre">swt</span></tt></a></li>
<li><a class="reference" href="#maximum-decomposition-level-swt-max-level" id="id33" name="id33">7.2. Maximum decomposition level - <tt class="docutils literal"><span class="pre">swt_max_level</span></tt></a></li>
</ul>
</li>
<li><a class="reference" href="#demo" id="id34" name="id34">8. Demo</a></li>
</ul>
</div>
<div class="section">
<h1><a class="toc-backref" href="#id2" id="introduction" name="introduction">1. Introduction</a></h1>
<div class="section">
<h2><a class="toc-backref" href="#id3" id="requirements" name="requirements">1.1. Requirements</a></h2>
<p><cite>PyWavelets</cite> was originally developed using <a class="reference" href="http://www.mingw.org/">MinGW</a> C compiler, <a class="reference" href="http://www.cosc.canterbury.ac.nz/~greg/python/Pyrex/">Pyrex</a> and
<a class="reference" href="http://python.org/">Python</a> 2.4 on 32-bit WindowsXP platform. Recent release adds support
for Python 2.5.</p>
<p>The only external requirement is a recent version of
<a class="reference" href="http://www.scipy.org/">NumPy</a> numeric array module.</p>
</div>
<div class="section">
<h2><a class="toc-backref" href="#id4" id="download" name="download">1.2. Download</a></h2>
<p>Current release, including source and binary versions for Windows, is available
for download from Python Cheese Shop directory at:</p>
<blockquote>
<a class="reference" href="http://cheeseshop.python.org/pypi/PyWavelets/">http://cheeseshop.python.org/pypi/PyWavelets/</a></blockquote>
<p>The latest <em>development</em> version can be downloaded from
<a class="reference" href="http://wavelets.scipy.org">wavelets.scipy.org</a> SVN <a class="reference" href="http://wavelets.scipy.org/svn/multiresolution/pywt/trunk">repository</a>:</p>
<pre class="literal-block">
svn co http://wavelets.scipy.org/svn/multiresolution/pywt/trunk pywt
</pre>
</div>
<div class="section">
<h2><a class="toc-backref" href="#id5" id="install" name="install">1.3. Install</a></h2>
<p>The most convenient way to install PyWavelets is to use <a class="reference" href="http://peak.telecommunity.com/DevCenter/setuptools">setuptools</a>
<a class="reference" href="http://peak.telecommunity.com/DevCenter/EasyInstall#using-easy-install">Easy Install</a> manager:</p>
<pre class="literal-block">
easy_install PyWavelets==0.1.6
</pre>
<p>Please note that in order to build <cite>PyWavelets</cite> from source code you will
need a working C compiler and, in case of source code modifications,
an <em>updated</em> version of Pyrex from</p>
<blockquote>
<a class="reference" href="http://codespeak.net/svn/lxml/pyrex/">http://codespeak.net/svn/lxml/pyrex/</a></blockquote>
<p>SVN repository, which includes features and bug fixes not yet available
in the regular Pyrex distribution.</p>
<p>Then in the shell prompt in the <cite>PyWavelets</cite> source code directory type:</p>
<pre class="literal-block">
python setupegg.py install
</pre>
<p>or if using the default distutils manager:</p>
<pre class="literal-block">
python setup.py install
</pre>
<p>For Windows users there is also standard binary installer available
in the Cheese Shop repository. Just download and execute it.</p>
<p>To verify the installation process try running tests and examples
from <cite>tests</cite> and <cite>demo</cite> directories included in the source distribution.
Note that some examples need <a class="reference" href="http://matplotlib.sourceforge.net">matplotlib</a> installed.</p>
</div>
<div class="section">
<h2><a class="toc-backref" href="#id6" id="license" name="license">1.4. License</a></h2>
<p><cite>PyWavelets</cite> is free Open Source software available under <a class="reference" href="COPYING.txt">MIT license</a>. Just do no evil.</p>
</div>
<div class="section">
<h2><a class="toc-backref" href="#id7" id="contact" name="contact">1.5. Contact</a></h2>
<p>Feel free to contact me directly at <a class="reference" href="mailto:filip.wasilewski@gmail.com">filip.wasilewski@gmail.com</a>.
Comments, bug reports and fixes are welcome.</p>
<p>You can also use the wiki and trac system available at <a class="reference" href="http://wavelets.scipy.org">wavelets.scipy.org</a>
to improve documentation, post cookbook recipes or submit enhancement proposals
or bug reports.</p>
</div>
</div>
<div class="section">
<h1><a class="toc-backref" href="#id8" id="wavelets" name="wavelets">2. Wavelets</a></h1>
<div class="section">
<h2><a class="toc-backref" href="#id9" id="wavelet-families" name="wavelet-families">2.1. Wavelet <tt class="docutils literal"><span class="pre">families</span></tt></a></h2>
<p>The <tt class="docutils literal"><span class="pre">families()</span></tt> function returns names of available built-in wavelet families.
Currently the following wavelet families with over seventy wavelets are available:</p>
<ul class="simple">
<li>Haar (<tt class="docutils literal"><span class="pre">haar</span></tt>)</li>
<li>Daubechies (<tt class="docutils literal"><span class="pre">db</span></tt>)</li>
<li>Symlets (<tt class="docutils literal"><span class="pre">sym</span></tt>)</li>
<li>Coiflets (<tt class="docutils literal"><span class="pre">coif</span></tt>)</li>
<li>Biorthogonal (<tt class="docutils literal"><span class="pre">bior</span></tt>)</li>
<li>Reverse biorthogonal (<tt class="docutils literal"><span class="pre">rbio</span></tt>)</li>
<li><cite>"Discrete"</cite> FIR approximation of Meyer wavelet (<tt class="docutils literal"><span class="pre">dmey</span></tt>)</li>
</ul>
<p class="example">Example:</p>
<div class="example"><div class="code"><span class="op">>>></span> <span class="kwd">import</span> <span class="name">pywt</span><br /><span class="op">>>></span> <span class="kwd">print</span> <span class="name">pywt</span><span class="op">.</span><span class="name">families</span><span class="op">(</span><span class="op">)</span><br /><span class="op">[</span><span class="str">'haar'</span><span class="op">,</span> <span class="str">'db'</span><span class="op">,</span> <span class="str">'sym'</span><span class="op">,</span> <span class="str">'coif'</span><span class="op">,</span> <span class="str">'bior'</span><span class="op">,</span> <span class="str">'rbio'</span><span class="op">,</span> <span class="str">'dmey'</span><span class="op">]</span>
</div></div></div>
<div class="section">
<h2><a class="toc-backref" href="#id10" id="built-in-wavelets-wavelist" name="built-in-wavelets-wavelist"><span id="wavelist"></span>2.2. Built-in wavelets - <tt class="docutils literal"><span class="pre">wavelist</span></tt></a></h2>
<p>The <tt class="docutils literal"><span class="pre">wavelist(short_name=None)</span></tt> function returns list of available
wavelet names.</p>
<p>If <tt class="docutils literal"><span class="pre">short_name</span></tt> is None, then names of all implemented wavelets is returned,
otherwise the function returns names of wavelets from given family name.</p>
<p class="example">Example:</p>
<div class="example"><div class="code"><span class="op">>>></span> <span class="kwd">import</span> <span class="name">pywt</span><br /><span class="op">>>></span> <span class="kwd">print</span> <span class="name">pywt</span><span class="op">.</span><span class="name">wavelist</span><span class="op">(</span><span class="str">'coif'</span><span class="op">)</span><br /><span class="op">[</span><span class="str">'coif1'</span><span class="op">,</span> <span class="str">'coif2'</span><span class="op">,</span> <span class="str">'coif3'</span><span class="op">,</span> <span class="str">'coif4'</span><span class="op">,</span> <span class="str">'coif5'</span><span class="op">]</span>
</div></div></div>
<div class="section">
<h2><a class="toc-backref" href="#id11" id="wavelet-object" name="wavelet-object"><span id="wavelet"></span>2.3. <tt class="docutils literal"><span class="pre">Wavelet</span></tt> object</a></h2>
<p><tt class="docutils literal"><span class="pre">Wavelet(name,</span> <span class="pre">filter_bank=None)</span></tt> object describe properties of a wavelet
identified by <tt class="docutils literal"><span class="pre">name</span></tt>.
In order to use a built-in wavelet the parameter <tt class="docutils literal"><span class="pre">name</span></tt> must be a valid
name from <a class="reference" href="#wavelist">wavelist()</a> list. Otherwise a <a class="reference" href="#filter-bank">filter_bank</a> argument must be provided.</p>
<dl class="docutils">
<dt>name</dt>
<dd>Wavelet name</dd>
</dl>
<dl class="docutils" id="filter-bank">
<dt>filter_bank</dt>
<dd>Use user supplied filter bank instead of built-in <tt class="docutils literal"><span class="pre">Wavelet</span></tt>.
The filter bank object must implement the
<a class="reference" href="#get-filters-coeffs">get_filters_coeffs()</a> method,
which returns a list of filters (dec_lo, dec_hi, rec_lo, rec_hi).
