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Diffstat (limited to 'cython/doc')
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diff --git a/cython/doc/Makefile.in b/cython/doc/Makefile.in new file mode 100644 index 00000000..526350b3 --- /dev/null +++ b/cython/doc/Makefile.in @@ -0,0 +1,44 @@ +# Makefile for Sphinx documentation +# + +# You can set these variables from the command line. +SPHINXOPTS = +SPHINXBUILD = @SPHINX_PATH@ +PAPER = +BUILDDIR = _build + +# User-friendly check for sphinx-build +ifeq ($(shell which $(SPHINXBUILD) >/dev/null 2>&1; echo $$?), 1) +$(error The '$(SPHINXBUILD)' command was not found. Make sure you have Sphinx installed, then set the SPHINXBUILD environment variable to point to the full path of the '$(SPHINXBUILD)' executable. Alternatively you can add the directory with the executable to your PATH. If you don't have Sphinx installed, grab it from http://sphinx-doc.org/) +endif + +# Internal variables. +PAPEROPT_a4 = -D latex_paper_size=a4 +PAPEROPT_letter = -D latex_paper_size=letter +ALLSPHINXOPTS = -d $(BUILDDIR)/doctrees $(PAPEROPT_$(PAPER)) $(SPHINXOPTS) . +# the i18n builder cannot share the environment and doctrees with the others +I18NSPHINXOPTS = $(PAPEROPT_$(PAPER)) $(SPHINXOPTS) . + +.PHONY: help clean html dirhtml singlehtml pickle json htmlhelp qthelp devhelp epub latex latexpdf text man changes linkcheck doctest gettext + +help: + @echo "Please use \`make <target>' where <target> is one of" + @echo " html to make standalone HTML files" + @echo " doctest to run all doctests embedded in the documentation (if enabled)" + +clean: + rm -f examples.inc + rm -rf $(BUILDDIR)/* + +# GUDHI specific : Examples.inc is generated with generate_examples.py (and deleted on clean) + +html: + ./generate_examples.py + $(SPHINXBUILD) -b html $(ALLSPHINXOPTS) $(BUILDDIR)/html + @echo + @echo "Build finished. The HTML pages are in $(BUILDDIR)/html." + +doctest: + $(SPHINXBUILD) -b doctest $(ALLSPHINXOPTS) $(BUILDDIR)/doctest + @echo "Testing of doctests in the sources finished, look at the " \ + "results in $(BUILDDIR)/doctest/output.txt." diff --git a/cython/doc/_templates/layout.html b/cython/doc/_templates/layout.html new file mode 100644 index 00000000..b11c1236 --- /dev/null +++ b/cython/doc/_templates/layout.html @@ -0,0 +1,270 @@ +{# + basic/layout.html + ~~~~~~~~~~~~~~~~~ + + Master layout template for Sphinx themes. + + :copyright: Copyright 2007-2016 by the Sphinx team, see AUTHORS. + :license: BSD, see LICENSE for details. +#} +{%- block doctype -%} +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" + "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> +{%- endblock %} +{%- set reldelim1 = reldelim1 is not defined and ' »' or reldelim1 %} +{%- set reldelim2 = reldelim2 is not defined and ' |' or reldelim2 %} +{%- set render_sidebar = (not embedded) and (not theme_nosidebar|tobool) and + (sidebars != []) %} +{%- set url_root = pathto('', 1) %} +{# XXX necessary? #} +{%- if url_root == '#' %}{% set url_root = '' %}{% endif %} +{%- if not embedded and docstitle %} + {%- set titlesuffix = " — "|safe + docstitle|e %} +{%- else %} + {%- set titlesuffix = "" %} +{%- endif %} + +{%- macro relbar() %} + <div class="related" role="navigation" aria-label="related navigation"> + <h3>{{ _('Navigation') }}</h3> + <ul> + {%- for rellink in rellinks %} + <li class="right" {% if loop.first %}style="margin-right: 10px"{% endif %}> + <a href="{{ pathto(rellink[0]) }}" title="{{ rellink[1]|striptags|e }}" + {{ accesskey(rellink[2]) }}>{{ rellink[3] }}</a> + {%- if not loop.first %}{{ reldelim2 }}{% endif %}</li> + {%- endfor %} + {%- block rootrellink %} + <li class="nav-item nav-item-0"><a href="{{ pathto(master_doc) }}">{{ shorttitle|e }}</a>{{ reldelim1 }}</li> + {%- endblock %} + {%- for parent in parents %} + <li class="nav-item nav-item-{{ loop.index }}"><a href="{{ parent.link|e }}" {% if loop.last %}{{ accesskey("U") }}{% endif %}>{{ parent.title }}</a>{{ reldelim1 }}</li> + {%- endfor %} + {%- block relbaritems %} {% endblock %} + </ul> + </div> +{%- endmacro %} + +{%- macro sidebar() %} + {%- if render_sidebar %} + <div class="sphinxsidebar" role="navigation" aria-label="main navigation"> + <div class="sphinxsidebarwrapper"> + {%- block sidebarlogo %} + {%- if logo %} + <p class="logo"><a href="{{ pathto(master_doc) }}"> + <img class="logo" src="{{ pathto('_static/' + logo, 1) }}" alt="Logo"/> + </a></p> + {%- endif %} + {%- endblock %} + {%- if sidebars != None %} + {#- new style sidebar: explicitly include/exclude templates #} + {%- for sidebartemplate in sidebars %} + {%- include sidebartemplate %} + {%- endfor %} + {%- else %} + {#- old style sidebars: using blocks -- should be deprecated #} + {%- block sidebartoc %} +<h2><a href="index.html">GUDHI</a></h2> +<h2><a href="installation.html">GUDHI installation</a></h2> +<h2><a href="citation.html">Acknowledging the GUDHI library</a></h2> +<h2><a href="genindex.html">Index</a></h2> +<h2><a href="examples.html">Examples</a></h2> + + {%- include "localtoc.html" %} + {%- endblock %} + {%- block sidebarrel %} + {%- include "relations.html" %} + {%- endblock %} + {%- block sidebarsourcelink %} + {%- include "sourcelink.html" %} + {%- endblock %} + {%- if customsidebar %} + {%- include customsidebar %} + {%- endif %} + {%- block sidebarsearch %} + {%- include "searchbox.html" %} + {%- endblock %} + {%- endif %} + </div> + </div> + {%- endif %} +{%- endmacro %} + +{%- macro script() %} + <script type="text/javascript"> + var DOCUMENTATION_OPTIONS = { + URL_ROOT: '{{ url_root }}', + VERSION: '{{ release|e }}', + COLLAPSE_INDEX: false, + FILE_SUFFIX: '{{ '' if no_search_suffix else file_suffix }}', + HAS_SOURCE: {{ has_source|lower }} + }; + </script> + {%- for scriptfile in script_files %} + <script type="text/javascript" src="{{ pathto(scriptfile, 1) }}"></script> + {%- endfor %} +{%- endmacro %} + +{%- macro css() %} +<!-- GUDHI website css for header BEGIN --> +<link rel="stylesheet" type="text/css" href="http://gudhi.gforge.inria.fr/assets/css/styles_feeling_responsive.css" /> +<!-- GUDHI website css for header END --> + <link rel="stylesheet" href="{{ pathto('_static/' + style, 1) }}" type="text/css" /> + <link rel="stylesheet" href="{{ pathto('_static/pygments.css', 1) }}" type="text/css" /> + {%- for cssfile in css_files %} + <link rel="stylesheet" href="{{ pathto(cssfile, 1) }}" type="text/css" /> + {%- endfor %} +{%- endmacro %} +<!-- GUDHI website html class for header BEGIN --> +<html xmlns="http://www.w3.org/1999/xhtml" class="no-js" lang="en"> +<!-- GUDHI website html class for header END --> + <head> + <meta http-equiv="Content-Type" content="text/html; charset={{ encoding }}" /> + {{ metatags }} + {%- block htmltitle %} + <title>{{ title|striptags|e }}{{ titlesuffix }}</title> + {%- endblock %} + {{ css() }} + {%- if not embedded %} + {{ script() }} + {%- if use_opensearch %} + <link rel="search" type="application/opensearchdescription+xml" + title="{% trans docstitle=docstitle|e %}Search within {{ docstitle }}{% endtrans %}" + href="{{ pathto('_static/opensearch.xml', 1) }}"/> + {%- endif %} + {%- if favicon %} + <link rel="shortcut icon" href="{{ pathto('_static/' + favicon, 1) }}"/> + {%- endif %} + {%- endif %} +{%- block linktags %} + {%- if hasdoc('about') %} + <link rel="author" title="{{ _('About these documents') }}" href="{{ pathto('about') }}" /> + {%- endif %} + {%- if hasdoc('genindex') %} + <link rel="index" title="{{ _('Index') }}" href="{{ pathto('genindex') }}" /> + {%- endif %} + {%- if hasdoc('search') %} + <link rel="search" title="{{ _('Search') }}" href="{{ pathto('search') }}" /> + {%- endif %} + {%- if hasdoc('copyright') %} + <link rel="copyright" title="{{ _('Copyright') }}" href="{{ pathto('copyright') }}" /> + {%- endif %} + <link rel="top" title="{{ docstitle|e }}" href="{{ pathto(master_doc) }}" /> + {%- if parents %} + <link rel="up" title="{{ parents[-1].title|striptags|e }}" href="{{ parents[-1].link|e }}" /> + {%- endif %} + {%- if next %} + <link rel="next" title="{{ next.title|striptags|e }}" href="{{ next.link|e }}" /> + {%- endif %} + {%- if prev %} + <link rel="prev" title="{{ prev.title|striptags|e }}" href="{{ prev.link|e }}" /> + {%- endif %} +{%- endblock %} +{%- block extrahead %} {% endblock %} + </head> + <body role="document"> + <!-- GUDHI website header BEGIN --> + <div id="navigation" class="sticky"> + <nav class="top-bar" role="navigation" data-topbar> + <ul class="title-area"> + <li class="name"> + <h1 class="show-for-small-only"><a href="http://gudhi.gforge.inria.fr" class="icon-tree"> GUDHI C++ library</a></h1> + </li> + <!