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"""
===========================================
Spectral clustering for image segmentation
===========================================
In this example, an image with connected circles is generated and
:ref:`spectral_clustering` is used to separate the circles.
In these settings, the spectral clustering 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.
As the algorithm tries to balance the volume (ie balance the region
sizes), if we take circles with different sizes, the segmentation fails.
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.
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.
"""
print __doc__
# Authors: Emmanuelle Gouillart <emmanuelle.gouillart@normalesup.org>
# Gael Varoquaux <gael.varoquaux@normalesup.org>
# License: BSD
import numpy as np
import pylab as pl
from sklearn.feature_extraction import image
from sklearn.cluster import spectral_clustering
###############################################################################
l = 100
x, y = np.indices((l, l))
center1 = (28, 24)
center2 = (40, 50)
center3 = (67, 58)
center4 = (24, 70)
radius1, radius2, radius3, radius4 = 16, 14, 15, 14
circle1 = (x - center1[0]) ** 2 + (y - center1[1]) ** 2 < radius1 ** 2
circle2 = (x - center2[0]) ** 2 + (y - center2[1]) ** 2 < radius2 ** 2
circle3 = (x - center3[0]) ** 2 + (y - center3[1]) ** 2 < radius3 ** 2
circle4 = (x - center4[0]) ** 2 + (y - center4[1]) ** 2 < radius4 ** 2
###############################################################################
# 4 circles
img = circle1 + circle2 + circle3 + circle4
mask = img.astype(bool)
img = img.astype(float)
img += 1 + 0.2 * np.random.randn(*img.shape)
# Convert the image into a graph with the value of the gradient on the
# edges.
graph = image.img_to_graph(img, mask=mask)
# Take a decreasing function of the gradient: we take it weakly
# dependant from the gradient the segmentation is close to a voronoi
graph.data = np.exp(-graph.data / graph.data.std())
# Force the solver to be arpack, since amg is numerically
# unstable on this example
labels = spectral_clustering(graph, k=4, mode='arpack')
label_im = -np.ones(mask.shape)
label_im[mask] = labels
pl.matshow(img)
pl.matshow(label_im)
###############################################################################
# 2 circles
img = circle1 + circle2
mask = img.astype(bool)
img = img.astype(float)
img += 1 + 0.2 * np.random.randn(*img.shape)
graph = image.img_to_graph(img, mask=mask)
graph.data = np.exp(-graph.data / graph.data.std())
labels = spectral_clustering(graph, k=2, mode='arpack')
label_im = -np.ones(mask.shape)
label_im[mask] = labels
pl.matshow(img)
pl.matshow(label_im)
pl.show()
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