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<section id="normal-distance-transform-ndt-map">
<span id="ndt-map"></span><h1>Normal Distance Transform (NDT) map<a class="headerlink" href="#normal-distance-transform-ndt-map" title="Permalink to this headline"></a></h1>
<p>This is a NDT mapping example.</p>
<p>Normal Distribution Transform (NDT) is a map representation that uses normal distribution for observation point modeling.</p>
<section id="normal-distribution">
<h2>Normal Distribution<a class="headerlink" href="#normal-distribution" title="Permalink to this headline"></a></h2>
<p>Normal distribution consists of two parameters: mean <span class="math notranslate nohighlight">\(\mu\)</span> and covariance <span class="math notranslate nohighlight">\(\Sigma\)</span>.</p>
<p><span class="math notranslate nohighlight">\(\mathbf{X} \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})\)</span></p>
<p>In the 2D case, <span class="math notranslate nohighlight">\(\boldsymbol{\mu}\)</span> is a 2D vector and <span class="math notranslate nohighlight">\(\boldsymbol{\Sigma}\)</span> is a 2x2 matrix.</p>
<p>In the matrix form, the probability density function of thr normal distribution is:</p>
<p><span class="math notranslate nohighlight">\(X=\frac{1}{\sqrt{(2 \pi)^2|\Sigma|}} \exp \left\{-\frac{1}{2}^t(x-\mu) \sum^{-1}(x-\mu)\right\}\)</span></p>
</section>
<section id="normal-distance-transform-mapping-steps">
<h2>Normal Distance Transform mapping steps<a class="headerlink" href="#normal-distance-transform-mapping-steps" title="Permalink to this headline"></a></h2>
<p>NDT mapping consists of two steps:</p>
<p>When we have a new observation like this:</p>
<figure class="align-default">
<img alt="../../../_images/raw_observations.png" src="../../../_images/raw_observations.png" />
</figure>
<p>First, we need to cluster the observation points.
This is done by using a grid based clustering algorithm.</p>
<p>The result is:</p>
<figure class="align-default">
<img alt="../../../_images/grid_clustering.png" src="../../../_images/grid_clustering.png" />
</figure>
<p>Then, we need to fit a normal distribution to each grid cluster.</p>
<p>Black ellipse shows each NDT grid like this:</p>
<figure class="align-default">
<img alt="../../../_images/ndt_map1.png" src="../../../_images/ndt_map1.png" />
</figure>
<figure class="align-default">
<img alt="../../../_images/ndt_map2.png" src="../../../_images/ndt_map2.png" />
</figure>
</section>
<section id="api">
<h2>API<a class="headerlink" href="#api" title="Permalink to this headline"></a></h2>
<dl class="py class">
<dt class="sig sig-object py" id="Mapping.ndt_map.ndt_map.NDTMap">
<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">Mapping.ndt_map.ndt_map.</span></span><span class="sig-name descname"><span class="pre">NDTMap</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">ox</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">oy</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">resolution</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../../_modules/Mapping/ndt_map/ndt_map.html#NDTMap"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#Mapping.ndt_map.ndt_map.NDTMap" title="Permalink to this definition"></a></dt>
<dd><p>Normal Distribution Transform (NDT) map class</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>ox</strong> – obstacle x position list</p></li>
<li><p><strong>oy</strong> – obstacle y position list</p></li>
<li><p><strong>resolution</strong> – grid resolution [m]</p></li>
</ul>
</dd>
</dl>
<dl class="py class">
<dt class="sig sig-object py" id="Mapping.ndt_map.ndt_map.NDTMap.NDTGrid">
<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">NDTGrid</span></span><a class="reference internal" href="../../../_modules/Mapping/ndt_map/ndt_map.html#NDTMap.NDTGrid"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#Mapping.ndt_map.ndt_map.NDTMap.NDTGrid" title="Permalink to this definition"></a></dt>
<dd><p>NDT grid</p>
<dl class="py attribute">