-- Remove the class "menu-icon" to get rid of menu icon. Take out "Menu" to just have icon alone --> + <li class="toggle-topbar menu-icon"><a href="#"><span>Navigation</span></a></li> + </ul> + <section class="top-bar-section"> + <ul class="right"> + <li class="divider"></li> + <li><a href="http://gudhi.gforge.inria.fr/contact/">Contact</a></li> + </ul> + <ul class="left"> + <li><a href="http://gudhi.gforge.inria.fr/"> <img src="http://gudhi.gforge.inria.fr/assets/img/home.png" alt=" GUDHI"> GUDHI </a></li> + <li class="divider"></li> + <li class="has-dropdown"> + <a href="#">Project</a> + <ul class="dropdown"> + <li><a href="http://gudhi.gforge.inria.fr/people/">People</a></li> + <li><a href="http://gudhi.gforge.inria.fr/keepintouch/">Keep in touch</a></li> + <li><a href="http://gudhi.gforge.inria.fr/partners/">Partners and Funding</a></li> + <li><a href="http://gudhi.gforge.inria.fr/relatedprojects/">Related projects</a></li> + <li><a href="http://gudhi.gforge.inria.fr/theyaretalkingaboutus/">They are talking about us</a></li> + </ul> + 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</div><!-- /#navigation --> + <!-- GUDHI website header BEGIN --> + + +{%- block header %}{% endblock %} + +{%- block relbar1 %}{% endblock %} + +{%- block content %} + {%- block sidebar1 %} {# possible location for sidebar #} {% endblock %} + + <div class="document"> + {%- block document %} + <div class="documentwrapper"> + {%- if render_sidebar %} + <div class="bodywrapper"> + {%- endif %} + <div class="body" role="main"> + {% block body %} {% endblock %} + </div> + {%- if render_sidebar %} + </div> + {%- endif %} + </div> + {%- endblock %} + + {%- block sidebar2 %}{{ sidebar() }}{% endblock %} + <div class="clearer"></div> + </div> +{%- endblock %} + +{%- block relbar2 %}{% endblock %} + +{%- block footer %} + <div class="footer" role="contentinfo"> + {%- if show_copyright %} + {%- if hasdoc('copyright') %} + {% trans path=pathto('copyright'), copyright=copyright|e %}© <a href="{{ path }}">Copyright</a> {{ copyright }}.{% endtrans %} + {%- else %} + {% trans copyright=copyright|e %}© Copyright {{ copyright }}.{% endtrans %} + {%- endif %} + {%- endif %} + {%- if last_updated %} + {% trans last_updated=last_updated|e %}Last updated on {{ last_updated }}.{% endtrans %} + {%- endif %} + {%- if show_sphinx %} + {% trans sphinx_version=sphinx_version|e %}Created using <a href="http://sphinx-doc.org/">Sphinx</a> {{ sphinx_version }}.{% endtrans %} + {%- endif %} + </div> +{%- endblock %} + </body> +</html> + diff --git a/cython/doc/alpha_complex_ref.rst b/cython/doc/alpha_complex_ref.rst new file mode 100644 index 00000000..6a122b09 --- /dev/null +++ b/cython/doc/alpha_complex_ref.rst @@ -0,0 +1,10 @@ +============================== +Alpha complex reference manual +============================== + +.. autoclass:: gudhi.AlphaComplex + :members: + :undoc-members: + :show-inheritance: + + .. automethod:: gudhi.AlphaComplex.__init__ diff --git a/cython/doc/alpha_complex_sum.rst b/cython/doc/alpha_complex_sum.rst new file mode 100644 index 00000000..8437e901 --- /dev/null +++ b/cython/doc/alpha_complex_sum.rst @@ -0,0 +1,22 @@ +================================================================= =================================== =================================== +:Author: Vincent Rouvreau :Introduced in: GUDHI 2.0.0 :Copyright: GPL v3 +:Requires: CGAL :math:`\geq` 4.7.0 Eigen3 +================================================================= =================================== =================================== + ++----------------------------------------------------------------+------------------------------------------------------------------------+ +| .. figure:: | Alpha_complex is a simplicial complex constructed from the finite | +| img/alpha_complex_representation.png | cells of a Delaunay Triangulation. | +| :alt: Alpha complex representation | | +| :figclass: align-center | The filtration value of each simplex is computed as the square of the | +| | circumradius of the simplex if the circumsphere is empty (the simplex | +| Alpha complex representation | is then said to be Gabriel), and as the minimum of the filtration | +| | values of the codimension 1 cofaces that make it not Gabriel | +| | otherwise. All simplices that have a filtration value strictly | +| | greater than a given alpha squared value are not inserted into the | +| | complex. | +| | | +| | This package requires having CGAL version 4.7 or higher (4.8.1 is | +| | advised for better perfomances). | ++----------------------------------------------------------------+------------------------------------------------------------------------+ +| :doc:`alpha_complex_user` | :doc:`alpha_complex_ref` | ++----------------------------------------------------------------+------------------------------------------------------------------------+ diff --git a/cython/doc/alpha_complex_user.rst b/cython/doc/alpha_complex_user.rst new file mode 100644 index 00000000..e8268ef1 --- /dev/null +++ b/cython/doc/alpha_complex_user.rst @@ -0,0 +1,205 @@ +Alpha complex user manual +========================= +Definition +---------- + +.. include:: alpha_complex_sum.rst + +Alpha_complex is constructing a :doc:`Simplex_tree <simplex_tree_sum>` using +`Delaunay Triangulation <http://doc.cgal.org/latest/Triangulation/index.html#Chapter_Triangulations>`_ +:cite:`cgal:hdj-t-15b` from `CGAL <http://www.cgal.org/>`_ (the Computational Geometry Algorithms Library +:cite:`cgal:eb-15b`). + +Remarks +^^^^^^^ +When Alpha_complex is constructed with an infinite value of :math:`\alpha`, the complex is a Delaunay complex. + +Example from points +------------------- + +This example builds the Delaunay triangulation from the given points, and initializes the alpha complex with it: + +.. 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(max_alpha_square=60.0) + 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 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:: + img/alpha_complex_doc.png + :figclass: align-center + :alt: Simplex tree structure construction example + + Simplex tree structure construction example + +Filtration value computation algorithm +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + + **for** i : dimension :math:`\rightarrow` 0 **do** + **for all** :math:`\sigma` of dimension i + **if** filtration(:math:`\sigma`) is NaN **then** + filtration(:math:`\sigma`) = :math:`\alpha^2(\sigma)` + **end if** + + *//propagate alpha filtration value* + + **for all** :math:`\tau` face of :math:`\sigma` + **if** filtration(:math:`\tau`) is not NaN **then** + filtration(:math:`\tau`) = filtration(:math:`\sigma`) + **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:: + img/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 alpha squared values do not +quite define a proper filtration (i.e. non-decreasing with respect to inclusion). +We fix that up by calling `Simplex_tree::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 alpha squared value (cf. `Simplex_tree::prune_above_filtration()` +int he `C++ version <http://gudhi.gforge.inria.fr/doc/latest/index.html>`_). +In the following example, the value is given by the user as argument of the program. + + +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='alphacomplexdoc.off') + simplex_tree = alpha_complex.create_simplex_tree(max_alpha_square=59.0) + 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 - 23 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 + +CGAL citations +============== + +.. bibliography:: how_to_cite_cgal.bib + :filter: docnames + :style: unsrt diff --git a/cython/doc/bottleneck_distance_sum.rst b/cython/doc/bottleneck_distance_sum.rst new file mode 100644 index 00000000..5c475d0d --- /dev/null +++ b/cython/doc/bottleneck_distance_sum.rst @@ -0,0 +1,15 @@ +================================================================= =================================== =================================== +:Author: François Godi :Introduced in: GUDHI 2.0.0 :Copyright: GPL v3 +:Requires: CGAL :math:`\geq` 4.8.0 +================================================================= =================================== =================================== + ++-----------------------------------------------------------------+----------------------------------------------------------------------+ +| .. figure:: | Bottleneck distance measures the similarity between two persistence | +| img/perturb_pd.png | diagrams. It's the shortest distance b for which there exists a | +| :figclass: align-center | perfect matching between the points of the two diagrams (+ all the | +| | diagonal points) such that any couple of matched points are at | +| Bottleneck distance is the length of | distance at most b. | +| the longest edge | | ++-----------------------------------------------------------------+----------------------------------------------------------------------+ +| :doc:`bottleneck_distance_user` | | ++-----------------------------------------------------------------+----------------------------------------------------------------------+ diff --git a/cython/doc/bottleneck_distance_user.rst b/cython/doc/bottleneck_distance_user.rst new file mode 100644 index 00000000..0066992f --- /dev/null +++ b/cython/doc/bottleneck_distance_user.rst @@ -0,0 +1,36 @@ +Bottleneck distance user manual +=============================== +Definition +---------- + +.. include:: bottleneck_distance_sum.rst + +Function +-------- +.. autofunction:: gudhi.bottleneck_distance + + +Basic example +------------- + +This example computes the bottleneck distance from 2 persistence diagrams: + +.. testcode:: + + import