Other Wavelet object can also be used as a filter bank. See section
on <a class="reference" href="#using-custom-wavelets">using custom wavelets</a> for more information.</dd>
<dt>dec_lo, dec_hi</dt>
<dd>Decomposition filters values.</dd>
<dt>rec_lo, rec_hi</dt>
<dd>Reconstruction filters values.</dd>
<dt>dec_len</dt>
<dd>Decomposition filter length.</dd>
<dt>rec_len</dt>
<dd>Reconstruction filter length.</dd>
</dl>
<dl class="docutils" id="get-filters-coeffs">
<dt>get_filters_coeffs()</dt>
<dd>Returns quadrature mirror filters list for current wavelet (dec_lo, dec_hi, rec_lo, rec_hi)</dd>
<dt>other properties:</dt>
<dd><ul class="first last simple">
<li>family_name</li>
<li>short_name</li>
<li>orthogonal</li>
<li>biorthogonal</li>
<li>symmetry - <tt class="docutils literal"><span class="pre">asymmetric</span></tt>, <tt class="docutils literal"><span class="pre">near</span> <span class="pre">symmetric</span></tt>, <tt class="docutils literal"><span class="pre">symmetric</span></tt></li>
<li>vanishing_moments_psi</li>
<li>vanishing_moments_phi</li>
</ul>
</dd>
</dl>
<p class="example">Example:</p>
<div class="example"><div class="code"><span class="op">>>></span> <span class="kwd">import</span> <span class="name">pywt</span><br /><span class="op">>>></span> <span class="name">wavelet</span> <span class="op">=</span> <span class="name">pywt</span><span class="op">.</span><span class="name">Wavelet</span><span class="op">(</span><span class="str">'db1'</span><span class="op">)</span><br /><span class="op">>>></span> <span class="kwd">print</span> <span class="name">wavelet</span><br /><span class="name">Wavelet</span> <span class="name">db1</span><br />    <span class="name">Family</span> <span class="name">name</span><span class="op">:</span>    <span class="name">Daubechies</span><br />      <span class="name">Short</span> <span class="name">name</span><span class="op">:</span>     <span class="name">db</span><br />      <span class="name">Filters</span> <span class="name">length</span><span class="op">:</span> <span class="num">2</span><br />      <span class="name">Orthogonal</span><span class="op">:</span>     <span class="name">True</span><br />      <span class="name">Biorthogonal</span><span class="op">:</span>   <span class="name">True</span><br />      <span class="name">Symmetry</span><span class="op">:</span>       <span class="name">asymmetric</span><br />    <span class="op">>>></span> <span class="kwd">print</span> <span class="name">wavelet</span><span class="op">.</span><span class="name">dec_lo</span><span class="op">,</span> <span class="name">wavelet</span><span class="op">.</span><span class="name">dec_hi</span><br /><span class="op">[</span><span class="num">0.70710678118654757</span><span class="op">,</span> <span class="num">0.70710678118654757</span><span class="op">]</span> <span class="op">[</span><span class="op">-</span><span class="num">0.70710678118654757</span><span class="op">,</span> <span class="num">0.70710678118654757</span><span class="op">]</span><br /><span class="op">>>></span> <span class="kwd">print</span> <span class="name">wavelet</span><span class="op">.</span><span class="name">rec_lo</span><span class="op">,</span> <span class="name">wavelet</span><span class="op">.</span><span class="name">rec_hi</span><br /><span class="op">[</span><span class="num">0.70710678118654757</span><span class="op">,</span> <span class="num">0.70710678118654757</span><span class="op">]</span> <span class="op">[</span><span class="num">0.70710678118654757</span><span class="op">,</span> <span class="op">-</span><span class="num">0.70710678118654757</span><span class="op">]</span>
</div></div><div class="section">
<h3><a id="wavefun" name="wavefun">2.3.1. <tt class="docutils literal"><span class="pre">wavefun</span></tt></a></h3>
<p>The <tt class="docutils literal"><span class="pre">wavefun(level)</span></tt> function can be used to calculates approximations of wavelet function (<em>psi</em>)
and associated scaling function (<em>phi</em>) at given level of refinement.</p>
<p>For orthogonal wavelet returns scaling and wavelet function.</p>
<div class="example"><div class="code"><span class="op">>>></span> <span class="kwd">import</span> <span class="name">pywt</span><br /><span class="op">>>></span> <span class="name">wavelet</span> <span class="op">=</span> <span class="name">pywt</span><span class="op">.</span><span class="name">Wavelet</span><span class="op">(</span><span class="str">'db2'</span><span class="op">)</span><br /><span class="op">>>></span> <span class="name">phi</span><span class="op">,</span> <span class="name">psi</span> <span class="op">=</span> <span class="name">wavelet</span><span class="op">.</span><span class="name">wavefun</span><span class="op">(</span><span class="name">level</span><span class="op">=</span><span class="num">5</span><span class="op">)</span>
</div></div><p>For biorthogonal wavelet returns scaling and wavelet function both for decomposition
and reconstruction.</p>
<div class="example"><div class="code"><span class="op">>>></span> <span class="kwd">import</span> <span class="name">pywt</span><br /><span class="op">>>></span> <span class="name">wavelet</span> <span class="op">=</span> <span class="name">pywt</span><span class="op">.</span><span class="name">Wavelet</span><span class="op">(</span><span class="str">'bior1.1'</span><span class="op">)</span><br /><span class="op">>>></span> <span class="name">phi_d</span><span class="op">,</span> <span class="name">psi_d</span><span class="op">,</span> <span class="name">phi_r</span><span class="op">,</span> <span class="name">psi_r</span> <span class="op">=</span> <span class="name">wavelet</span><span class="op">.</span><span class="name">wavefun</span><span class="op">(</span><span class="name">level</span><span class="op">=</span><span class="num">5</span><span class="op">)</span>
</div></div><!-- See also plots of Daubechies and Symlets wavelet familes generated with ``wavefun`` function:
- `db.png`_
- `sym.png`_ -->
</div>
</div>
<div class="section">
<h2><a class="toc-backref" href="#id12" id="id1" name="id1"><span id="using-custom-wavelets"></span>2.4. Using custom wavelets</a></h2>
<p><cite>PyWavelets</cite> comes with <a class="reference" href="#wavelist">long list</a> of the most popular wavelets built-in and ready to use.
If there is a need of using a specific wavelet which is not included in the list it is
very easy to create one.
Just pass an object of a class implementing <tt class="docutils literal"><span class="pre">get_filters_coeffs()</span></tt> method
as a <a class="reference" href="#filter-bank">filter_bank</a> argument of <a class="reference" href="#wavelet">Wavelet</a> constructor.</p>
<p>The <tt class="docutils literal"><span class="pre">get_filters_coeffs()</span></tt> method must return a list of four filters:
lowpass decomposition, highpass decomposition, lowpass reconstruction and
highpass reconstruction filter, just as the <a class="reference" href="#get-filters-coeffs">get_filters_coeffs()</a> method
of the <a class="reference" href="#wavelet">Wavelet</a> class.</p>
<p>A Wavelet object created in this way is a standard <a class="reference" href="#wavelet">Wavelet</a> object and can be used
as any other <a class="reference" href="#wavelet">Wavelet</a> object.</p>
<p class="example">Example:</p>
<div class="example"><div class="code"><span class="op">>>></span> <span class="kwd">import</span> <span class="name">pywt</span><span class="op">,</span> <span class="name">math</span><br /><span class="op">>>></span> <span class="kwd">class</span> <span class="name">HaarFilterBank</span><span class="op">(</span><span class="name">object</span><span class="op">)</span><span class="op">:</span><br /><span class="op">.</span><span class="op">.</span><span class="op">.</span>     <span class="kwd">def</span> <span class="name">get_filters_coeffs</span><span class="op">(</span><span class="name">self</span><span class="op">)</span><span class="op">:</span><br /><span class="op">.</span><span class="op">.</span><span class="op">.</span>         <span class="name">c</span> <span class="op">=</span> <span class="name">math</span><span class="op">.</span><span class="name">sqrt</span><span class="op">(</span><span class="num">2</span><span class="op">)</span><span class="op">/</span><span class="num">2</span><br /><span class="op">.</span><span class="op">.</span><span class="op">.</span>         <span class="name">dec_lo</span><span class="op">,</span> <span class="name">dec_hi</span><span class="op">,</span> <span class="name">rec_lo</span><span class="op">,</span> <span class="name">rec_hi</span> <span class="op">=</span> <span class="op">[</span><span class="name">c</span><span class="op">,</span> <span class="name">c</span><span class="op">]</span><span class="op">,</span> <span class="op">[</span><span class="op">-</span><span class="name">c</span><span class="op">,</span> <span class="name">c</span><span class="op">]</span><span class="op">,</span> <span class="op">[</span><span class="name">c</span><span class="op">,</span> <span class="name">c</span><span class="op">]</span><span class="op">,</span> <span class="op">[</span><span class="name">c</span><span class="op">,</span> <span class="op">-</span><span class="name">c</span><span class="op">]</span><br /><span class="op">.</span><span class="op">.</span><span class="op">.</span>         <span class="kwd">return</span> <span class="op">[</span><span class="name">dec_lo</span><span class="op">,</span> <span class="name">dec_hi</span><span class="op">,</span> <span class="name">rec_lo</span><span class="op">,</span> <span class="name">rec_hi</span><span class="op">]</span><br /><span class="op">>>></span> <span class="name">myWavelet</span> <span class="op">=</span> <span class="name">pywt</span><span class="op">.</span><span class="name">Wavelet</span><span class="op">(</span><span class="name">name</span><span class="op">=</span><span class="str">"myHaarWavelet"</span><span class="op">,</span> <span class="name">filter_bank</span><span class="op">=</span><span class="name">HaarFilterBank</span><span class="op">(</span><span class="op">)</span><span class="op">)</span>
</div></div></div>
</div>
<div class="section">
<h1><a class="toc-backref" href="#id13" id="discrete-wavelet-transform-dwt" name="discrete-wavelet-transform-dwt">3. Discrete Wavelet Transform (DWT)</a></h1>
<p>Wavelet transform has recently became very popular
when it comes to analysis, denoising and compression of
signals and images.</p>
<div class="section">
<h2><a class="toc-backref" href="#id14" id="single-level-dwt" name="single-level-dwt"><span id="dwt"></span>3.1. Single level <tt class="docutils literal"><span class="pre">dwt</span></tt></a></h2>
<p>The <tt class="docutils literal"><span class="pre">dwt</span></tt> function is used to perform single level,
one dimensional Discrete Wavelet Transform.</p>
<pre class="literal-block">
(cA, cD) = dwt(data, wavelet, mode='sym')
</pre>
<dl class="docutils">
<dt>data</dt>
<dd>Input signal can be numeric array, python list or other iterable object.