<dt class="sig sig-object py" id="Mapping.ndt_map.ndt_map.NDTMap.NDTGrid.center_grid_x">
<span class="sig-name descname"><span class="pre">center_grid_x</span></span><a class="headerlink" href="#Mapping.ndt_map.ndt_map.NDTMap.NDTGrid.center_grid_x" title="Permalink to this definition"></a></dt>
<dd><p>Center x position of the NDT grid</p>
</dd></dl>
<dl class="py attribute">
<dt class="sig sig-object py" id="Mapping.ndt_map.ndt_map.NDTMap.NDTGrid.center_grid_y">
<span class="sig-name descname"><span class="pre">center_grid_y</span></span><a class="headerlink" href="#Mapping.ndt_map.ndt_map.NDTMap.NDTGrid.center_grid_y" title="Permalink to this definition"></a></dt>
<dd><p>Center y position of the NDT grid</p>
</dd></dl>
<dl class="py attribute">
<dt class="sig sig-object py" id="Mapping.ndt_map.ndt_map.NDTMap.NDTGrid.covariance">
<span class="sig-name descname"><span class="pre">covariance</span></span><a class="headerlink" href="#Mapping.ndt_map.ndt_map.NDTMap.NDTGrid.covariance" title="Permalink to this definition"></a></dt>
<dd><p>Covariance matrix of the NDT grid</p>
</dd></dl>
<dl class="py attribute">
<dt class="sig sig-object py" id="Mapping.ndt_map.ndt_map.NDTMap.NDTGrid.eig_values">
<span class="sig-name descname"><span class="pre">eig_values</span></span><a class="headerlink" href="#Mapping.ndt_map.ndt_map.NDTMap.NDTGrid.eig_values" title="Permalink to this definition"></a></dt>
<dd><p>Eigen values of the NDT grid</p>
</dd></dl>
<dl class="py attribute">
<dt class="sig sig-object py" id="Mapping.ndt_map.ndt_map.NDTMap.NDTGrid.eig_vec">
<span class="sig-name descname"><span class="pre">eig_vec</span></span><a class="headerlink" href="#Mapping.ndt_map.ndt_map.NDTMap.NDTGrid.eig_vec" title="Permalink to this definition"></a></dt>
<dd><p>Eigen vectors of the NDT grid</p>
</dd></dl>
<dl class="py attribute">
<dt class="sig sig-object py" id="Mapping.ndt_map.ndt_map.NDTMap.NDTGrid.mean_x">
<span class="sig-name descname"><span class="pre">mean_x</span></span><a class="headerlink" href="#Mapping.ndt_map.ndt_map.NDTMap.NDTGrid.mean_x" title="Permalink to this definition"></a></dt>
<dd><p>Mean x position of points in the NDTGrid cell</p>
</dd></dl>
<dl class="py attribute">
<dt class="sig sig-object py" id="Mapping.ndt_map.ndt_map.NDTMap.NDTGrid.mean_y">
<span class="sig-name descname"><span class="pre">mean_y</span></span><a class="headerlink" href="#Mapping.ndt_map.ndt_map.NDTMap.NDTGrid.mean_y" title="Permalink to this definition"></a></dt>
<dd><p>Mean y position of points in the NDTGrid cell</p>
</dd></dl>
<dl class="py attribute">
<dt class="sig sig-object py" id="Mapping.ndt_map.ndt_map.NDTMap.NDTGrid.n_points">
<span class="sig-name descname"><span class="pre">n_points</span></span><a class="headerlink" href="#Mapping.ndt_map.ndt_map.NDTMap.NDTGrid.n_points" title="Permalink to this definition"></a></dt>
<dd><p>Number of points in the NDTGrid grid</p>
</dd></dl>
</dd></dl>
<dl class="py attribute">
<dt class="sig sig-object py" id="Mapping.ndt_map.ndt_map.NDTMap.grid_index_map">
<span class="sig-name descname"><span class="pre">grid_index_map</span></span><a class="headerlink" href="#Mapping.ndt_map.ndt_map.NDTMap.grid_index_map" title="Permalink to this definition"></a></dt>
<dd><p>NDT grid index map</p>
</dd></dl>
<dl class="py attribute">
<dt class="sig sig-object py" id="Mapping.ndt_map.ndt_map.NDTMap.min_n_points">
<span class="sig-name descname"><span class="pre">min_n_points</span></span><a class="headerlink" href="#Mapping.ndt_map.ndt_map.NDTMap.min_n_points" title="Permalink to this definition"></a></dt>
<dd><p>Minimum number of points in the NDT grid</p>
</dd></dl>
<dl class="py attribute">
<dt class="sig sig-object py" id="Mapping.ndt_map.ndt_map.NDTMap.resolution">
<span class="sig-name descname"><span class="pre">resolution</span></span><a class="headerlink" href="#Mapping.ndt_map.ndt_map.NDTMap.resolution" title="Permalink to this definition"></a></dt>
<dd><p>Resolution of the NDT grid [m]</p>
</dd></dl>
</dd></dl>
</section>
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