gudhi + + diag1 = [[2.7, 3.7],[9.6, 14.],[34.2, 34.974], [3.,float('Inf')]] + diag2 = [[2.8, 4.45],[9.5, 14.1],[3.2,float('Inf')]] + + message = "Bottleneck distance approximation=" + '%.2f' % gudhi.bottleneck_distance(diag1, diag2, 0.1) + print(message) + + message = "Bottleneck distance exact value=" + '%.2f' % gudhi.bottleneck_distance(diag1, diag2, 0) + print(message) + +The output is: + +.. testoutput:: + + Bottleneck distance approximation=0.81 + Bottleneck distance exact value=0.75 diff --git a/cython/doc/citation.rst b/cython/doc/citation.rst new file mode 100644 index 00000000..6cdfb7cc --- /dev/null +++ b/cython/doc/citation.rst @@ -0,0 +1,15 @@ +Acknowledging the GUDHI library +############################### + +We kindly ask users to cite the GUDHI library as appropriately as possible in +their papers, and to mention the use of the GUDHI library on the web pages of +their projects using GUDHI and provide us with links to these web pages. Feel +free to contact us in case you have any question or remark on this topic. + +We provide GUDHI bibtex entries for the modules of the User and Reference +Manual, as well as for publications directly related to the GUDHI library. + +GUDHI bibtex +************ + +.. literalinclude:: how_to_cite_gudhi.bib diff --git a/cython/doc/conf.py b/cython/doc/conf.py new file mode 100755 index 00000000..42bfd59c --- /dev/null +++ b/cython/doc/conf.py @@ -0,0 +1,278 @@ +# -*- coding: utf-8 -*- +# +# GUDHI documentation build configuration file, created by +# sphinx-quickstart on Thu Jun 30 09:55:51 2016. +# +# This file is execfile()d with the current directory set to its +# containing dir. +# +# Note that not all possible configuration values are present in this +# autogenerated file. +# +# All configuration values have a default; values that are commented out +# serve to show the default. + +import sys +import os + +# If extensions (or modules to document with autodoc) are in another directory, +# add these directories to sys.path here. If the directory is relative to the +# documentation root, use os.path.abspath to make it absolute, like shown here. +#sys.path.insert(0, os.path.abspath('.')) + +# Path to Gudhi.so from source path +sys.path.insert(0, os.path.abspath('..')) + +# -- General configuration ------------------------------------------------ + +# If your documentation needs a minimal Sphinx version, state it here. +#needs_sphinx = '1.0' + +# Add any Sphinx extension module names here, as strings. They can be +# extensions coming with Sphinx (named 'sphinx.ext.*') or your custom +# ones. +extensions = [ + 'matplotlib.sphinxext.plot_directive', + 'sphinx.ext.autodoc', + 'sphinx.ext.doctest', + 'sphinx.ext.todo', + 'sphinx.ext.mathjax', + 'sphinx.ext.ifconfig', + 'sphinx.ext.viewcode', + 'sphinxcontrib.bibtex', +] + +todo_include_todos = True +# plot option : do not show hyperlinks (Source code, png, hires.png, pdf) +plot_html_show_source_link = False +plot_html_show_formats = False +# Add any paths that contain templates here, relative to this directory. +templates_path = ['_templates'] + +# The suffix of source filenames. +source_suffix = '.rst' + +# The encoding of source files. +#source_encoding = 'utf-8-sig' + +# The master toctree document. +master_doc = 'index' + +# General information about the project. +project = u'GUDHI' +copyright = u'2016, GUDHI Editorial Board' + +# The version info for the project you're documenting, acts as replacement for +# |version| and |release|, also used in various other places throughout the +# built documents. +# +# The short X.Y version. +version = '2.0' +# The full version, including alpha/beta/rc tags. +release = '2.0.0' + +# The language for content autogenerated by Sphinx. Refer to documentation +# for a list of supported languages. +#language = None + +# There are two options for replacing |today|: either, you set today to some +# non-false value, then it is used: +#today = '' +# Else, today_fmt is used as the format for a strftime call. +#today_fmt = '%B %d, %Y' + +# List of patterns, relative to source directory, that match files and +# directories to ignore when looking for source files. +exclude_patterns = ['_build'] + +# The reST default role (used for this markup: `text`) to use for all +# documents. +#default_role = None + +# If true, '()' will be appended to :func: etc. cross-reference text. +#add_function_parentheses = True + +# If true, the current module name will be prepended to all description +# unit titles (such as .. function::). +#add_module_names = True + +# If true, sectionauthor and moduleauthor directives will be shown in the +# output. They are ignored by default. +#show_authors = False + +# The name of the Pygments (syntax highlighting) style to use. +pygments_style = 'sphinx' + +# A list of ignored prefixes for module index sorting. +#modindex_common_prefix = [] + +# If true, keep warnings as "system message" paragraphs in the built documents. +#keep_warnings = False + + +# -- Options for HTML output ---------------------------------------------- + +# The theme to use for HTML and HTML Help pages. See the documentation for +# a list of builtin themes. +html_theme = 'classic' + +# Theme options are theme-specific and customize the look and feel of a theme +# further. For a list of options available for each theme, see the +# documentation. +html_theme_options = { + "sidebarbgcolor": "#A1ADCD", + "sidebartextcolor": "black", + "sidebarlinkcolor": "#334D5C", +} + +# Add any paths that contain custom themes here, relative to this directory. +#html_theme_path = [] + +# The name for this set of Sphinx documents. If None, it defaults to +# "<project> v<release> documentation". +#html_title = None + +# A shorter title for the navigation bar. Default is the same as html_title. +#html_short_title = None + +# The name of an image file (relative to this directory) to place at the top +# of the sidebar. +#html_logo = + +# The name of an image file (within the static path) to use as favicon of the +# docs. This file should be a Windows icon file (.ico) being 16x16 or 32x32 +# pixels large. +#html_favicon = + +# Add any paths that contain custom static files (such as style sheets) here, +# relative to this directory. They are copied after the builtin static files, +# so a file named "default.css" will overwrite the builtin "default.css". +html_static_path = ['_static'] + +# Add any extra paths that contain custom files (such as robots.txt or +# .htaccess) here, relative to this directory. These files are copied +# directly to the root of the documentation. +#html_extra_path = [] + +# If not '', a 'Last updated on:' timestamp is inserted at every page bottom, +# using the given strftime format. +#html_last_updated_fmt = '%b %d, %Y' + +# If true, SmartyPants will be used to convert quotes and dashes to +# typographically correct entities. +#html_use_smartypants = True + +# Custom sidebar templates, maps document names to template names. +#html_sidebars = {} + +# Additional templates that should be rendered to pages, maps page names to +# template names. +#html_additional_pages = {'installation': 'installation.html'} + +# If false, no module index is generated. +#html_domain_indices = True + +# If false, no index is generated. +#html_use_index = True + +# If true, the index is split into individual pages for each letter. +#html_split_index = False + +# If true, links to the reST sources are added to the pages. +#html_show_sourcelink = True + +# If true, "Created using Sphinx" is shown in the HTML footer. Default is True. +#html_show_sphinx = True + +# If true, "(C) Copyright ..." is shown in the HTML footer. Default is True. +#html_show_copyright = True + +# If true, an OpenSearch description file will be output, and all pages will +# contain a <link> tag referring to it. The value of this option must be the +# base URL from which the finished HTML is served. +#html_use_opensearch = '' + +# This is the file name suffix for HTML files (e.g. ".xhtml"). +#html_file_suffix = None + +# Output file base name for HTML help builder. +htmlhelp_basename = 'GUDHIdoc' + + +# -- Options for LaTeX output --------------------------------------------- + +latex_elements = { +# The paper size ('letterpaper' or 'a4paper'). +#'papersize': 'letterpaper', + +# The font size ('10pt', '11pt' or '12pt'). +#'pointsize': '10pt', + +# Additional stuff for the LaTeX preamble. +#'preamble': '', +} + +# Grouping the document tree into LaTeX files. List of tuples +# (source start file, target name, title, +# author, documentclass [howto, manual, or own class]). +latex_documents = [ + ('index', 'GUDHI.tex', u'GUDHI Documentation', + u'Vincent Rouvreau', 'manual'), +] + +# The name of an image file (relative to this directory) to place at the top of +# the title page. +#latex_logo = None + +# For "manual" documents, if this is true, then toplevel headings are parts, +# not chapters. +#latex_use_parts = False + +# If true, show page references after internal links. +#latex_show_pagerefs = False + +# If true, show URL addresses after external links. +#latex_show_urls = False + +# Documents to append as