If data is not in <em>double</em> format it will be converted to that type
before performing computation.</dd>
<dt>wavelet</dt>
<dd>Wavelet to use in transform. This can be name of wavelet from <a class="reference" href="#wavelist">wavelist()</a> or <a class="reference" href="#wavelet">Wavelet</a> object.</dd>
<dt>mode</dt>
<dd>Signal extension mode, see <a class="reference" href="#modes">MODES</a>.</dd>
</dl>
<p>The transform coefficients are returned as two arrays containing
approximation (cA) and detail (cD) coefficients respectively.
Length of returned arrays depends on selected <a class="reference" href="#mode">mode</a> - see <a class="reference" href="#dwt-coeff-len">dwt_coeff_len</a>:</p>
<ul>
<li><p class="first">for all <a class="reference" href="#modes">modes</a> except <a class="reference" href="#periodization">periodization</a>:</p>
<pre class="literal-block">
len(cA) == len(cD) == floor((len(data) + wavelet.dec_len - 1) / 2)
</pre>
</li>
<li><p class="first">for <a class="reference" href="#periodization">periodization</a> mode (<cite>"per"</cite>):</p>
<pre class="literal-block">
len(cA) == len(cD) == ceil(len(data) / 2)
</pre>
</li>
</ul>
<p class="example">Example:</p>
<div class="example"><div class="code"><span class="op">>>></span> <span class="kwd">import</span> <span class="name">pywt</span><br /><span class="op">>>></span> <span class="op">(</span><span class="name">cA</span><span class="op">,</span> <span class="name">cD</span><span class="op">)</span> <span class="op">=</span> <span class="name">pywt</span><span class="op">.</span><span class="name">dwt</span><span class="op">(</span><span class="op">[</span><span class="num">1</span><span class="op">,</span><span class="num">2</span><span class="op">,</span><span class="num">3</span><span class="op">,</span><span class="num">4</span><span class="op">,</span><span class="num">5</span><span class="op">,</span><span class="num">6</span><span class="op">]</span><span class="op">,</span> <span class="str">'db1'</span><span class="op">)</span><br /><span class="op">>>></span> <span class="kwd">print</span> <span class="name">cA</span><br /><span class="op">[</span> <span class="num">2.12132034</span>  <span class="num">4.94974747</span>  <span class="num">7.77817459</span><span class="op">]</span><br /><span class="op">>>></span> <span class="kwd">print</span> <span class="name">cD</span><br /><span class="op">[</span><span class="op">-</span><span class="num">0.70710678</span> <span class="op">-</span><span class="num">0.70710678</span> <span class="op">-</span><span class="num">0.70710678</span><span class="op">]</span>
</div></div></div>
<div class="section">
<h2><a class="toc-backref" href="#id15" id="multilevel-decomposition-using-wavedec" name="multilevel-decomposition-using-wavedec"><span id="wavedec"></span>3.2. Multilevel decomposition using <tt class="docutils literal"><span class="pre">wavedec</span></tt></a></h2>
<p><cite>(Please note the mode and level arguments order change in 0.1.6 version.)</cite></p>
<p>The <tt class="docutils literal"><span class="pre">wavedec</span></tt> function performs 1D multilevel Discrete Wavelet Transform
decomposition of given signal and returns ordered list of coefficients arrays
<tt class="docutils literal"><span class="pre">[cAn,</span> <span class="pre">cDn,</span> <span class="pre">cDn-1,</span> <span class="pre">...,</span> <span class="pre">cD2,</span> <span class="pre">cD1]</span></tt>, where <tt class="docutils literal"><span class="pre">n</span></tt> denotes the level of decomposition.
The first element (<tt class="docutils literal"><span class="pre">cAn</span></tt>) of the result is approximation coefficients array and
the following elements (<tt class="docutils literal"><span class="pre">cDn</span></tt> - <tt class="docutils literal"><span class="pre">cD1</span></tt>) are details coefficients arrays.</p>
<pre class="literal-block">
wavedec(data, wavelet, mode='sym', level=None)
</pre>
<dl class="docutils">
<dt>data</dt>
<dd>Input signal can be numeric array, python list or other iterable object.
If data is not in <em>double</em> format it will be converted to that type
before performing computation.</dd>
<dt>wavelet</dt>
<dd>Wavelet to use in transform. This can be name of wavelet from <a class="reference" href="#wavelist">wavelist()</a> or <a class="reference" href="#wavelet">Wavelet</a> object.</dd>
<dt>mode</dt>
<dd>Signal extension mode, see <a class="reference" href="#modes">MODES</a>.</dd>
<dt>level</dt>
<dd>Decomposition levels count.
If the level is None, then full decomposition up to
level computed with <a class="reference" href="#dwt-max-level">dwt_max_level</a> function for corresponding
data and wavelet lengths is performed.</dd>
</dl>
<p class="example">Example:</p>
<div class="example"><div class="code"><span class="op">>>></span> <span class="kwd">import</span> <span class="name">pywt</span><br /><span class="op">>>></span> <span class="name">coeffs</span> <span class="op">=</span> <span class="name">pywt</span><span class="op">.</span><span class="name">wavedec</span><span class="op">(</span><span class="op">[</span><span class="num">1</span><span class="op">,</span><span class="num">2</span><span class="op">,</span><span class="num">3</span><span class="op">,</span><span class="num">4</span><span class="op">,</span><span class="num">5</span><span class="op">,</span><span class="num">6</span><span class="op">,</span><span class="num">7</span><span class="op">,</span><span class="num">8</span><span class="op">]</span><span class="op">,</span> <span class="str">'db1'</span><span class="op">,</span> <span class="name">level</span><span class="op">=</span><span class="num">2</span><span class="op">)</span><br /><span class="op">>>></span> <span class="name">cA2</span><span class="op">,</span> <span class="name">cD2</span><span class="op">,</span> <span class="name">cD1</span> <span class="op">=</span> <span class="name">coeffs</span><br /><span class="op">>>></span> <span class="kwd">print</span> <span class="name">cD1</span><br /><span class="op">[</span><span class="op">-</span><span class="num">0.70710678</span> <span class="op">-</span><span class="num">0.70710678</span> <span class="op">-</span><span class="num">0.70710678</span> <span class="op">-</span><span class="num">0.70710678</span><span class="op">]</span><br /><span class="op">>>></span> <span class="kwd">print</span> <span class="name">cD2</span><br /><span class="op">[</span><span class="op">-</span><span class="num">2.</span> <span class="op">-</span><span class="num">2.</span><span class="op">]</span><br /><span class="op">>>></span> <span class="kwd">print</span> <span class="name">cA2</span><br /><span class="op">[</span>  <span class="num">5.</span>  <span class="num">13.</span><span class="op">]</span>
</div></div></div>
<div class="section">
<h2><a class="toc-backref" href="#id16" id="maximum-decomposition-level-dwt-max-level" name="maximum-decomposition-level-dwt-max-level"><span id="dwt-max-level"></span>3.3. Maximum decomposition level - <tt class="docutils literal"><span class="pre">dwt_max_level</span></tt></a></h2>
<p>The <tt class="docutils literal"><span class="pre">dwt_max_level</span></tt> function can be used to
compute the maximum useful level of decomposition
for given <tt class="docutils literal"><span class="pre">input</span> <span class="pre">data</span> <span class="pre">length</span></tt> and <tt class="docutils literal"><span class="pre">wavelet</span> <span class="pre">filter</span> <span class="pre">length</span></tt>.</p>
<pre class="literal-block">
dwt_max_level(data_len, filter_len)
</pre>
<p>The returned value equals to:</p>
<pre class="literal-block">
floor(log(data_len/(filter_len-1))/log(2))
</pre>
<p>Although the maximum decomposition level can be quite high for long signals,
usually smaller values are chosen.</p>
<p class="example">Example:</p>
<div class="example"><div class="code"><span class="op">>>></span> <span class="kwd">import</span> <span class="name">pywt</span><br /><span class="op">>>></span> <span class="name">w</span> <span class="op">=</span> <span class="name">pywt</span><span class="op">.</span><span class="name">Wavelet</span><span class="op">(</span><span class="str">'sym5'</span><span class="op">)</span><br /><span class="op">>>></span> <span class="kwd">print</span> <span class="name">pywt</span><span class="op">.</span><span class="name">dwt_max_level</span><span class="op">(</span><span class="name">data_len</span> <span class="op">=</span> <span class="num">1000</span><span class="op">,</span> <span class="name">filter_len</span> <span class="op">=</span> <span class="name">w</span><span class="op">.</span><span class="name">dec_len</span><span class="op">)</span><br /><span class="num">6</span>
