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Cover complexes user manual
===========================
Definition
----------
.. include:: nerve_gic_complex_sum.rst
Visualizations of the simplicial complexes can be done with either
neato (from `graphviz <http://www.graphviz.org/>`_),
`geomview <http://www.geomview.org/>`_,
`KeplerMapper <https://github.com/MLWave/kepler-mapper>`_.
Input point clouds are assumed to be
`OFF files <http://www.geomview.org/docs/html/OFF.html>`_.
Covers
------
Nerves and Graph Induced Complexes require a cover C of the input point cloud P,
that is a set of subsets of P whose union is P itself.
Very often, this cover is obtained from the preimage of a family of intervals covering
the image of some scalar-valued function f defined on P. This family is parameterized
by its resolution, which can be either the number or the length of the intervals,
and its gain, which is the overlap percentage between consecutive intervals (ordered by their first values).
Nerves
------
Nerve definition
^^^^^^^^^^^^^^^^
Assume you are given a cover C of your point cloud P. Then, the Nerve of this cover
is the simplicial complex that has one k-simplex per k-fold intersection of cover elements.
See also `Wikipedia <https://en.wikipedia.org/wiki/Nerve_of_a_covering>`_.
.. figure::
../../doc/Nerve_GIC/nerve.png
:figclass: align-center
:alt: Nerve of a double torus
Nerve of a double torus
Example
^^^^^^^
This example builds the Nerve of a point cloud sampled on a 3D human shape (human.off).
The cover C comes from the preimages of intervals (10 intervals with gain 0.3)
covering the height function (coordinate 2),
which are then refined into their connected components using the triangulation of the .OFF file.
.. testcode::
import gudhi
nerve_complex = gudhi.CoverComplex()
nerve_complex.set_verbose(True)
if (nerve_complex.read_point_cloud(gudhi.__root_source_dir__ + \
'/data/points/human.off')):
nerve_complex.set_type('Nerve')
nerve_complex.set_color_from_coordinate(2)
nerve_complex.set_function_from_coordinate(2)
nerve_complex.set_graph_from_OFF()
nerve_complex.set_resolution_with_interval_number(10)
nerve_complex.set_gain(0.3)
nerve_complex.set_cover_from_function()
nerve_complex.find_simplices()
nerve_complex.write_info()
simplex_tree = nerve_complex.create_simplex_tree()
nerve_complex.compute_PD()
result_str = 'Nerve is of dimension ' + repr(simplex_tree.dimension()) + ' - ' + \
repr(simplex_tree.num_simplices()) + ' simplices - ' + \
repr(simplex_tree.num_vertices()) + ' vertices.'
print(result_str)
for filtered_value in simplex_tree.get_filtration():
print(filtered_value[0])
the program output is:
.. code-block:: none
Min function value = -0.979672 and Max function value = 0.816414
Interval 0 = [-0.979672, -0.761576]
Interval 1 = [-0.838551, -0.581967]
Interval 2 = [-0.658942, -0.402359]
Interval 3 = [-0.479334, -0.22275]
Interval 4 = [-0.299725, -0.0431414]
Interval 5 = [-0.120117, 0.136467]
Interval 6 = [0.059492, 0.316076]
Interval 7 = [0.239101, 0.495684]
Interval 8 = [0.418709, 0.675293]
Interval 9 = [0.598318, 0.816414]
Computing preimages...
Computing connected components...
5 interval(s) in dimension 0:
[-0.909111, 0.0081753]
[-0.171433, 0.367393]
[-0.171433, 0.367393]
[-0.909111, 0.745853]
0 interval(s) in dimension 1:
.. testoutput::
Nerve is of dimension 1 - 41 simplices - 21 vertices.
[1]
[0]
[4]
[0, 4]
[2]
[1, 2]
[8]
[2, 8]
[5]
[4, 5]
[9]
[8, 9]
[13]
[5, 13]
[14]
[9, 14]
[19]
[13, 19]
[25]
[32]
[20]
[20, 32]
[33]
[25, 33]
[26]
[14, 26]
[19, 26]
[42]
[26, 42]
[34]
[33, 34]
[27]
[20, 27]
[35]
[27, 35]
[34, 35]
[35, 42]
[44]
[35, 44]
[54]
[44, 54]
The program also writes a file ../../data/points/human.off_sc.txt. The first
three lines in this file are the location of the input point cloud and the
function used to compute the cover.
The fourth line contains the number of vertices nv and edges ne of the Nerve.
The next nv lines represent the vertices. Each line contains the vertex ID,
the number of data points it contains, and their average color function value.
Finally, the next ne lines represent the edges, characterized by the ID of
their vertices.
Using KeplerMapper, one can obtain the following visualization:
.. figure::
../../doc/Nerve_GIC/nervevisu.jpg
:figclass: align-center
:alt: Visualization with KeplerMapper
Visualization with KeplerMapper
Graph Induced Complexes (GIC)
-----------------------------
GIC definition
^^^^^^^^^^^^^^
Again, assume you are given a cover C of your point cloud P. Moreover, assume
you are also given a graph G built on top of P. Then, for any clique in G
whose nodes all belong to different elements of C, the GIC includes a
corresponding simplex, whose dimension is the number of nodes in the clique
minus one.
See :cite:`Dey13` for more details.
.. figure::
../../doc/Nerve_GIC/GIC.jpg
:figclass: align-center
:alt: GIC of a point cloud
GIC of a point cloud
Example with cover from Voronoï
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
This example builds the GIC of a point cloud sampled on a 3D human shape
(human.off).
We randomly subsampled 100 points in the point cloud, which act as seeds of
a geodesic Voronoï diagram. Each cell of the diagram is then an element of C.
The graph G (used to compute both the geodesics for Voronoï and the GIC)
comes from the triangulation of the human shape. Note that the resulting
simplicial complex is in dimension 3 in this example.
.. testcode::
import gudhi
nerve_complex = gudhi.CoverComplex()
if (nerve_complex.read_point_cloud(gudhi.__root_source_dir__ + \
'/data/points/human.off')):
nerve_complex.set_type('GIC')
nerve_complex.set_color_from_coordinate()
nerve_complex.set_graph_from_OFF()
nerve_complex.set_cover_from_Voronoi(700)
nerve_complex.find_simplices()
nerve_complex.plot_off()
the program outputs SC.off. Using e.g.
.. code-block:: none
geomview ../../data/points/human.off_sc.off
one can obtain the following visualization:
.. figure::
../../doc/Nerve_GIC/gicvoronoivisu.jpg
:figclass: align-center
:alt: Visualization with Geomview
Visualization with Geomview
|