an appendix to all manuals. +#latex_appendices = [] + +# If false, no module index is generated. +#latex_domain_indices = True + + +# -- Options for manual page output --------------------------------------- + +# One entry per manual page. List of tuples +# (source start file, name, description, authors, manual section). +man_pages = [ + ('index', 'gudhi', u'GUDHI Documentation', + [u'Vincent Rouvreau'], 1) +] + +# If true, show URL addresses after external links. +#man_show_urls = False + + +# -- Options for Texinfo output ------------------------------------------- + +# Grouping the document tree into Texinfo files. List of tuples +# (source start file, target name, title, author, +# dir menu entry, description, category) +texinfo_documents = [ + ('index', 'GUDHI', u'GUDHI Documentation', + u'Vincent Rouvreau', 'GUDHI', 'One line description of project.', + 'Miscellaneous'), +] + +# Documents to append as an appendix to all manuals. +#texinfo_appendices = [] + +# If false, no module index is generated. +#texinfo_domain_indices = True + +# How to display URL addresses: 'footnote', 'no', or 'inline'. +#texinfo_show_urls = 'footnote' + +# If true, do not generate a @detailmenu in the "Top" node's menu. +#texinfo_no_detailmenu = False diff --git a/cython/doc/cubical_complex_ref.rst b/cython/doc/cubical_complex_ref.rst new file mode 100644 index 00000000..84aa4223 --- /dev/null +++ b/cython/doc/cubical_complex_ref.rst @@ -0,0 +1,9 @@ +Cubical complex reference manual +################################ + +.. autoclass:: gudhi.CubicalComplex + :members: + :undoc-members: + :show-inheritance: + + .. automethod:: gudhi.CubicalComplex.__init__ diff --git a/cython/doc/cubical_complex_sum.rst b/cython/doc/cubical_complex_sum.rst new file mode 100644 index 00000000..3ddf6375 --- /dev/null +++ b/cython/doc/cubical_complex_sum.rst @@ -0,0 +1,15 @@ +================================================================= =================================== =================================== +:Author: Pawel Dlotko :Introduced in: GUDHI 2.0.0 :Copyright: GPL v3 +================================================================= =================================== =================================== + ++-----------------------------------------------------------------+----------------------------------------------------------------------+ +| .. figure:: | The cubical complex is an example of a structured complex useful in | +| img/Cubical_complex_representation.png | computational mathematics (specially rigorous numerics) and image | +| :alt: Cubical complex representation | analysis. | +| :figclass: align-center | | +| | | +| Cubical complex representation | | ++-----------------------------------------------------------------+----------------------------------------------------------------------+ +| :doc:`cubical_complex_user` | * :doc:`cubical_complex_ref` | +| | * :doc:`periodic_cubical_complex_ref` | ++-----------------------------------------------------------------+----------------------------------------------------------------------+ diff --git a/cython/doc/cubical_complex_user.rst b/cython/doc/cubical_complex_user.rst new file mode 100644 index 00000000..344b9554 --- /dev/null +++ b/cython/doc/cubical_complex_user.rst @@ -0,0 +1,160 @@ +Cubical complex user manual +=========================== +Definition +---------- + +===================================== ===================================== ===================================== +:Author: Pawel Dlotko :Introduced in: GUDHI PYTHON 2.0.0 :Copyright: GPL v3 +===================================== ===================================== ===================================== + ++---------------------------------------------+----------------------------------------------------------------------+ +| :doc:`cubical_complex_user` | * :doc:`cubical_complex_ref` | +| | * :doc:`periodic_cubical_complex_ref` | ++---------------------------------------------+----------------------------------------------------------------------+ + +The cubical complex is an example of a structured complex useful in computational mathematics (specially rigorous +numerics) and image analysis. + +An *elementary interval* is an interval of a form :math:`[n,n+1]`, or :math:`[n,n]`, for :math:`n \in \mathcal{Z}`. +The first one is called *non-degenerate*, while the second one is a *degenerate* interval. A +*boundary of a elementary interval* is a chain :math:`\partial [n,n+1] = [n+1,n+1]-[n,n]` in case of +non-degenerated elementary interval and :math:`\partial [n,n] = 0` in case of degenerate elementary interval. An +*elementary cube* :math:`C` is a product of elementary intervals, :math:`C=I_1 \times \ldots \times I_n`. +*Embedding dimension* of a cube is n, the number of elementary intervals (degenerate or not) in the product. +A *dimension of a cube* :math:`C=I_1 \times ... \times I_n` is the number of non degenerate elementary +intervals in the product. A *boundary of a cube* :math:`C=I_1 \times \ldots \times I_n` is a chain obtained +in the following way: + +.. math:: + + \partial C = (\partial I_1 \times \ldots \times I_n) + (I_1 \times \partial I_2 \times \ldots \times I_n) + + \ldots + (I_1 \times I_2 \times \ldots \times \partial I_n). + +A *cubical complex* :math:`\mathcal{K}` is a collection of cubes closed under operation of taking boundary +(i.e. boundary of every cube from the collection is in the collection). A cube :math:`C` in cubical complex +:math:`\mathcal{K}` is *maximal* if it is not in a boundary of any other cube in :math:`\mathcal{K}`. A +*support* of a cube :math:`C` is the set in :math:`\mathbb{R}^n` occupied by :math:`C` (:math:`n` is the embedding +dimension of :math:`C`). + +Cubes may be equipped with a filtration values in which case we have filtered cubical complex. All the cubical +complexes considered in this implementation are filtered cubical complexes (although, the range of a filtration may +be a set of two elements). + +For further details and theory of cubical complexes, please consult :cite:`kaczynski2004computational` as well as the +following paper :cite:`peikert2012topological`. + +Data structure. +--------------- + +The implementation of Cubical complex provides a representation of complexes that occupy a rectangular region in +:math:`\mathbb{R}^n`. This extra assumption allows for a memory efficient way of storing cubical complexes in a form +of so called bitmaps. Let +:math:`R = [b_1,e_1] \times \ldots \times [b_n,e_n]`, for :math:`b_1,...b_n,e_1,...,e_n \in \mathbb{Z}`, +:math:`b_i \leq d_i` be the considered rectangular region and let :math:`\mathcal{K}` be a filtered +cubical complex having the rectangle :math:`R` as its support. Note that the structure of the coordinate system gives +a way a lexicographical ordering of cells of :math:`\mathcal{K}`. This ordering is a base of the presented +bitmap-based implementation. In this implementation, the whole cubical complex is stored as a vector of the values +of filtration. This, together with dimension of :math:`\mathcal{K}` and the sizes of :math:`\mathcal{K}` in all +directions, allows to determine, dimension, neighborhood, boundary and coboundary of every cube +:math:`C \in \mathcal{K}`. + +.. figure:: + img/Cubical_complex_representation.png + :alt: Cubical complex. + :figclass: align-center + + Cubical complex. + +Note that the cubical complex in the figure above is, in a natural way, a product of one dimensional cubical +complexes in :math:`\mathbb{R}`. The number of all cubes in each direction is equal :math:`2n+1`, where :math:`n` is +the number of maximal cubes in the considered direction. Let us consider a cube at the position :math:`k` in the +bitmap. +Knowing the sizes of the bitmap, by a series of modulo operation, we can determine which elementary intervals are +present in the product that gives the cube :math:`C`. In a similar way, we can compute boundary and the coboundary of +each cube. Further details can be found in the literature. + +Input Format. +------------- + +In the current implantation, filtration is given at the maximal cubes, and it is then extended by the lower star +filtration to all cubes. There are a number of constructors that can be used to construct cubical complex by users +who want to use the code directly. They can be found in the :doc:`cubical_complex_ref`. +Currently one input from a text file is used. It uses a format used already in +`Perseus software <http://www.sas.upenn.edu/~vnanda/perseus/>`_ by Vidit Nanda. +Below we are providing a description of the format. The first line contains a number d begin the dimension of the +bitmap (2 in the example below). Next d lines are the numbers of top dimensional cubes in each dimensions (3 and 3 +in the example below). Next, in lexicographical order, the filtration of top dimensional cubes is given (1 4 6 8 +20 4 7 6 5 in the example below). + +.. figure:: + img/exampleBitmap.png + :alt: Example of a input data. + :figclass: align-center + + Example of a input data. + +The input file for the following complex is: + +.. literalinclude:: cubicalcomplexdoc.txt + +.. centered:: cubicalcomplexdoc.txt + +.. testcode:: + + import gudhi + cubical_complex = gudhi.CubicalComplex(perseus_file='cubicalcomplexdoc.txt') + result_str = 'Cubical complex is of dimension ' + repr(cubical_complex.dimension()) + ' - ' + \ + repr(cubical_complex.num_simplices()) + ' simplices.' + print(result_str) + +the program output is: + +.. testoutput:: + + Cubical complex is of dimension 2 - 49 simplices. + +Periodic boundary conditions. +----------------------------- + +Often one would like to impose periodic boundary conditions to the cubical complex (cf. +:doc:`periodic_cubical_complex_ref`). +Let :math:`I_1\times ... \times I_n` be a box that is decomposed with a cubical complex :math:`\mathcal{K}`. +Imposing periodic boundary conditions in the direction i, means that the left and the right side of a complex +:math:`\mathcal{K}` are considered the same. In particular, if for a bitmap :math:`\mathcal{K}` periodic boundary +conditions are imposed in all directions, then complex :math:`\mathcal{K}` became n-dimensional torus. One can use +various constructors from the file Bitmap_cubical_complex_periodic_boundary_conditions_base.h to construct cubical +complex with periodic boundary conditions. One can also use Perseus style input files. To indicate periodic boundary +conditions in a given direction, then number of top dimensional cells in this direction have to be multiplied by -1. +For instance: + +.. literalinclude:: periodiccubicalcomplexdoc.txt + +.. centered:: periodiccubicalcomplexdoc.txt + +Indicate that we have imposed periodic boundary conditions in the direction x, but not in the direction y. + +.. testcode:: + + import gudhi + periodic_cc = gudhi.PeriodicCubicalComplex(perseus_file='periodiccubicalcomplexdoc.txt') + result_str = 'Periodic cubical complex is of dimension ' + repr(periodic_cc.dimension()) + ' - ' + \ + repr(periodic_cc.num_simplices()) + ' simplices.' + print(result_str) + +the program output is: + +.. testoutput:: + + Periodic cubical complex is of dimension 2 - 42 simplices. + +Examples. +--------- + +End user programs are available in cython/example/ folder. + +Bibliography +============ + +.. bibliography:: bibliography.bib + :filter: docnames + :style: unsrt diff --git a/cython/doc/euclidean_strong_witness_complex_ref.rst b/cython/doc/euclidean_strong_witness_complex_ref.rst new file mode 100644 index 00000000..bebf0f9a --- /dev/null +++ b/cython/doc/euclidean_strong_witness_complex_ref.rst @@ -0,0 +1,10 @@ +================================================= +Euclidean strong witness complex reference manual +================================================= + +.. autoclass:: gudhi.EuclideanStrongWitnessComplex + :members: + :undoc-members: + :show-inheritance: + + .. automethod:: gudhi.EuclideanStrongWitnessComplex.__init__ diff --git a/cython/doc/euclidean_witness_complex_ref.rst b/cython/doc/euclidean_witness_complex_ref.rst new file mode 100644 index 00000000..29b8806f --- /dev/null +++ b/cython/doc/euclidean_witness_complex_ref.rst @@ -0,0 +1,10 @@ +========================================== +Euclidean witness complex reference manual +========================================== + +.. autoclass:: gudhi.EuclideanWitnessComplex + :members: + :undoc-members: + :show-inheritance: + + .. automethod:: gudhi.EuclideanWitnessComplex.__init__ diff --git a/cython/doc/examples.rst b/cython/doc/examples.rst new file mode 100644 index 00000000..a89e0596 --- /dev/null +++ b/cython/doc/examples.rst @@ -0,0 +1,4 @@ +Examples +######## + +.. include:: examples.inc diff --git a/cython/doc/generate_examples.py b/cython/doc/generate_examples.py new file mode 100755 index 00000000..d64d506c --- /dev/null +++ b/cython/doc/generate_examples.py @@ -0,0 +1,43 @@ +#!/usr/bin/env python + +from os import listdir + +"""This file is part of the Gudhi Library. The Gudhi library + (Geometric Understanding in Higher Dimensions) is a generic C++ + library for computational topology. + + Author(s): Vincent Rouvreau + + Copyright (C) 2017 INRIA + + This program is free software: you can redistribute it and/or modify + it under the terms of the GNU General Public License as published by + the Free Software Foundation, either version 3 of the License, or + (at your option) any later version. + + This program is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + GNU General Public License for more details. + + You should have received a copy of the GNU General Public License + along with this program. If not, see <http://www.gnu.org/licenses/>. +""" + +__author__ = "Vincent Rouvreau" +__copyright__ = "Copyright (C) 2017 INRIA" +__license__ = "GPL v3" + +""" +generate_examples.py generates examples.inc to be included in examples.rst. +Refer to Makefile and make.bat to see if it is correctly launched. +""" + +output_file = open('examples.inc','w') + +output_file.write('.. only:: builder_html\n\n') + +for file in listdir('../example/'): + output_file.write(" * :download:`" + file + " <../example/" + file + ">`\n") + +output_file.close() diff --git a/cython/doc/img/graphical_tools_representation.png b/cython/doc/img/graphical_tools_representation.png Binary files differnew file mode 100644 index 00000000..9759f7ba --- /dev/null +++ b/cython/doc/img/graphical_tools_representation.png diff --git a/cython/doc/index.rst b/cython/doc/index.rst new file mode 100644 index 00000000..94dbc65a --- /dev/null +++ b/cython/doc/index.rst @@ -0,0 +1,87 @@ +GUDHI documentation +################### + +.. image:: img/Gudhi_banner.png + :align: center + +Introduction +************ + +The Gudhi library (Geometry Understanding in Higher Dimensions) is a generic +open source `C++ library <http://gudhi.gforge.inria.fr/doc/latest/>`_, with a +`Python interface <http://gudhi.gforge.inria.fr/python/latest/>`_, for +Computational Topology and Topological Data Analysis +(`TDA <https://en.wikipedia.org/wiki/Topological_data_analysis>`_). +The GUDHI library intends to help the development of new algorithmic solutions +in TDA and their transfer to applications. It provides robust, efficient, +flexible and easy to use implementations of state-of-the-art algorithms and +data structures. + +The current release of the GUDHI library includes: + +* Data structures to represent, construct and manipulate simplicial complexes. +* Algorithms to compute persistent homology and multi-field persistent homology. +* Simplication of simplicial complexes by edge contraction. + +All data-structures are generic and several of their aspects can be +parameterized via template classes. We refer to :cite:`gudhilibrary_ICMS14` +for a detailed description of the design of the library. + +Data structures +*************** + +Alpha complex +============= + +.. include:: alpha_complex_sum.rst + +Cubical complex +=============== + +.. include:: cubical_complex_sum.rst + +Rips complex +============ + +.. include:: rips_complex_sum.rst + +Simplex tree +============ + +.. include:: simplex_tree_sum.rst + +Tangential complex +================== + +.. include:: tangential_complex_sum.rst + +Witness complex +=============== + +.. include:: witness_complex_sum.rst + + +Toolbox +******* + +Bottleneck distance +=================== + +.. include:: bottleneck_distance_sum.rst + +Persistence cohomology +====================== + +.. include:: persistent_cohomology_sum.rst + +Persistence graphical tools +=========================== + +.. include:: persistence_graphical_tools_sum.rst + +Bibliography +************ + +.. bibliography:: bibliography.bib + :filter: docnames + :style: unsrt diff --git a/cython/doc/installation.rst b/cython/doc/installation.rst new file mode 100644 index 00000000..f98a5039 --- /dev/null +++ b/cython/doc/installation.rst @@ -0,0 +1,176 @@ +Installation +############ + +Compiling +********* + +The library uses c++11 and requires `Boost <http://www.boost.org/>`_ with +version 1.48.0 or more recent. It is a multi-platform library and compiles on +Linux, Mac OSX and Visual Studio 2015. +It also requires cmake to generate makefiles, and cython to compile the +library. + +On `Windows <https://wiki.python.org/moin/WindowsCompilers>`_ , only Python +3.5 and 3.6 are available because of the required Visual Studio version. + +On other systems, if you have several Python/cython installed, the version 2.X +will be used by default, but you can force it by adding +:code:`-DPython_ADDITIONAL_VERSIONS=3` to the cmake command. + +GUDHI Cythonization +=================== + +To build the GUDHI cython module, run the following commands in a terminal: + +.. code-block:: bash + + cd /path-to-gudhi/ + mkdir build + cd build/ + cmake .. + make cython + +Test suites +=========== + +To test your build, `py.test <http://doc.pytest.org>`_ is optional. Run the +following command in a terminal: + +.. code-block:: bash + + cd /path-to-gudhi/build/cython + # For windows, you have to set PYTHONPATH environment variable + export PYTHONPATH='$PYTHONPATH:/path-to-gudhi/build/cython' + ctest -R py_test + +Documentation +============= + +To build the documentation, `sphinx-doc <http://http://www.sphinx-doc.org>`_ is +required. Please refer to *conf.py* file to see which +`sphinx-doc <http://http://www.sphinx-doc.org>`_ modules are required to +generate the documentation. Run the following commands in a terminal: + +.. code-block:: bash + + make sphinx + +Optional third-party library +**************************** + +CGAL +==== + +The :doc:`Alpha complex </alpha_complex_user>`, +:doc:`Tangential complex </tangential_complex_user>` and +:doc:`Witness complex </witness_complex_user>` data structures, and +:doc:`Bottleneck distance </bottleneck_distance_user>` requires CGAL, which is a +C++ library which provides easy access to efficient and reliable geometric +algorithms. + +Having CGAL version 4.6.0 or higher installed is recommended. The procedure to +install this library according to your operating system is detailed +`here <http://doc.cgal.org/latest/Manual/installation.html>`_. + +The following examples require the Computational Geometry Algorithms Library: + +.. only:: builder_html + + * :download:`euclidean_strong_witness_complex_diagram_persistence_from_off_file_example.py <../example/euclidean_strong_witness_complex_diagram_persistence_from_off_file_example.py>` + * :download:`euclidean_witness_complex_diagram_persistence_from_off_file_example.py <../example/euclidean_witness_complex_diagram_persistence_from_off_file_example.py>` + +The following example requires CGAL version ≥ 4.7.0: + +.. only:: builder_html + + * :download:`alpha_complex_diagram_persistence_from_off_file_example.py <../example/alpha_complex_diagram_persistence_from_off_file_example.py>` + * :download:`alpha_complex_from_points_example.py <../example/alpha_complex_from_points_example.py>` + +The following example requires CGAL version ≥ 4.8.0: + +.. only:: builder_html + + * :download:`bottleneck_basic_example.py <../example/bottleneck_basic_example.py>` + * :download:`tangential_complex_plain_homology_from_off_file_example.py <../example/tangential_complex_plain_homology_from_off_file_example.py>` + +Eigen3 +====== + +The :doc:`Alpha complex </alpha_complex_user>`, +:doc:`Tangential complex </tangential_complex_user>` and +:doc:`Witness complex </witness_complex_user>` data structures and few +examples requires `Eigen3 <http://eigen.tuxfamily.org/>`_, a C++ template +library for linear algebra: matrices, vectors, numerical solvers, and related +algorithms. + +The following examples require the `Eigen3 <http://eigen.tuxfamily.org/>`_: + +.. only:: builder_html + + * :download:`alpha_complex_diagram_persistence_from_off_file_example.py <../example/alpha_complex_diagram_persistence_from_off_file_example.py>` + * :download:`alpha_complex_from_points_example.py <../example/alpha_complex_from_points_example.py>` + * :download:`tangential_complex_plain_homology_from_off_file_example.py <../example/tangential_complex_plain_homology_from_off_file_example.py>` + * :download:`euclidean_strong_witness_complex_diagram_persistence_from_off_file_example.py <../example/euclidean_strong_witness_complex_diagram_persistence_from_off_file_example.py>` + * :download:`euclidean_witness_complex_diagram_persistence_from_off_file_example.py <../example/euclidean_witness_complex_diagram_persistence_from_off_file_example.py>` + +Matplotlib +========== + +The :doc:`persistence graphical tools </persistence_graphical_tools_user>` +module requires `Matplotlib <http://matplotlib.org>`_, a Python 2D plotting +library which produces publication quality figures in a variety of hardcopy +formats and interactive environments across platforms. + +The following examples require the `Matplotlib <http://matplotlib.org>`_: + +.. only:: builder_html + + * :download:`alpha_complex_diagram_persistence_from_off_file_example.py <../example/alpha_complex_diagram_persistence_from_off_file_example.py>` + * :download:`gudhi_graphical_tools_example.py <../example/gudhi_graphical_tools_example.py>` + * :download:`periodic_cubical_complex_barcode_persistence_from_perseus_file_example.py <../example/periodic_cubical_complex_barcode_persistence_from_perseus_file_example.py>` + * :download:`rips_complex_diagram_persistence_from_off_file_example.py <../example/rips_complex_diagram_persistence_from_off_file_example.py>` + * :download:`rips_persistence_diagram.py <../example/rips_persistence_diagram.py>` + * :download:`rips_complex_diagram_persistence_from_distance_matrix_file_example.py <../example/rips_complex_diagram_persistence_from_distance_matrix_file_example.py>` + * :download:`tangential_complex_plain_homology_from_off_file_example.py <../example/tangential_complex_plain_homology_from_off_file_example.py>` + * :download:`euclidean_strong_witness_complex_diagram_persistence_from_off_file_example.py <../example/euclidean_strong_witness_complex_diagram_persistence_from_off_file_example.py>` + * :download:`euclidean_witness_complex_diagram_persistence_from_off_file_example.py <../example/euclidean_witness_complex_diagram_persistence_from_off_file_example.py>` + +Numpy +===== + +The :doc:`persistence graphical tools </persistence_graphical_tools_user>` +module requires `NumPy <http://numpy.org>`_, a fundamental package for +scientific computing with Python. + +The following examples require the `NumPy <http://numpy.org>`_: + +.. only:: builder_html + + * :download:`alpha_complex_diagram_persistence_from_off_file_example.py <../example/alpha_complex_diagram_persistence_from_off_file_example.py>` + * :download:`gudhi_graphical_tools_example.py <../example/gudhi_graphical_tools_example.py>` + * :download:`periodic_cubical_complex_barcode_persistence_from_perseus_file_example.py <../example/periodic_cubical_complex_barcode_persistence_from_perseus_file_example.py>` + * :download:`rips_complex_diagram_persistence_from_off_file_example.py <../example/rips_complex_diagram_persistence_from_off_file_example.py>` + * :download:`rips_persistence_diagram.py <../example/rips_persistence_diagram.py>` + * :download:`rips_complex_diagram_persistence_from_distance_matrix_file_example.py <../example/rips_complex_diagram_persistence_from_distance_matrix_file_example.py>` + * :download:`tangential_complex_plain_homology_from_off_file_example.py <../example/tangential_complex_plain_homology_from_off_file_example.py>` + * :download:`euclidean_strong_witness_complex_diagram_persistence_from_off_file_example.py <../example/euclidean_strong_witness_complex_diagram_persistence_from_off_file_example.py>` + * :download:`euclidean_witness_complex_diagram_persistence_from_off_file_example.py <../example/euclidean_witness_complex_diagram_persistence_from_off_file_example.py>` + +Threading Building Blocks +========================= + +`Intel® TBB <https://www.threadingbuildingblocks.org/>`_ lets you easily write +parallel C++ programs that take full advantage of multicore performance, that +are portable and composable, and that have future-proof scalability. + +Having Intel® TBB installed is recommended to parallelize and accelerate some +GUDHI computations. + +Bug reports and contributions +***************************** + +Please help us improving the quality of the GUDHI library. You may report bugs or suggestions to: + + Contact: gudhi-users@lists.gforge.inria.fr + +GUDHI is open to external contributions. If you want to join our development team, please contact us. diff --git a/cython/doc/make.bat.in b/cython/doc/make.bat.in new file mode 100644 index 00000000..ff1a6d56 --- /dev/null +++ b/cython/doc/make.bat.in @@ -0,0 +1,67 @@ +@ECHO OFF
+
+REM Command file for Sphinx documentation
+
+if "%SPHINXBUILD%" == "" (
+ set SPHINXBUILD=@SPHINX_PATH@
+)
+set BUILDDIR=_build
+set ALLSPHINXOPTS=-d %BUILDDIR%/doctrees %SPHINXOPTS% .
+set I18NSPHINXOPTS=%SPHINXOPTS% .
+if NOT "%PAPER%" == "" (
+ set ALLSPHINXOPTS=-D latex_paper_size=%PAPER% %ALLSPHINXOPTS%
+ set I18NSPHINXOPTS=-D latex_paper_size=%PAPER% %I18NSPHINXOPTS%
+)
+
+if "%1" == "" goto help
+
+if "%1" == "help" (
+ :help
+ echo.Please use `make ^<target^>` where ^<target^> is one of
+ echo. html to make standalone HTML files
+ echo. doctest to run all doctests embedded in the documentation if enabled
+ goto end
+)
+
+if "%1" == "clean" (
+ del examples.inc
+ for /d %%i in (%BUILDDIR%\*) do rmdir /q /s %%i
+ del /q /s %BUILDDIR%\*
+ goto end
+)
+
+
+%SPHINXBUILD% 2> nul
+if errorlevel 9009 (
+ echo.
+ echo.The 'sphinx-build' command was not found. Make sure you have Sphinx
+ echo.installed, then set the SPHINXBUILD environment variable to point
+ echo.to the full path of the 'sphinx-build' executable. Alternatively you
+ echo.may add the Sphinx directory to PATH.
+ echo.
+ echo.If you don't have Sphinx installed, grab it from
+ echo.http://sphinx-doc.org/
+ exit /b 1
+)
+
+:: GUDHI specific : Examples.inc is generated with generate_examples.py (and deleted on clean)
+
+if "%1" == "html" (
+ generate_examples.py
+ %SPHINXBUILD% -b html %ALLSPHINXOPTS% %BUILDDIR%/html
+ if errorlevel 1 exit /b 1
+ echo.
+ echo.Build finished. The HTML pages are in %BUILDDIR%/html.
+ goto end
+)
+
+if "%1" == "doctest" (
+ %SPHINXBUILD% -b doctest %ALLSPHINXOPTS% %BUILDDIR%/doctest
+ if errorlevel 1 exit /b 1
+ echo.
+ echo.Testing of doctests in the sources finished, look at the ^
+results in %BUILDDIR%/doctest/output.txt.