</div></div></div>
<div class="section">
<h2><a class="toc-backref" href="#id17" id="result-coefficients-length-dwt-coeff-len" name="result-coefficients-length-dwt-coeff-len"><span id="dwt-coeff-len"></span>3.4. Result coefficients length - <tt class="docutils literal"><span class="pre">dwt_coeff_len</span></tt></a></h2>
<p>Based on input data length, Wavelet decomposition filter length and signal extension <a class="reference" href="#mode">mode</a>,
the <tt class="docutils literal"><span class="pre">dwt_coeff_len</span></tt> function calculates length of result coefficients arrays after <a class="reference" href="#dwt">dwt</a>.</p>
<pre class="literal-block">
dwt_coeff_len(data_len, filter_len, mode)
</pre>
<p>For <a class="reference" href="#periodization">periodization</a> mode this equals:</p>
<pre class="literal-block">
ceil(data_len / 2)
</pre>
<p>which is the lowest possible length guaranteeing perfect reconstruction.</p>
<p>For other <a class="reference" href="#modes">modes</a>:</p>
<pre class="literal-block">
floor((data_len + filter_len - 1) / 2)
</pre>
</div>
<div class="section">
<h2><a class="toc-backref" href="#id18" id="signal-extension-modes-modes" name="signal-extension-modes-modes"><span id="modes"></span><span id="mode"></span>3.5. Signal extension modes - <tt class="docutils literal"><span class="pre">MODES</span></tt></a></h2>
<p>To handle problem of border distortion while performing <a class="reference" href="#dwt">DWT</a>,
one of several signal extension modes can be selected.</p>
<ul>
<li><p class="first"><tt class="docutils literal"><span class="pre">zpd</span></tt> - <strong>zero-padding</strong> - signal is extended by adding zero samples:</p>
<pre class="literal-block">
0 0 | x1 x2 ... xn | 0 0
</pre>
</li>
<li><p class="first"><tt class="docutils literal"><span class="pre">cpd</span></tt> - <strong>constant-padding</strong> - edge values are used:</p>
<pre class="literal-block">
x1 x1 | x1 x2 ... xn | xn xn
</pre>
</li>
<li><p class="first"><tt class="docutils literal"><span class="pre">sym</span></tt> - <strong>symmetric-padding</strong> - signal is extended by <em>mirroring</em> samples:</p>
<pre class="literal-block">
x2 x1 | x1 x2 ... xn | xn xn-1
</pre>
</li>
</ul>
<ul id="periodic-padding">
<li><p class="first"><tt class="docutils literal"><span class="pre">ppd</span></tt> - <strong>periodic-padding</strong> - signal is treated as periodic:</p>
<pre class="literal-block">
xn-1 xn | x1 x2 ... xn | x1 x2
</pre>
</li>
<li><p class="first"><tt class="docutils literal"><span class="pre">sp1</span></tt> - <strong>smooth-padding</strong> - signal is extended according to first derivatives
calculated on the edges</p>
</li>
</ul>
<p><a class="reference" href="#dwt">DWT</a> performed for these extension modes is slightly redundant, but ensure
the perfect reconstruction. To receive the smallest number of coefficients,
<a class="reference" href="#dwt">DWT</a> can be computed with <a class="reference" href="#periodization">periodization</a> mode</p>
<ul class="simple" id="periodization">
<li><tt class="docutils literal"><span class="pre">per</span></tt> - <strong>periodization</strong> - is like <a class="reference" href="#periodic-padding">periodic-padding</a> but gives the smallest possible
number of decomposition coefficients. <a class="reference" href="#idwt">IDWT</a> must be performed with the same mode to
ensure perfect reconstruction.</li>
</ul>
<p class="example">Example:</p>
<div class="example"><div class="code"><span class="op">>>></span> <span class="kwd">import</span> <span class="name">pywt</span><br /><span class="op">>>></span> <span class="kwd">print</span> <span class="name">pywt</span><span class="op">.</span><span class="name">MODES</span><span class="op">.</span><span class="name">modes</span><br /><span class="op">[</span><span class="str">'zpd'</span><span class="op">,</span> <span class="str">'cpd'</span><span class="op">,</span> <span class="str">'sym'</span><span class="op">,</span> <span class="str">'ppd'</span><span class="op">,</span> <span class="str">'sp1'</span><span class="op">,</span> <span class="str">'per'</span><span class="op">]</span>
</div></div><p>Notice that you can use either of the following forms:</p>
<div class="code"><span class="op">>>></span> <span class="kwd">import</span> <span class="name">pywt</span><br /><span class="op">>>></span> <span class="op">(</span><span class="name">a</span><span class="op">,</span> <span class="name">d</span><span class="op">)</span> <span class="op">=</span> <span class="name">pywt</span><span class="op">.</span><span class="name">dwt</span><span class="op">(</span><span class="op">[</span><span class="num">1</span><span class="op">,</span><span class="num">2</span><span class="op">,</span><span class="num">3</span><span class="op">,</span><span class="num">4</span><span class="op">,</span><span class="num">5</span><span class="op">,</span><span class="num">6</span><span class="op">]</span><span class="op">,</span> <span class="str">'db2'</span><span class="op">,</span> <span class="str">'sp1'</span><span class="op">)</span><br /><span class="op">>>></span> <span class="op">(</span><span class="name">a</span><span class="op">,</span> <span class="name">d</span><span class="op">)</span> <span class="op">=</span> <span class="name">pywt</span><span class="op">.</span><span class="name">dwt</span><span class="op">(</span><span class="op">[</span><span class="num">1</span><span class="op">,</span><span class="num">2</span><span class="op">,</span><span class="num">3</span><span class="op">,</span><span class="num">4</span><span class="op">,</span><span class="num">5</span><span class="op">,</span><span class="num">6</span><span class="op">]</span><span class="op">,</span> <span class="name">pywt</span><span class="op">.</span><span class="name">Wavelet</span><span class="op">(</span><span class="str">'db2'</span><span class="op">)</span><span class="op">,</span> <span class="name">pywt</span><span class="op">.</span><span class="name">MODES</span><span class="op">.</span><span class="name">sp1</span><span class="op">)</span>
</div><p>Note that extending data in context of <cite>PyWavelets</cite> does not really mean reallocating
memory and copying values. Instead of that the extra values are computed only
when needed. This feature saves extra memory and CPU resources and helps to avoid
page swapping when handling relatively big data arrays on computers with low
physical memory.</p>
</div>
</div>
<div class="section">
<h1><a class="toc-backref" href="#id19" id="inverse-discrete-wavelet-transform-idwt" name="inverse-discrete-wavelet-transform-idwt">4. Inverse Discrete Wavelet Transform (IDWT)</a></h1>
<div class="section">
<h2><a class="toc-backref" href="#id20" id="single-level-idwt" name="single-level-idwt"><span id="idwt"></span>4.1. Single level <tt class="docutils literal"><span class="pre">idwt</span></tt></a></h2>
<p>The <tt class="docutils literal"><span class="pre">idwt</span></tt> function reconstructs data from given coefficients by performing
single level Inverse Discrete Wavelet Transform.</p>
<pre class="literal-block">
idwt(cA, cD, wavelet, mode='sym', correct_size=0)
</pre>
<dl class="docutils">
<dt>cA</dt>
<dd>approximation coefficients.</dd>
<dt>cD</dt>
<dd>detail coefficients.</dd>
<dt>wavelet</dt>
<dd>Wavelet to use in transform. This can be name of wavelet from <a class="reference" href="#wavelist">wavelist()</a> or <a class="reference" href="#wavelet">Wavelet</a> object.</dd>
<dt>mode</dt>
<dd>Signal extension mode, see <a class="reference" href="#modes">MODES</a>. This is only important when DWT was performed in <a class="reference" href="#periodization">periodization</a> mode.</dd>
<dt>correct_size</dt>
<dd>additional option. Under normal conditions (all data lengths dyadic) Ca and cD
coefficients lists must have the same lengths. With correct_size set to True,
length of cA may be greater by one than length of cA.
