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<li><a class="reference internal" href="#">Spectral clustering for image segmentation</a><ul>
<li><a class="reference internal" href="#generate-the-data">Generate the data</a></li>
<li><a class="reference internal" href="#plotting-four-circles">Plotting four circles</a></li>
<li><a class="reference internal" href="#plotting-two-circles">Plotting two circles</a></li>
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to download the full example code or to run this example in your browser via Binder</p>
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<section class="sphx-glr-example-title" id="spectral-clustering-for-image-segmentation">
<span id="sphx-glr-auto-examples-cluster-plot-segmentation-toy-py"></span><h1>Spectral clustering for image segmentation<a class="headerlink" href="#spectral-clustering-for-image-segmentation" title="Permalink to this heading">¶</a></h1>
<p>In this example, an image with connected circles is generated and
spectral clustering is used to separate the circles.</p>
<p>In these settings, the <a class="reference internal" href="../../modules/clustering.html#spectral-clustering"><span class="std std-ref">Spectral clustering</span></a> approach solves the problem
know as ‘normalized graph cuts’: the image is seen as a graph of
connected voxels, and the spectral clustering algorithm amounts to
choosing graph cuts defining regions while minimizing the ratio of the
gradient along the cut, and the volume of the region.</p>
<p>As the algorithm tries to balance the volume (ie balance the region
sizes), if we take circles with different sizes, the segmentation fails.</p>
<p>In addition, as there is no useful information in the intensity of the image,
or its gradient, we choose to perform the spectral clustering on a graph
that is only weakly informed by the gradient. This is close to performing
a Voronoi partition of the graph.</p>
<p>In addition, we use the mask of the objects to restrict the graph to the
outline of the objects. In this example, we are interested in
separating the objects one from the other, and not from the background.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="c1"># Authors: Emmanuelle Gouillart <[email protected]></span>
<span class="c1"># Gael Varoquaux <[email protected]></span>
<span class="c1"># License: BSD 3 clause</span>
</pre></div>
</div>
<section id="generate-the-data">
<h2>Generate the data<a class="headerlink" href="#generate-the-data" title="Permalink to this heading">¶</a></h2>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">l</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.indices.html#numpy.indices" title="numpy.indices" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">indices</span></a><span class="p">((</span><span class="n">l</span><span class="p">,</span> <span class="n">l</span><span class="p">))</span>
<span class="n">center1</span> <span class="o">=</span> <span class="p">(</span><span class="mi">28</span><span class="p">,</span> <span class="mi">24</span><span class="p">)</span>
<span class="n">center2</span> <span class="o">=</span> <span class="p">(</span><span class="mi">40</span><span class="p">,</span> <span class="mi">50</span><span class="p">)</span>
<span class="n">center3</span> <span class="o">=</span> <span class="p">(</span><span class="mi">67</span><span class="p">,</span> <span class="mi">58</span><span class="p">)</span>
<span class="n">center4</span> <span class="o">=</span> <span class="p">(</span><span class="mi">24</span><span class="p">,</span> <span class="mi">70</span><span class="p">)</span>
<span class="n">radius1</span><span class="p">,</span> <span class="n">radius2</span><span class="p">,</span> <span class="n">radius3</span><span class="p">,</span> <span class="n">radius4</span> <span class="o">=</span> <span class="mi">16</span><span class="p">,</span> <span class="mi">14</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">14</span>
<span class="n">circle1</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">center1</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">+</span> <span class="p">(</span><span class="n">y</span> <span class="o">-</span> <span class="n">center1</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="o">**</span> <span class="mi">2</span> <span class="o"><</span> <span class="n">radius1</span><span class="o">**</span><span class="mi">2</span>
<span class="n">circle2</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">center2</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">+</span> <span class="p">(</span><span class="n">y</span> <span class="o">-</span> <span class="n">center2</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="o">**</span> <span class="mi">2</span> <span class="o"><</span> <span class="n">radius2</span><span class="o">**</span><span class="mi">2</span>
<span class="n">circle3</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">center3</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">+</span> <span class="p">(</span><span class="n">y</span> <span class="o">-</span> <span class="n">center3</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="o">**</span> <span class="mi">2</span> <span class="o"><</span> <span class="n">radius3</span><span class="o">**</span><span class="mi">2</span>
<span class="n">circle4</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">center4</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">+</span> <span class="p">(</span><span class="n">y</span> <span class="o">-</span> <span class="n">center4</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="o">**</span> <span class="mi">2</span> <span class="o"><</span> <span class="n">radius4</span><span class="o">**</span><span class="mi">2</span>
</pre></div>
</div>
</section>
<section id="plotting-four-circles">
<h2>Plotting four circles<a class="headerlink" href="#plotting-four-circles" title="Permalink to this heading">¶</a></h2>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">img</span> <span class="o">=</span> <span class="n">circle1</span> <span class="o">+</span> <span class="n">circle2</span> <span class="o">+</span> <span class="n">circle3</span> <span class="o">+</span> <span class="n">circle4</span>
<span class="c1"># We use a mask that limits to the foreground: the problem that we are</span>
<span class="c1"># interested in here is not separating the objects from the background,</span>
<span class="c1"># but separating them one from the other.</span>
<span class="n">mask</span> <span class="o">=</span> <span class="n">img</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">bool</span><span class="p">)</span>
<span class="n">img</span> <span class="o">=</span> <span class="n">img</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">float</span><span class="p">)</span>
<span class="n">img</span> <span class="o">+=</span> <span class="mi">1</span> <span class="o">+</span> <span class="mf">0.2</span> <span class="o">*</span> <a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.randn.html#numpy.random.randn" title="numpy.random.randn" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span></a><span class="p">(</span><span class="o">*</span><span class="n">img</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
