:orphan:

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Alpha complex user manual
=========================
Definition
----------

.. include:: alpha_complex_sum.inc

`AlphaComplex` is constructing a :doc:`SimplexTree <simplex_tree_ref>` using
`Delaunay Triangulation  <http://doc.cgal.org/latest/Triangulation/index.html#Chapter_Triangulations>`_
:cite:`cgal:hdj-t-19b` from the `Computational Geometry Algorithms Library <http://www.cgal.org/>`_
:cite:`cgal:eb-19b`.

Remarks
^^^^^^^
When an :math:`\alpha`-complex is constructed with an infinite value of :math:`\alpha^2`,
the complex is a Delaunay complex (with special filtration values).

Example from points
-------------------

This example builds the alpha-complex from the given points:

.. testcode::

    import gudhi
    alpha_complex = gudhi.AlphaComplex(points=[[1, 1], [7, 0], [4, 6], [9, 6], [0, 14], [2, 19], [9, 17]])

    simplex_tree = alpha_complex.create_simplex_tree()
    result_str = 'Alpha complex is of dimension ' + repr(simplex_tree.dimension()) + ' - ' + \
        repr(simplex_tree.num_simplices()) + ' simplices - ' + \
        repr(simplex_tree.num_vertices()) + ' vertices.'
    print(result_str)
    fmt = '%s -> %.2f'
    for filtered_value in simplex_tree.get_filtration():
        print(fmt % tuple(filtered_value))

The output is:

.. testoutput::

   Alpha complex is of dimension 2 - 25 simplices - 7 vertices.
   [0] -> 0.00
   [1] -> 0.00
   [2] -> 0.00
   [3] -> 0.00
   [4] -> 0.00
   [5] -> 0.00
   [6] -> 0.00
   [2, 3] -> 6.25
   [4, 5] -> 7.25
   [0, 2] -> 8.50
   [0, 1] -> 9.25
   [1, 3] -> 10.00
   [1, 2] -> 11.25
   [1, 2, 3] -> 12.50
   [0, 1, 2] -> 13.00
   [5, 6] -> 13.25
   [2, 4] -> 20.00
   [4, 6] -> 22.74
   [4, 5, 6] -> 22.74
   [3, 6] -> 30.25
   [2, 6] -> 36.50
   [2, 3, 6] -> 36.50
   [2, 4, 6] -> 37.24
   [0, 4] -> 59.71
   [0, 2, 4] -> 59.71


Algorithm
---------

Data structure
^^^^^^^^^^^^^^

In order to build the alpha complex, first, a Simplex tree is built from the cells of a Delaunay Triangulation.
(The filtration value is set to NaN, which stands for unknown value):

.. figure::
    ../../doc/Alpha_complex/alpha_complex_doc.png
    :figclass: align-center
    :alt: Simplex tree structure construction example

    Simplex tree structure construction example

Filtration value computation algorithm
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.. code-block:: vim

    for i : dimension → 0 do
      for all σ of dimension i
        if filtration(σ) is NaN then
          filtration(σ) = α²(σ)
        end if
        for all τ face of σ do // propagate alpha filtration value
          if filtration(τ) is not NaN then
            filtration(τ) = min( filtration(τ), filtration(σ) )
          else
            if τ is not Gabriel for σ then
              filtration(τ) = filtration(σ)
            end if
          end if
        end for
      end for
    end for
    
    make_filtration_non_decreasing()
    prune_above_filtration()


Dimension 2
^^^^^^^^^^^

From the example above, it means the algorithm looks into each triangle ([0,1,2], [0,2,4], [1,2,3], ...),
computes the filtration value of the triangle, and then propagates the filtration value as described
here:

.. figure::
    ../../doc/Alpha_complex/alpha_complex_doc_420.png
    :figclass: align-center
    :alt: Filtration value propagation example

    Filtration value propagation example

Dimension 1
^^^^^^^^^^^

Then, the algorithm looks into each edge ([0,1], [0,2], [1,2], ...),
computes the filtration value of the edge (in this case, propagation will have no effect).

Dimension 0
^^^^^^^^^^^

Finally, the algorithm looks into each vertex ([0], [1], [2], [3], [4], [5] and [6]) and
sets the filtration value (0 in case of a vertex - propagation will have no effect).

Non decreasing filtration values
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

As the squared radii computed by CGAL are an approximation, it might happen that these
:math:`\alpha^2` values do not quite define a proper filtration (i.e. non-decreasing with
respect to inclusion).
We fix that up by calling :func:`~gudhi.SimplexTree.make_filtration_non_decreasing` (cf.
`C++ version <http://gudhi.gforge.inria.fr/doc/latest/index.html>`_).

Prune above given filtration value
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The simplex tree is pruned from the given maximum :math:`\alpha^2` value (cf.
:func:`~gudhi.SimplexTree.prune_above_filtration`). Note that this does not provide any kind
of speed-up, since we always first build the full filtered complex, so it is recommended not to use
:paramref:`~gudhi.AlphaComplex.create_simplex_tree.max_alpha_square`.
In the following example, a threshold of :math:`\alpha^2 = 32.0` is used.


Example from OFF file
^^^^^^^^^^^^^^^^^^^^^

This example builds the Delaunay triangulation from the points given by an OFF file, and initializes the alpha complex
with it.


Then, it is asked to display information about the alpha complex:

.. testcode::

    import gudhi
    alpha_complex = gudhi.AlphaComplex(off_file=gudhi.__root_source_dir__ + \
        '/data/points/alphacomplexdoc.off')
    simplex_tree = alpha_complex.create_simplex_tree(max_alpha_square=32.0)
    result_str = 'Alpha complex is of dimension ' + repr(simplex_tree.dimension()) + ' - ' + \
        repr(simplex_tree.num_simplices()) + ' simplices - ' + \
        repr(simplex_tree.num_vertices()) + ' vertices.'
    print(result_str)
    fmt = '%s -> %.2f'
    for filtered_value in simplex_tree.get_filtration():
        print(fmt % tuple(filtered_value))

the program output is:

.. testoutput::

   Alpha complex is of dimension 2 - 20 simplices - 7 vertices.
   [0] -> 0.00
   [1] -> 0.00
   [2] -> 0.00
   [3] -> 0.00
   [4] -> 0.00
   [5] -> 0.00
   [6] -> 0.00
   [2, 3] -> 6.25
   [4, 5] -> 7.25
   [0, 2] -> 8.50
   [0, 1] -> 9.25
   [1, 3] -> 10.00
   [1, 2] -> 11.25
   [1, 2, 3] -> 12.50
   [0, 1, 2] -> 13.00
   [5, 6] -> 13.25
   [2, 4] -> 20.00
   [4, 6] -> 22.74
   [4, 5, 6] -> 22.74
   [3, 6] -> 30.25