This option is very useful when doing multilevel decomposition and reconstruction
of non-dyadic length signals.</dd>
</dl>
<p class="example">Example:</p>
<div class="example"><div class="code"><span class="op">>>></span> <span class="kwd">import</span> <span class="name">pywt</span><br /><span class="op">>>></span> <span class="op">(</span><span class="name">cA</span><span class="op">,</span> <span class="name">cD</span><span class="op">)</span> <span class="op">=</span> <span class="name">pywt</span><span class="op">.</span><span class="name">dwt</span><span class="op">(</span><span class="op">[</span><span class="num">1</span><span class="op">,</span><span class="num">2</span><span class="op">,</span><span class="num">3</span><span class="op">,</span><span class="num">4</span><span class="op">,</span><span class="num">5</span><span class="op">,</span><span class="num">6</span><span class="op">]</span><span class="op">,</span> <span class="str">'db2'</span><span class="op">,</span> <span class="str">'sp1'</span><span class="op">)</span><br /><span class="op">>>></span> <span class="kwd">print</span> <span class="name">pywt</span><span class="op">.</span><span class="name">idwt</span><span class="op">(</span><span class="name">cA</span><span class="op">,</span> <span class="name">cD</span><span class="op">,</span> <span class="str">'db2'</span><span class="op">,</span> <span class="str">'sp1'</span><span class="op">)</span><br /><span class="op">[</span> <span class="num">1.</span>  <span class="num">2.</span>  <span class="num">3.</span>  <span class="num">4.</span>  <span class="num">5.</span>  <span class="num">6.</span><span class="op">]</span>
</div></div><p>One of the <em>cA</em> and <em>cD</em> arguments can be <em>None</em>. In that situation
the reconstruction will be performed using only the other one.</p>
<p class="example">Example:</p>
<div class="example"><div class="code"><span class="op">>>></span> <span class="kwd">import</span> <span class="name">pywt</span><br /><span class="op">>>></span> <span class="op">(</span><span class="name">cA</span><span class="op">,</span> <span class="name">cD</span><span class="op">)</span> <span class="op">=</span> <span class="name">pywt</span><span class="op">.</span><span class="name">dwt</span><span class="op">(</span><span class="op">[</span><span class="num">1</span><span class="op">,</span><span class="num">2</span><span class="op">,</span><span class="num">3</span><span class="op">,</span><span class="num">4</span><span class="op">,</span><span class="num">5</span><span class="op">,</span><span class="num">6</span><span class="op">]</span><span class="op">,</span> <span class="str">'db2'</span><span class="op">,</span> <span class="str">'sp1'</span><span class="op">)</span><br /><span class="op">>>></span> <span class="name">A</span> <span class="op">=</span> <span class="name">pywt</span><span class="op">.</span><span class="name">idwt</span><span class="op">(</span><span class="name">cA</span><span class="op">,</span> <span class="name">None</span><span class="op">,</span> <span class="str">'db2'</span><span class="op">,</span> <span class="str">'sp1'</span><span class="op">)</span><br /><span class="op">>>></span> <span class="name">D</span> <span class="op">=</span> <span class="name">pywt</span><span class="op">.</span><span class="name">idwt</span><span class="op">(</span><span class="name">None</span><span class="op">,</span> <span class="name">cD</span><span class="op">,</span> <span class="str">'db2'</span><span class="op">,</span> <span class="str">'sp1'</span><span class="op">)</span><br /><span class="op">>>></span> <span class="kwd">print</span> <span class="name">A</span> <span class="op">+</span> <span class="name">D</span><br /><span class="op">[</span> <span class="num">1.</span>  <span class="num">2.</span>  <span class="num">3.</span>  <span class="num">4.</span>  <span class="num">5.</span>  <span class="num">6.</span><span class="op">]</span>
</div></div></div>
<div class="section">
<h2><a class="toc-backref" href="#id21" id="multilevel-reconstruction-using-waverec" name="multilevel-reconstruction-using-waverec"><span id="waverec"></span>4.2. Multilevel reconstruction using <tt class="docutils literal"><span class="pre">waverec</span></tt></a></h2>
<p>Performs multilevel reconstruction of signal from given coefficient list.</p>
<pre class="literal-block">
waverec(coeffs, wavelet, mode='sym')
</pre>
<dl class="docutils">
<dt>coeffs</dt>
<dd><p class="first">coefficients list must be in the form like returned from <a class="reference" href="#wavedec">wavedec</a> decomposition:</p>
<pre class="last literal-block">
[cAn, cDn, cDn-1, ..., cD2, cD1]
</pre>
</dd>
<dt>wavelet</dt>
<dd>Wavelet to use in transform. This can be name of wavelet from <a class="reference" href="#wavelist">wavelist()</a> or <a class="reference" href="#wavelet">Wavelet</a> object.</dd>
<dt>mode</dt>
<dd>Signal extension mode, see <a class="reference" href="#modes">MODES</a>.</dd>
</dl>
<p class="example">Example:</p>
<div class="example"><div class="code"><span class="op">>>></span> <span class="kwd">import</span> <span class="name">pywt</span><br /><span class="op">>>></span> <span class="name">coeffs</span> <span class="op">=</span> <span class="name">pywt</span><span class="op">.</span><span class="name">wavedec</span><span class="op">(</span><span class="op">[</span><span class="num">1</span><span class="op">,</span><span class="num">2</span><span class="op">,</span><span class="num">3</span><span class="op">,</span><span class="num">4</span><span class="op">,</span><span class="num">5</span><span class="op">,</span><span class="num">6</span><span class="op">,</span><span class="num">7</span><span class="op">,</span><span class="num">8</span><span class="op">]</span><span class="op">,</span> <span class="str">'db2'</span><span class="op">,</span> <span class="name">level</span><span class="op">=</span><span class="num">2</span><span class="op">)</span><br /><span class="op">>>></span> <span class="kwd">print</span> <span class="name">pywt</span><span class="op">.</span><span class="name">waverec</span><span class="op">(</span><span class="name">coeffs</span><span class="op">,</span> <span class="str">'db2'</span><span class="op">)</span><br /><span class="op">[</span> <span class="num">1.</span>  <span class="num">2.</span>  <span class="num">3.</span>  <span class="num">4.</span>  <span class="num">5.</span>  <span class="num">6.</span>  <span class="num">7.</span>  <span class="num">8.</span><span class="op">]</span>
</div></div></div>
<div class="section">
<h2><a class="toc-backref" href="#id22" id="direct-reconstruction-with-upcoef" name="direct-reconstruction-with-upcoef"><span id="upcoef"></span>4.3. Direct reconstruction with <tt class="docutils literal"><span class="pre">upcoef</span></tt></a></h2>
<p>Direct reconstruction from coefficients.</p>
<pre class="literal-block">
upcoef(part, coeffs, wavelet, level=1, take=0)
</pre>
<dl class="docutils">
<dt>part</dt>
<dd><p class="first">defines coefficients type:</p>
<ul class="last simple">
<li><strong>'a'</strong> - approximations reconstruction is performed</li>
<li><strong>'d'</strong> - details reconstruction is performed</li>
</ul>
</dd>
<dt>coeffs</dt>
<dd>coefficients array.</dd>
<dt>wavele</dt>
<dd><tt class="docutils literal"><span class="pre">Wavelet</span></tt></dd>
<dt>level</dt>
<dd>if <em>level</em> is specified then multilevel reconstruction is performed</dd>
<dt>take</dt>
<dd>if <em>take</em> is specified then only the central part of length equal to
<em>'take'</em> is returned.</dd>
</dl>
<p class="example">Example:</p>
<div class="example"><div class="code"><span class="op">>>></span> <span class="kwd">import</span> <span class="name">pywt</span><br /><span class="op">>>></span> <span class="name">data</span> <span class="op">=</span> <span class="op">[</span><span class="num">1</span><span class="op">,</span><span class="num">2</span><span class="op">,</span><span class="num">3</span><span class="op">,</span><span class="num">4</span><span class="op">,</span><span class="num">5</span><span class="op">,</span><span class="num">6</span><span class="op">]</span><br /><span class="op">>>></span> <span class="op">(</span><span class="name">cA</span><span class="op">,</span> <span class="name">cD</span><span class="op">)</span> <span class="op">=</span> <span class="name">pywt</span><span class="op">.</span><span class="name">dwt</span><span class="op">(</span><span class="name">data</span><span class="op">,</span> <span class="str">'db2'</span><span class="op">,</span> <span class="str">'sp1'</span><span class="op">)</span><br /><span class="op">>>></span> <span class="kwd">print</span> <span class="name">pywt</span><span class="op">.</span><span class="name">upcoef</span><span class="op">(</span><span class="str">'a'</span><span class="op">,</span> <span class="name">cA</span><span class="op">,</span> <span class="str">'db2'</span><span class="op">)</span> <span class="op">+</span> <span class="name">pywt</span><span class="op">.</span><span class="name">upcoef</span><span class="op">(</span><span class="str">'d'</span><span class="op">,</span> <span class="name">cD</span><span class="op">,</span> <span class="str">'db2'</span><span class="op">)</span><br /><span class="op">[</span><span class="op">-</span><span class="num">0.25</span>       <span class="op">-</span><span class="num">0.4330127</span>   <span class="num">1.