</pre></div>
</div>
<p>Convert the image into a graph with the value of the gradient on the
edges.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.feature_extraction</span> <span class="kn">import</span> <span class="n">image</span>
<span class="n">graph</span> <span class="o">=</span> <span class="n">image</span><span class="o">.</span><span class="n">img_to_graph</span><span class="p">(</span><span class="n">img</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">mask</span><span class="p">)</span>
</pre></div>
</div>
<p>Take a decreasing function of the gradient resulting in a segmentation
that is close to a Voronoi partition</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">graph</span><span class="o">.</span><span class="n">data</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.exp.html#numpy.exp" title="numpy.exp" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">exp</span></a><span class="p">(</span><span class="o">-</span><span class="n">graph</span><span class="o">.</span><span class="n">data</span> <span class="o">/</span> <span class="n">graph</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">std</span><span class="p">())</span>
</pre></div>
</div>
<p>Here we perform spectral clustering using the arpack solver since amg is
numerically unstable on this example. We then plot the results.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.cluster</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.cluster.spectral_clustering.html#sklearn.cluster.spectral_clustering" title="sklearn.cluster.spectral_clustering" class="sphx-glr-backref-module-sklearn-cluster sphx-glr-backref-type-py-function"><span class="n">spectral_clustering</span></a>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="n">labels</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.cluster.spectral_clustering.html#sklearn.cluster.spectral_clustering" title="sklearn.cluster.spectral_clustering" class="sphx-glr-backref-module-sklearn-cluster sphx-glr-backref-type-py-function"><span class="n">spectral_clustering</span></a><span class="p">(</span><span class="n">graph</span><span class="p">,</span> <span class="n">n_clusters</span><span class="o">=</span><span class="mi">4</span><span class="p">,</span> <span class="n">eigen_solver</span><span class="o">=</span><span class="s2">"arpack"</span><span class="p">)</span>
<span class="n">label_im</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.full.html#numpy.full" title="numpy.full" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">full</span></a><span class="p">(</span><span class="n">mask</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.0</span><span class="p">)</span>
<span class="n">label_im</span><span class="p">[</span><span class="n">mask</span><span class="p">]</span> <span class="o">=</span> <span class="n">labels</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">matshow</span><span class="p">(</span><span class="n">img</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">matshow</span><span class="p">(</span><span class="n">label_im</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="matplotlib.pyplot.show" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_segmentation_toy_001.png" srcset="../../_images/sphx_glr_plot_segmentation_toy_001.png" alt="plot segmentation toy" class = "sphx-glr-single-img"/></section>
<section id="plotting-two-circles">
<h2>Plotting two circles<a class="headerlink" href="#plotting-two-circles" title="Permalink to this heading">¶</a></h2>
<p>Here we repeat the above process but only consider the first two circles
we generated. Note that this results in a cleaner separation between the
circles as the region sizes are easier to balance in this case.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">img</span> <span class="o">=</span> <span class="n">circle1</span> <span class="o">+</span> <span class="n">circle2</span>
<span class="n">mask</span> <span class="o">=</span> <span class="n">img</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">bool</span><span class="p">)</span>
<span class="n">img</span> <span class="o">=</span> <span class="n">img</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">float</span><span class="p">)</span>
<span class="n">img</span> <span class="o">+=</span> <span class="mi">1</span> <span class="o">+</span> <span class="mf">0.2</span> <span class="o">*</span> <a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.randn.html#numpy.random.randn" title="numpy.random.randn" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span></a><span class="p">(</span><span class="o">*</span><span class="n">img</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">graph</span> <span class="o">=</span> <span class="n">image</span><span class="o">.</span><span class="n">img_to_graph</span><span class="p">(</span><span class="n">img</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">mask</span><span class="p">)</span>
<span class="n">graph</span><span class="o">.</span><span class="n">data</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.exp.html#numpy.exp" title="numpy.exp" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">exp</span></a><span class="p">(</span><span class="o">-</span><span class="n">graph</span><span class="o">.</span><span class="n">data</span> <span class="o">/</span> <span class="n">graph</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">std</span><span class="p">())</span>
<span class="n">labels</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.cluster.spectral_clustering.html#sklearn.cluster.spectral_clustering" title="sklearn.cluster.spectral_clustering" class="sphx-glr-backref-module-sklearn-cluster sphx-glr-backref-type-py-function"><span class="n">spectral_clustering</span></a><span class="p">(</span><span class="n">graph</span><span class="p">,</span> <span class="n">n_clusters</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">eigen_solver</span><span class="o">=</span><span class="s2">"arpack"</span><span class="p">)</span>
<span class="n">label_im</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.full.html#numpy.full" title="numpy.full" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">full</span></a><span class="p">(</span><span class="n">mask</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.0</span><span class="p">)</span>
<span class="n">label_im</span><span class="p">[</span><span class="n">mask</span><span class="p">]</span> <span class="o">=</span> <span class="n">labels</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">matshow</span><span class="p">(</span><span class="n">img</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">matshow</span><span class="p">(</span><span class="n">label_im</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="matplotlib.pyplot.show" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_segmentation_toy_002.png" srcset="../../_images/sphx_glr_plot_segmentation_toy_002.png" alt="plot segmentation toy" class = "sphx-glr-single-img"/><p class="sphx-glr-timing"><strong>Total running time of the script:</strong> ( 0 minutes 0.541 seconds)</p>
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