+ goto end
+)
+
+:end
diff --git a/cython/doc/periodic_cubical_complex_ref.rst b/cython/doc/periodic_cubical_complex_ref.rst new file mode 100644 index 00000000..c6190a1b --- /dev/null +++ b/cython/doc/periodic_cubical_complex_ref.rst @@ -0,0 +1,9 @@ +Periodic cubical complex reference manual +######################################### + +.. autoclass:: gudhi.PeriodicCubicalComplex + :members: + :undoc-members: + :show-inheritance: + + .. automethod:: gudhi.PeriodicCubicalComplex.__init__ diff --git a/cython/doc/persistence_graphical_tools_ref.rst b/cython/doc/persistence_graphical_tools_ref.rst new file mode 100644 index 00000000..27c2f68a --- /dev/null +++ b/cython/doc/persistence_graphical_tools_ref.rst @@ -0,0 +1,8 @@ +============================================ +Persistence graphical tools reference manual +============================================ + +.. autofunction:: gudhi.__min_birth_max_death +.. autofunction:: gudhi.show_palette_values +.. autofunction:: gudhi.plot_persistence_barcode +.. autofunction:: gudhi.plot_persistence_diagram diff --git a/cython/doc/persistence_graphical_tools_sum.rst b/cython/doc/persistence_graphical_tools_sum.rst new file mode 100644 index 00000000..d602daa7 --- /dev/null +++ b/cython/doc/persistence_graphical_tools_sum.rst @@ -0,0 +1,12 @@ +================================================================= =================================== =================================== +:Author: Vincent Rouvreau :Introduced in: GUDHI 2.0.0 :Copyright: GPL v3 +:Requires: Matplotlib Numpy +================================================================= =================================== =================================== + ++-----------------------------------------------------------------+-----------------------------------------------------------------------+ +| .. figure:: | These graphical tools comes on top of persistence results and allows | +| img/graphical_tools_representation.png | the user to build easily barcode and persistence diagram. | +| | | ++-----------------------------------------------------------------+-----------------------------------------------------------------------+ +| :doc:`persistence_graphical_tools_user` | :doc:`persistence_graphical_tools_ref` | ++-----------------------------------------------------------------+-----------------------------------------------------------------------+ diff --git a/cython/doc/persistence_graphical_tools_user.rst b/cython/doc/persistence_graphical_tools_user.rst new file mode 100644 index 00000000..cae18323 --- /dev/null +++ b/cython/doc/persistence_graphical_tools_user.rst @@ -0,0 +1,66 @@ +Persistence graphical tools user manual +======================================= +Definition +---------- +.. include:: persistence_graphical_tools_sum.rst + + +Show palette values +------------------- + +This function is useful to show the color palette values of dimension: + + +.. testcode:: + + import gudhi + gudhi.show_palette_values(alpha=1.0) + +.. plot:: + + import gudhi + gudhi.show_palette_values(alpha=1.0) + +Show persistence as a barcode +----------------------------- + +This function can display the persistence result as a barcode: + +.. testcode:: + + import gudhi + + periodic_cc = gudhi.PeriodicCubicalComplex(perseus_file='3d_torus.txt') + diag = periodic_cc.persistence() + gudhi.plot_persistence_barcode(diag) + +.. plot:: + + import gudhi + + periodic_cc = gudhi.PeriodicCubicalComplex(perseus_file='3d_torus.txt') + diag = periodic_cc.persistence() + gudhi.plot_persistence_barcode(diag) + +Show persistence as a diagram +----------------------------- + +This function can display the persistence result as a diagram: + +.. testcode:: + + import gudhi + + rips_complex = gudhi.RipsComplex(off_file='tore3D_300.off', max_edge_length=2.0) + simplex_tree = rips_complex.create_simplex_tree(max_dimension=3) + diag = simplex_tree.persistence() + gudhi.plot_persistence_diagram(diag) + +.. plot:: + + import gudhi + + rips_complex = gudhi.RipsComplex(off_file='tore3D_300.off', max_edge_length=2.0) + simplex_tree = rips_complex.create_simplex_tree(max_dimension=3) + diag = simplex_tree.persistence() + gudhi.plot_persistence_diagram(diag) diff --git a/cython/doc/persistent_cohomology_sum.rst b/cython/doc/persistent_cohomology_sum.rst new file mode 100644 index 00000000..d1f79cb4 --- /dev/null +++ b/cython/doc/persistent_cohomology_sum.rst @@ -0,0 +1,27 @@ +================================================================= =================================== =================================== +:Author: Clément Maria :Introduced in: GUDHI 2.0.0 :Copyright: GPL v3 +================================================================= =================================== =================================== + ++-----------------------------------------------------------------+-----------------------------------------------------------------------+ +| .. figure:: | The theory of homology consists in attaching to a topological space | +| img/3DTorus_poch.png | a sequence of (homology) groups, capturing global topological | +| :figclass: align-center | features like connected components, holes, cavities, etc. Persistent | +| | homology studies the evolution -- birth, life and death -- of these | +| Rips Persistent Cohomology on a 3D | features when the topological space is changing. Consequently, the | +| Torus | theory is essentially composed of three elements: topological spaces, | +| | their homology groups and an evolution scheme. | +| | | +| | Computation of persistent cohomology using the algorithm of | +| | :cite:`DBLP:journals/dcg/SilvaMV11` and | +| | :cite:`DBLP:journals/corr/abs-1208-5018` and the Compressed | +| | Annotation Matrix implementation of | +| | :cite:`DBLP:conf/esa/BoissonnatDM13`. | +| | | ++-----------------------------------------------------------------+-----------------------------------------------------------------------+ +| :doc:`persistent_cohomology_user` | Please refer to each data structure that contains persistence | +| | feature for reference: | +| | | +| | * :doc:`simplex_tree_ref` | +| | * :doc:`cubical_complex_ref` | +| | * :doc:`periodic_cubical_complex_ref` | ++-----------------------------------------------------------------+-----------------------------------------------------------------------+ diff --git a/cython/doc/persistent_cohomology_user.rst b/cython/doc/persistent_cohomology_user.rst new file mode 100644 index 00000000..72f1a7f7 --- /dev/null +++ b/cython/doc/persistent_cohomology_user.rst @@ -0,0 +1,114 @@ +Persistent cohomology user manual +================================= +Definition +---------- +===================================== ===================================== ===================================== +:Author: Clément Maria :Introduced in: GUDHI PYTHON 2.0.0 :Copyright: GPL v3 +===================================== ===================================== ===================================== + ++---------------------------------------------+----------------------------------------------------------------------+ +| :doc:`persistent_cohomology_user` | Please refer to each data structure that contains persistence | +| | feature for reference: | +| | | +| | * :doc:`simplex_tree_ref` | ++---------------------------------------------+----------------------------------------------------------------------+ + + +Computation of persistent cohomology using the algorithm of :cite:`DBLP:journals/dcg/SilvaMV11` and +:cite:`DBLP:journals/corr/abs-1208-5018` and the Compressed Annotation Matrix implementation of +:cite:`DBLP:conf/esa/BoissonnatDM13`. + +The theory of homology consists in attaching to a topological space a sequence of (homology) groups, capturing global +topological features like connected components, holes, cavities, etc. Persistent homology studies the evolution -- +birth, life and death -- of these features when the topological space is changing. Consequently, the theory is +essentially composed of three elements: + +* topological spaces +* their homology groups +* an evolution scheme. + +Topological Spaces +------------------ + +Topological spaces are represented by simplicial complexes. +Let :math:`V = \{1, \cdots ,|V|\}` be a set of *vertices*. +A *simplex* :math:`\sigma` is a subset of vertices :math:`\sigma \subseteq V`. +A *simplicial complex* :math:`\mathbf{K}` on :math:`V` is a collection of simplices :math:`\{\sigma\}`, +:math:`\sigma \subseteq V`, such that :math:`\tau \subseteq \sigma \in \mathbf{K} \Rightarrow \tau \in \mathbf{K}`. +The dimension :math:`n=|\sigma|-1` of :math:`\sigma` is its number of elements minus 1. +A *filtration* of a simplicial complex is a function :math:`f:\mathbf{K} \rightarrow \mathbb{R}` satisfying +:math:`f(\tau)\leq f(\sigma)` whenever :math:`\tau \subseteq \sigma`. + +Homology +-------- + +For a ring :math:`\mathcal{R}`, the group of *n-chains*, denoted :math:`\mathbf{C}_n(\mathbf{K},\mathcal{R})`, of +:math:`\mathbf{K}` is the group of formal sums of n-simplices with :math:`\mathcal{R}` coefficients. The +*boundary operator* is a linear operator +:math:`\partial_n: \mathbf{C}_n(\mathbf{K},\mathcal{R}) \rightarrow \mathbf{C}_{n-1}(\mathbf{K},\mathcal{R})` +such that :math:`\partial_n \sigma = \partial_n [v_0, \cdots , v_n] = \sum_{i=0}^n (-1)^{i}[v_0,\cdots ,\widehat{v_i}, \cdots,v_n]`, +where :math:`\widehat{v_i}` means :math:`v_i` is omitted