</span>          <span class="num">2.</span>          <span class="num">3.</span>          <span class="num">4.</span>          <span class="num">5.</span><br />  <span class="num">6.</span>          <span class="num">1.78589838</span> <span class="op">-</span><span class="num">1.03108891</span><span class="op">]</span><br /><span class="op">>>></span> <span class="name">n</span> <span class="op">=</span> <span class="name">len</span><span class="op">(</span><span class="name">data</span><span class="op">)</span><br /><span class="op">>>></span> <span class="kwd">print</span> <span class="name">pywt</span><span class="op">.</span><span class="name">upcoef</span><span class="op">(</span><span class="str">'a'</span><span class="op">,</span><span class="name">cA</span><span class="op">,</span><span class="str">'db2'</span><span class="op">,</span><span class="name">take</span><span class="op">=</span><span class="name">n</span><span class="op">)</span> <span class="op">+</span> <span class="name">pywt</span><span class="op">.</span><span class="name">upcoef</span><span class="op">(</span><span class="str">'d'</span><span class="op">,</span><span class="name">cD</span><span class="op">,</span><span class="str">'db2'</span><span class="op">,</span><span class="name">take</span><span class="op">=</span><span class="name">n</span><span class="op">)</span><br /><span class="op">[</span> <span class="num">1.</span>  <span class="num">2.</span>  <span class="num">3.</span>  <span class="num">4.</span>  <span class="num">5.</span>  <span class="num">6.</span><span class="op">]</span>
</div></div></div>
</div>
<div class="section">
<h1><a class="toc-backref" href="#id23" id="d-dwt-and-idwt" name="d-dwt-and-idwt">5. 2D DWT and IDWT</a></h1>
<div class="section">
<h2><a class="toc-backref" href="#id24" id="single-level-dwt2" name="single-level-dwt2"><span id="dwt2"></span>5.1. Single level <tt class="docutils literal"><span class="pre">dwt2</span></tt></a></h2>
<p>The <tt class="docutils literal"><span class="pre">dwt2</span></tt> function performs single level 2D Discrete Wavelet Transform.</p>
<pre class="literal-block">
dwt2(data, wavelet, mode='sym')
</pre>
<dl class="docutils">
<dt>data</dt>
<dd>2D input data</dd>
<dt>wavelet</dt>
<dd>Wavelet to use in transform. This can be name of wavelet from <a class="reference" href="#wavelist">wavelist()</a> or <a class="reference" href="#wavelet">Wavelet</a> object.</dd>
<dt>mode</dt>
<dd>Signal extension mode, see <a class="reference" href="#modes">MODES</a>. This is only important when DWT was performed in <a class="reference" href="#periodization">periodization</a> mode.</dd>
</dl>
<p>Returns one average and three details 2D coefficients arrays. The coefficients
arrays are organized in tuples in the following form:</p>
<pre class="literal-block">
(cA, (cH, cV, cD)),
</pre>
<p>where <tt class="docutils literal"><span class="pre">cA</span></tt>, <tt class="docutils literal"><span class="pre">cH</span></tt>, <tt class="docutils literal"><span class="pre">cV</span></tt>, <tt class="docutils literal"><span class="pre">cD</span></tt> denotes approximation,
horizontal detail, vertical detail
and diagonal detail coefficients respectively.</p>
<p class="example">Example:</p>
<div class="example"><div class="code"><span class="op">>>></span> <span class="kwd">import</span> <span class="name">pywt</span><span class="op">,</span> <span class="name">numpy</span><br /><span class="op">>>></span> <span class="name">data</span> <span class="op">=</span> <span class="name">numpy</span><span class="op">.</span><span class="name">ones</span><span class="op">(</span><span class="op">(</span><span class="num">4</span><span class="op">,</span><span class="num">4</span><span class="op">)</span><span class="op">,</span> <span class="name">dtype</span><span class="op">=</span><span class="name">numpy</span><span class="op">.</span><span class="name">float64</span><span class="op">)</span><br /><span class="op">>>></span> <span class="name">coeffs</span> <span class="op">=</span> <span class="name">pywt</span><span class="op">.</span><span class="name">dwt2</span><span class="op">(</span><span class="name">data</span><span class="op">,</span> <span class="str">'haar'</span><span class="op">)</span><br /><span class="op">>>></span> <span class="name">cA</span><span class="op">,</span> <span class="op">(</span><span class="name">cH</span><span class="op">,</span> <span class="name">cV</span><span class="op">,</span> <span class="name">cD</span><span class="op">)</span> <span class="op">=</span> <span class="name">coeffs</span><br /><span class="op">>>></span> <span class="kwd">print</span> <span class="name">cA</span><br /><span class="op">[</span><span class="op">[</span> <span class="num">2.</span>  <span class="num">2.</span><span class="op">]</span><br /> <span class="op">[</span> <span class="num">2.</span>  <span class="num">2.</span><span class="op">]</span><span class="op">]</span><br /><span class="op">>>></span> <span class="kwd">print</span> <span class="name">cV</span><br /><span class="op">[</span><span class="op">[</span> <span class="num">0.</span>  <span class="num">0.</span><span class="op">]</span><br /> <span class="op">[</span> <span class="num">0.</span>  <span class="num">0.</span><span class="op">]</span><span class="op">]</span>
</div></div></div>
<div class="section">
<h2><a class="toc-backref" href="#id25" id="single-level-idwt2" name="single-level-idwt2"><span id="idwt2"></span>5.2. Single level <tt class="docutils literal"><span class="pre">idwt2</span></tt></a></h2>
<p>The <tt class="docutils literal"><span class="pre">idwt2</span></tt> function reconstructs data from given coefficients by performing
single level 2D Inverse Discrete Wavelet Transform.</p>
<pre class="literal-block">
idwt2(coeffs, wavelet, mode='sym')
</pre>
<dl class="docutils">
<dt>coeffs</dt>
<dd><p class="first">A tuple with approximation coefficients and three details coefficients 2D arrays
like from <a class="reference" href="#dwt2">dwt2</a>:</p>
<pre class="last literal-block">
(cA, (cH, cV, cD))
</pre>
</dd>
<dt>wavelet</dt>
<dd>Wavelet to use in transform. This can be name of wavelet from <a class="reference" href="#wavelist">wavelist()</a> or <a class="reference" href="#wavelet">Wavelet</a> object.</dd>
<dt>mode</dt>
<dd>Signal extension mode, see <a class="reference" href="#modes">MODES</a>. This is only important when DWT was performed in <a class="reference" href="#periodization">periodization</a> mode.</dd>
</dl>
<p class="example">Example:</p>
<div class="example"><div class="code"><span class="op">>>></span> <span class="kwd">import</span> <span class="name">pywt</span><span class="op">,</span> <span class="name">numpy</span><br /><span class="op">>>></span> <span class="name">data</span> <span class="op">=</span> <span class="name">numpy</span><span class="op">.</span><span class="name">array</span><span class="op">(</span><span class="op">[</span><span class="op">[</span><span class="num">1</span><span class="op">,</span><span class="num">2</span><span class="op">]</span><span class="op">,</span> <span class="op">[</span><span class="num">3</span><span class="op">,</span><span class="num">4</span><span class="op">]</span><span class="op">]</span><span class="op">,</span> <span class="name">dtype</span><span class="op">=</span><span class="name">numpy</span><span class="op">.</span><span class="name">float64</span><span class="op">)</span><br /><span class="op">>>></span> <span class="name">coeffs</span> <span class="op">=</span> <span class="name">pywt</span><span class="op">.</span><span class="name">dwt2</span><span class="op">(</span><span class="name">data</span><span class="op">,</span> <span class="str">'haar'</span><span class="op">)</span><br /><span class="op">>>></span> <span class="kwd">print</span> <span class="name">pywt</span><span class="op">.</span><span class="name">idwt2</span><span class="op">(</span><span class="name">coeffs</span><span class="op">,</span> <span class="str">'haar'</span><span class="op">)</span><br /><span class="op">[</span><span class="op">[</span> <span class="num">1.</span>  <span class="num">2.</span><span class="op">]</span><br /> <span class="op">[</span> <span class="num">3.</span>  <span class="num">4.</span><span class="op">]</span><span class="op">]</span>
</div></div></div>
<div class="section">
<h2><a class="toc-backref" href="#id26" id="d-multilevel-decomposition-using-wavedec2" name="d-multilevel-decomposition-using-wavedec2"><span id="wavedec2"></span>5.3. 2D multilevel decomposition using <tt class="docutils literal"><span class="pre">wavedec2</span></tt></a></h2>
<p>Performs multilevel 2D Discrete Wavelet Transform decomposition
and returns coefficients list <tt class="docutils literal"><span class="pre">[cAn,</span> <span class="pre">(cHn,</span> <span class="pre">cVn,</span> <span class="pre">cDn),</span> <span class="pre">...,</span> <span class="pre">(cH1,</span> <span class="pre">cV1,</span> <span class="pre">cD1)]</span></tt>,
where <tt class="docutils literal"><span class="pre">n</span></tt> denotes the level of decomposition and cA, cH, cV and cD are
approximation, horizontal detail, vertical detail and diagonal detail coefficients arrays.</p>
<pre class="literal-block">
wavedec2(data, wavelet, mode='sym', level=None)
</pre>
<dl class="docutils">
<dt>data</dt>
<dd>Input signal can be numeric array, python list or other iterable object.
If data is not in <em>double</em> format it will be converted to that type
before performing computation.</dd>
<dt>wavelet</dt>
<dd>Wavelet to use in transform. This can be name of wavelet from <a class="reference" href="#wavelist">wavelist()</a> or <a class="reference" href="#wavelet">Wavelet</a> object.</dd>
<dt>level</dt>
<dd>Decomposition level. This should not be greater than value
from the <a class="reference" href="#dwt-max-level">dwt_max_level</a> function for smallest dimension.</dd>
<dt>mode</dt>
<dd>Signal extension mode, see <a class="reference" href="#modes">MODES</a>.</dd>
</dl>
<p class="example">Example:</p>
<div class="example"><div class="code"><span class="op">>>></span> <span class="kwd">import</span> <span class="name">pywt</span><span class="op">,</span> <span class="name">numpy</span><br /><span class="op">>>></span> <span class="name">coeffs</span> <span class="op">=</span> <span class="name">pywt</span><span class="op">.</span><span class="name">wavedec2</span><span class="op">(</span><span class="name">numpy</span><span class="op">.</span><span class="name">ones</span><span class="op">(</span><span class="op">(</span><span class="num">8</span><span class="op">,</span><span class="num">8</span><span class="op">)</span><span class="op">)</span><span class="op">,</span> <span class="str">'db1'</span><span class="op">,</span> <span class="name">level</span><span class="op">=</span><span class="num">2</span><span class="op">)</span><br /><span class="op">>>></span> <span class="name">cA2</span><span class="op">,</span> <span class="op">(</span><span class="name">cH2</span><span class="op">,</span> <span class="name">cV2</span><span class="op">,</span> <span class="name">cD2</span><span class="op">)</span><span class="op">,</span> <span class="op">(</span><span class="name">cH1</span><span class="op">,</span> <span class="name">cV1</span><span class="op">,</span> <span class="name">cD1</span><span class="op">)</span> <span class="op">=</span> <span class="name">coeffs</span><br /><span class="op">>>></span> <span class="kwd">print</span> <span class="name">cA2</span><br /><span class="op">[</span><span class="op">[</span> <span class="num">4.</span>  <span class="num">4.</span><span class="op">]</span><br /> <span class="op">[</span> <span class="num">4.</span>  <span class="num">4.</span><span class="op">]</span><span class="op">]</span>
</div></div></div>
<div class="section">
<h2><a class="toc-backref" href="#id27" id="d-multilevel-reconstruction-using-waverec2" name="d-multilevel-reconstruction-using-waverec2"><span id="waverec2"></span>5.4. 2D multilevel reconstruction using <tt class="docutils literal"><span class="pre">waverec2</span></tt></a></h2>
<p>Performs multilevel reconstruction from given coefficient list.</p>
<pre class="literal-block">
waverec2(coeffs, wavelet, mode='sym')
</pre>
<dl class="docutils">
<dt>coeffs</dt>
<dd><p class="first">coefficients list must be in form like that from <a class="reference" href="#wavedec2">wavedec2</a> decomposition:</p>
<pre class="last literal-block">
[cAn, (cHn, cVn, cDn), ..., (cH1, cV1, cD1)]
</pre>
</dd>
<dt>wavelet</dt>
<dd>Wavelet to use in transform. This can be name of wavelet from <a class="reference" href="#wavelist">wavelist()</a> or <a class="reference" href="#wavelet">Wavelet</a> object.</dd>
<dt>mode</dt>
<dd>Signal extension mode, see <a class="reference" href="#modes">MODES</a>.</dd>
</dl>
<p class="example">Example:</p>
<div class="example"><div class="code"><span class="op">>>></span> <span class="kwd">import</span> <span class="name">pywt</span><span class="op">,</span> <span class="name">numpy</span><br /><span class="op">>>></span> <span class="name">coeffs</span> <span class="op">=</span> <span class="name">pywt</span><span class="op">.</span><span class="name">wavedec2</span><span class="op">(</span><span class="name">numpy</span><span class="op">.</span><span class="name">ones</span><span class="op">(</span><span class="op">(</span><span class="num">4</span><span class="op">,</span><span class="num">4</span><span class="op">)</span><span class="op">)</span><span class="op">,</span> <span class="str">'db1'</span><span class="op">)</span><br /><span class="op">>>></span> <span class="kwd">print</span> <span class="str">"levels:"</span><span class="op">,</span> <span class="name">len</span><span class="op">(</span><span class="name">coeffs</span><span class="op">)</span><span class="op">-</span><span class="num">1</span><br /><span class="name">levels</span><span class="op">:</span> <span class="num">2</span><br /><span class="op">>>></span> <span class="kwd">print</span> <span class="name">pywt</span><span class="op">.</span><span class="name">waverec2</span><span class="op">(</span><span class="name">coeffs</span><span class="op">,</span> <span class="str">'db1'</span><span class="op">)</span><br /><span class="op">[</span><span class="op">[</span> <span class="num">1.</span>  <span class="num">1.</span>  <span class="num">1.</span>  <span class="num">1.</span><span class="op">]</span><br /> <span class="op">[</span> <span class="num">1.</span>  <span class="num">1.</span>  <span class="num">1.</span>  <span class="num">1.</span><span class="op">]</span><br /> <span class="op">[</span> <span class="num">1.</span>  <span class="num">1.</span>  <span class="num">1.</span>  <span class="num">1.</span><span class="op">]</span><br /> <span class="op">[</span> <span class="num">1.</span>  <span class="num">1.</span>  <span class="num">1.</span>  <span class="num">1.</span><span class="op">]</span><span class="op">]</span>
</div></div></div>
</div>
<div class="section">
<h1><a class="toc-backref" href="#id28" id="wavelet-packets" name="wavelet-packets">6. Wavelet Packets</a></h1>
<div class="section">
<h2><a class="toc-backref" href="#id29" id="wavelet-packet" name="wavelet-packet">6.1. Wavelet Packet</a></h2>
<p>Tree structure simplifying operations on Wavelet Packet decomposition coefficients.
It consists of <a class="reference" href="#node">Node</a> elements.</p>
<pre class="literal-block">
WaveletPacket(data, wavelet, mode='sp1', maxlevel=None)
</pre>
<dl class="docutils">
<dt>data</dt>
<dd>Input signal can be numeric array, python list or other iterable object.
If data is not in <em>double</em> format it will be converted to that type
before performing computation.</dd>
<dt>wavelet</dt>
<dd>Wavelet to use in transform. This can be name of wavelet from <a class="reference" href="#wavelist">wavelist()</a> or <a class="reference" href="#wavelet">Wavelet</a> object.</dd>
<dt>mode</dt>
<dd>Signal extension mode, see <a class="reference" href="#modes">MODES</a>.</dd>
<dt>maxlevel</dt>
<dd>Maximum level of decomposition. If <em>maxlevel</em> is None it will be computed with
<a class="reference" href="#dwt-max-level">dwt_max_level</a> function.</dd>
</dl>
<p>wp = WaveletPacket(range(16), 'db1', maxlevel=3)</p>
<div class="section">
<h3><a id="access-nodes-get-node-path" name="access-nodes-get-node-path"><span id="get-node-path"></span>6.1.1. Access nodes - <tt class="docutils literal"><span class="pre">get_node(path)</span></tt></a></h3>
<p>Find node of given path in tree.</p>
<dl class="docutils">
<dt>path</dt>
<dd>string composed of "a" and "d", of total length not greater than maxlevel.</dd>
</dl>
<p>If node does not exist yet, it will be created by decomposition of its
parent node.</p>
</div>
<div class="section">
<h3><a id="access-node-data-wp-path" name="access-node-data-wp-path">6.1.2. Access node data - <tt class="docutils literal"><span class="pre">wp[path]</span></tt></a></h3>
<p>Calls <a class="reference" href="#get-node-path">get_node(path)</a> and returns data associated with node under given path.</p>
</div>
<div class="section">
<h3><a id="set-node-data-wp-path-data" name="set-node-data-wp-path-data">6.1.3. Set node data - <tt class="docutils literal"><span class="pre">wp[path]</span> <span class="pre">=</span> <span class="pre">data</span></tt></a></h3>
<p>Calls <a class="reference" href="#get-node-path">get_node(path)</a> and sets data of node under given path.</p>
</div>
<div class="section">
<h3><a id="delete-node-del-wp-path" name="delete-node-del-wp-path">6.1.4. Delete node - <tt class="docutils literal"><span class="pre">del</span> <span class="pre">wp[path]</span></tt></a></h3>
<p>Marks node under given path in tree as ZeroTree root.</p>
<dl class="docutils">
<dt>path</dt>
<dd>string composed of "a" and "d", of total length not greater than maxlevel.</dd>
</dl>
<p>If node does not exist yet, it will be created by decomposition of its
parent node.</p>
</div>
<div class="section">
<h3><a id="reconstruct-signal-reconstruct-update-true" name="reconstruct-signal-reconstruct-update-true">6.1.5. Reconstruct signal - <tt class="docutils literal"><span class="pre">reconstruct(update=True)</span></tt></a></h3>
<p>Returns data reconstruction using coefficients from subnodes.</p>
<p>If update is True, then node's data values will be replaced by
reconstruction values (also in subnodes).</p>
</div>
<div class="section">
<h3><a id="get-nodes-by-level-get-level-level-order-natural" name="get-nodes-by-level-get-level-level-order-natural">6.1.6. Get nodes by level - <tt class="docutils literal"><span class="pre">get_level(level,</span> <span class="pre">order="natural")</span></tt></a></h3>
<p>Returns all nodes from specified level.</p>
<dl class="docutils">
<dt>order</dt>
<dd><ul class="first last simple">
<li>"natural" - left to right in tree</li>
<li>"freq" - frequency ordered nodes</li>
</ul>
</dd>
</dl>
</div>
<div class="section">
<h3><a id="get-terminal-nodes-get-nonzero-decompose-false" name="get-terminal-nodes-get-nonzero-decompose-false">6.1.7. Get terminal nodes - <tt class="docutils literal"><span class="pre">get_nonzero(decompose=False)</span></tt></a></h3>
<p>Returns non-zero terminal nodes.</p>
</div>
<div class="section">
<h3><a id="walk-tree-walk-func-args-tuple" name="walk-tree-walk-func-args-tuple">6.1.8. Walk tree - <tt class="docutils literal"><span class="pre">walk(func,</span> <span class="pre">args=tuple())</span></tt></a></h3>
<p>Walks tree and calls func on every node - <tt class="docutils literal"><span class="pre">func(node,</span> <span class="pre">*args)</span></tt>.
If func returns True, descending to subnodes will proceed.</p>
<dl class="docutils">
<dt>func</dt>
<dd>callable object</dd>
<dt>args</dt>
<dd>additional func parms</dd>
</dl>
</div>
<div class="section">
<h3><a id="walk-tree-postorder-walk-depth-func-args-tuple" name="walk-tree-postorder-walk-depth-func-args-tuple">6.1.9. Walk tree postorder - <tt class="docutils literal"><span class="pre">walk_depth(func,</span> <span class="pre">args=tuple())</span></tt></a></h3>
<p>Walks tree and calls func on every node starting from bottom most nodes.</p>
<dl class="docutils">
<dt>func</dt>
<dd>callable object</dd>
<dt>args</dt>
<dd>additional func parms</dd>
</dl>
</div>
</div>
<div class="section">
<h2><a class="toc-backref" href="#id30" id="node" name="node">6.2. Node</a></h2>
<p>WaveletPacket tree node.</p>
<p>Subnodes are called <strong>'a'</strong> and <strong>'d'</strong>, like approximation and detail coefficients
in Discrete Wavelet Transform</p>
<div class="section">
<h3><a id="path" name="path">6.2.1. <tt class="docutils literal"><span class="pre">path</span></tt></a></h3>
<p>Path under node is accessible in Wavelet Packet tree.</p>
</div>
<div class="section">
<h3><a id="data" name="data">6.2.2. <tt class="docutils literal"><span class="pre">data</span></tt></a></h3>
<p>Data associated with node.</p>
</div>
<div class="section">
<h3><a id="markzerotree-flag-true-remove-sub-true" name="markzerotree-flag-true-remove-sub-true">6.2.3. <tt class="docutils literal"><span class="pre">markZeroTree(flag=True,</span> <span class="pre">remove_sub=True)</span></tt></a></h3>
<p>Mark <em>node</em> as root of ZeroTree, which means that current node and all subnodes
don't take part in reconstruction (all coefficients equals 0).</p>
<dl class="docutils">
<dt>flag</dt>
<dd>True/False - mark/unmark node.</dd>
<dt>remove_sub</dt>
<dd>If remove_sub and flag is True, subnodes of current node will be removed.</dd>
</dl>
</div>
<div class="section">
<h3><a id="iszerotree" name="iszerotree">6.2.4. <tt class="docutils literal"><span class="pre">isZeroTree</span></tt></a></h3>
<p>Field - like markZeroTree.</p>
</div>
<div class="section">
<h3><a id="getchild-part-decompose-true" name="getchild-part-decompose-true">6.2.5. <tt class="docutils literal"><span class="pre">getChild(part,</span> <span class="pre">decompose=True)</span></tt></a></h3>
<p>Returns chosen subnode.</p>
<dl class="docutils">
<dt>part</dt>
<dd>subnode name ('a' or 'd')</dd>
<dt>decompose</dt>
<dd>if True and subnodes don't exist, they will be created by
decomposition of current node (lazy evaluation).</dd>
</dl>
</div>
</div>
</div>
<div class="section">
<h1><a class="toc-backref" href="#id31" id="stationary-wavelet-transform" name="stationary-wavelet-transform">7. Stationary Wavelet Transform</a></h1>
<div class="section">
<h2><a class="toc-backref" href="#id32" id="multilevel-swt" name="multilevel-swt">7.1. Multilevel <tt class="docutils literal"><span class="pre">swt</span></tt></a></h2>
<p>Performs multilevel Stationary Wavelet Transform.</p>
<pre class="literal-block">
swt(data, wavelet, level)
</pre>
<dl class="docutils">
<dt>data</dt>
<dd>Input signal can be numeric array, python list or other iterable object.
If data is not in <em>double</em> format it will be converted to that type
before performing computation. Data length must be divisible by <tt class="docutils literal"><span class="pre">2^level</span></tt>.</dd>
<dt>wavelet</dt>
<dd>Wavelet to use in transform. This can be name of wavelet from <a class="reference" href="#wavelist">wavelist()</a> or <a class="reference" href="#wavelet">Wavelet</a> object.</dd>
<dt>level</dt>
<dd>Required transform level. See <a class="reference" href="#swt-max-level">swt_max_level</a>.</dd>
</dl>
<p>Returned list of coefficient pairs is in form
<tt class="docutils literal"><span class="pre">[(cA1,</span> <span class="pre">cD1),</span> <span class="pre">(cA2,</span> <span class="pre">cD2),</span> <span class="pre">...,</span> <span class="pre">(cAn,</span> <span class="pre">cDn)]</span></tt>, where n = level</p>
</div>
<div class="section">
<h2><a class="toc-backref" href="#id33" id="maximum-decomposition-level-swt-max-level" name="maximum-decomposition-level-swt-max-level"><span id="swt-max-level"></span>7.2. Maximum decomposition level - <tt class="docutils literal"><span class="pre">swt_max_level</span></tt></a></h2>
<p>Returns maximum level of Stationary Wavelet Transform for data of given length.</p>
<pre class="literal-block">
swt_max_level(input_len)
</pre>
<dl class="docutils">
<dt>input_len</dt>
<dd>input data length.</dd>
</dl>
</div>
</div>
<div class="section">
<h1><a class="toc-backref" href="#id34" id="demo" name="demo">8. Demo</a></h1>
<ul class="simple">
<li>Multilevet wavelet decomposition and reconstruction - <a class="reference" href="./demo/wavedec.py">wavedec.py</a></li>
<li>Plot wavelet families - <a class="reference" href="./demo/plot_wavelets.py">plot_wavelets.py</a> - <a class="reference" href="./img/db.png">db.png</a> <a class="reference" href="./img/sym.png">sym.png</a></li>
<li>Plot wavelet and scaling functions - <a class="reference" href="./demo/waveinfo.py">waveinfo.py</a></li>
<li>Plot coefficients from DWT and SWT for 3 different signals - <a class="reference" href="./demo/dwt_swt_show_coeffs.py">dwt_swt_show_coeffs.py</a></li>
<li>Multilevel signal decomposition with DWT - <a class="reference" href="./demo/dwt_signal_decomposition.py">dwt_signal_decomposition.py</a></li>
<li>Simple compression with Wavelet Packet - <a class="reference" href="./demo/wp_simple_compression.py">wp_simple_compression.py</a></li>
<li>Coefficient distribution for several Wavelet Packet Transform levels - <a class="reference" href="./demo/wp_visualize_coeffs_distribution.py">wp_visualize_coeffs_distribution.py</a> - <a class="reference" href="./img/wp_distrib.png">wp_distrib.png</a></li>
<li>Signal frequency analysis using Wavelet Packet - <a class="reference" href="./demo/wp_scalogram.py">wp_scalogram.py</a> - <a class="reference" href="./img/linchirp.png">linchirp.png</a>. See also output of some orca sound scalogram with WP - <a class="reference" href="./img/orca.png">orca.png</a>.</li>
<li>Benchmark <a class="reference" href="#dwt">dwt</a> and <a class="reference" href="#idwt">idwt</a> computation - <a class="reference" href="./demo/benchmark.py">benchmark.py</a> - results achieved on Centrino 1,8GHz laptop - <a class="reference" href="./img/benchmark_dwt.png">benchmark_dwt.png</a>, <a class="reference" href="./img/benchmark_idwt.png">benchmark_idwt.png</a></li>
<li>Creating Wavelet objects from user supplied filter banks - <a class="reference" href="./demo/user_filter_banks.py">user_filter_banks.py</a></li>
<li>Blending image textures in wavelet space - <a class="reference" href="./demo/image_blender.py">image_blender.py</a></li>
</ul>
</div>
</div>
</body>
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