from the list. The chain groups form a sequence: + +.. math:: + + \cdots \ \ \mathbf{C}_n(\mathbf{K},\mathcal{R}) \xrightarrow{\ \partial_n\ } + \mathbf{C}_{n-1}(\mathbf{K},\mathcal{R}) \xrightarrow{\partial_{n-1}} \cdots \xrightarrow{\ \partial_2 \ } + \mathbf{C}_1(\mathbf{K},\mathcal{R}) \xrightarrow{\ \partial_1 \ } \mathbf{C}_0(\mathbf{K},\mathcal{R}) + +of finitely many groups :math:`\mathbf{C}_n(\mathbf{K},\mathcal{R})` and homomorphisms :math:`\partial_n`, indexed by +the dimension :math:`n \geq 0`. The boundary operators satisfy the property :math:`\partial_n \circ \partial_{n+1}=0` +for every :math:`n > 0` and we define the homology groups: + +.. math:: + + \mathbf{H}_n(\mathbf{K},\mathcal{R}) = \ker \partial_n / \mathrm{im} \ \partial_{n+1} + +We refer to :cite:`Munkres-elementsalgtop1984` for an introduction to homology +theory and to :cite:`DBLP:books/daglib/0025666` for an introduction to persistent homology. + +Indexing Scheme +--------------- + +"Changing" a simplicial complex consists in applying a simplicial map. An *indexing scheme* is a directed graph +together with a traversal order, such that two consecutive nodes in the graph are connected by an arrow (either forward +or backward). +The nodes represent simplicial complexes and the directed edges simplicial maps. + +From the computational point of view, there are two types of indexing schemes of interest in persistent homology: + +* linear ones + :math:`\bullet \longrightarrow \bullet \longrightarrow \cdots \longrightarrow \bullet \longrightarrow \bullet` + in persistent homology :cite:`DBLP:journals/dcg/ZomorodianC05`, +* zigzag ones + :math:`\bullet \longrightarrow \bullet \longleftarrow \cdots \longrightarrow \bullet \longleftarrow \bullet` + in zigzag persistent homology :cite:`DBLP:journals/focm/CarlssonS10`. + +These indexing schemes have a natural left-to-right traversal order, and we describe them with ranges and iterators. +In the current release of the Gudhi library, only the linear case is implemented. + +In the following, we consider the case where the indexing scheme is induced by a filtration. + +Ordering the simplices by increasing filtration values (breaking ties so as a simplex appears after its subsimplices of +same filtration value) provides an indexing scheme. + +Examples +-------- + +We provide several example files: run these examples with -h for details on their use. + +.. only:: builder_html + + * :download:`alpha_complex_diagram_persistence_from_off_file_example.py <../example/alpha_complex_diagram_persistence_from_off_file_example.py>` + * :download:`periodic_cubical_complex_barcode_persistence_from_perseus_file_example.py <../example/periodic_cubical_complex_barcode_persistence_from_perseus_file_example.py>` + * :download:`rips_complex_diagram_persistence_from_off_file_example.py <../example/rips_complex_diagram_persistence_from_off_file_example.py>` + * :download:`rips_persistence_diagram.py <../example/rips_persistence_diagram.py>` + * :download:`rips_complex_diagram_persistence_from_distance_matrix_file_example.py <../example/rips_complex_diagram_persistence_from_distance_matrix_file_example.py>` + * :download:`random_cubical_complex_persistence_example.py <../example/random_cubical_complex_persistence_example.py>` + * :download:`tangential_complex_plain_homology_from_off_file_example.py <../example/tangential_complex_plain_homology_from_off_file_example.py>` + +Bibliography +============ + +.. bibliography:: bibliography.bib + :filter: docnames + :style: unsrt diff --git a/cython/doc/pyplots/barcode_persistence.py b/cython/doc/pyplots/barcode_persistence.py new file mode 100755 index 00000000..c06ac5a7 --- /dev/null +++ b/cython/doc/pyplots/barcode_persistence.py @@ -0,0 +1,5 @@ +import gudhi + +periodic_cc = gudhi.PeriodicCubicalComplex(perseus_file='../3d_torus.txt') +diag = periodic_cc.persistence() +gudhi.plot_persistence_barcode(diag) diff --git a/cython/doc/pyplots/diagram_persistence.py b/cython/doc/pyplots/diagram_persistence.py new file mode 100755 index 00000000..b4714fe3 --- /dev/null +++ b/cython/doc/pyplots/diagram_persistence.py @@ -0,0 +1,5 @@ +import gudhi + +alpha_complex = gudhi.AlphaComplex(off_file='../tore3D_300.off') +diag = alpha_complex.persistence() +gudhi.plot_persistence_diagram(diag) diff --git a/cython/doc/pyplots/show_palette_values.py b/cython/doc/pyplots/show_palette_values.py new file mode 100755 index 00000000..e72a55fd --- /dev/null +++ b/cython/doc/pyplots/show_palette_values.py @@ -0,0 +1,2 @@ +import gudhi +gudhi.show_palette_values(alpha=1.0) diff --git a/cython/doc/python3-sphinx-build b/cython/doc/python3-sphinx-build new file mode 100755 index 00000000..44b94169 --- /dev/null +++ b/cython/doc/python3-sphinx-build @@ -0,0 +1,11 @@ +#!/usr/bin/python3 + +""" +Emulate sphinx-build for python3 +""" + +from sys import exit, argv +from sphinx import main + +if __name__ == '__main__': + exit(main(argv)) diff --git a/cython/doc/rips_complex_ref.rst b/cython/doc/rips_complex_ref.rst new file mode 100644 index 00000000..b17dc4e0 --- /dev/null +++ b/cython/doc/rips_complex_ref.rst @@ -0,0 +1,10 @@ +============================= +Rips complex reference manual +============================= + +.. autoclass:: gudhi.RipsComplex + :members: + :undoc-members: + :show-inheritance: + + .. automethod:: gudhi.RipsComplex.__init__ diff --git a/cython/doc/rips_complex_sum.rst b/cython/doc/rips_complex_sum.rst new file mode 100644 index 00000000..2b65fc19 --- /dev/null +++ b/cython/doc/rips_complex_sum.rst @@ -0,0 +1,17 @@ +================================================================= =================================== =================================== +:Author: Clément Maria, Pawel Dlotko, Vincent Rouvreau :Introduced in: GUDHI 2.0.0 :Copyright: GPL v3 +================================================================= =================================== =================================== + ++----------------------------------------------------------------+------------------------------------------------------------------------+ +| .. figure:: | Rips complex is a simplicial complex constructed from a one skeleton | +| img/rips_complex_representation.png | graph. | +| :figclass: align-center | | +| | The filtration value of each edge is computed from a user-given | +| Rips complex representation | distance function and is inserted until a user-given threshold | +| | value. | +| | | +| | This complex can be built from a point cloud and a distance function, | +| | or from a distance matrix. | ++----------------------------------------------------------------+------------------------------------------------------------------------+ +| :doc:`rips_complex_user` | :doc:`rips_complex_ref` | ++----------------------------------------------------------------+------------------------------------------------------------------------+ diff --git a/cython/doc/rips_complex_user.rst b/cython/doc/rips_complex_user.rst new file mode 100644 index 00000000..f9760976 --- /dev/null +++ b/cython/doc/rips_complex_user.rst @@ -0,0 +1,240 @@ +Rips complex user manual +========================= +Definition +---------- + +======================================================= ===================================== ===================================== +:Authors: Clément Maria, Pawel Dlotko, Vincent Rouvreau :Introduced in: GUDHI 2.0.0 :Copyright: GPL v3 +======================================================= ===================================== ===================================== + ++-------------------------------------------+----------------------------------------------------------------------+ +| :doc:`rips_complex_user` | :doc:`rips_complex_ref` | ++-------------------------------------------+----------------------------------------------------------------------+ + +`Rips complex <https://en.wikipedia.org/wiki/Vietoris%E2%80%93Rips_complex>`_ is a one skeleton graph that allows to +construct a simplicial complex from it. The input can be a point cloud with a given distance function, or a distance +matrix. + +The filtration value of each edge is computed from a user-given distance function, or directly from the distance +matrix. + +All edges that have a filtration value strictly greater than a given threshold value are not inserted into the complex. + +When creating a simplicial complex from this one skeleton graph, Rips inserts the one skeleton graph into the data +structure, and then expands the simplicial complex when required. + +Vertex name correspond to the index of the point in the given range (aka. the point cloud). + +.. figure:: + img/rips_complex_representation.png + :align: center + + Rips-complex one skeleton graph representation + +On this example, as edges (4,5), (4,6) and (5,6) are in the complex, simplex (4,5,6) is added with the filtration value +set with :math:`max(filtration(4,5), filtration(4,6), filtration(5,6))`. And so on for simplex (0,1,2,3). + +If the Rips_complex interfaces are not detailed enough for your need, please refer to rips_persistence_step_by_step.cpp +example, where the graph construction over the Simplex_tree is more detailed. + +Point cloud +----------- + +Example from a point cloud +^^^^^^^^^^^^^^^^^^^^^^^^^^ + +This example builds the one skeleton graph from the given points, and max_edge_length value. +Then it creates a :doc:`Simplex_tree <simplex_tree_ref>` with it. + +Finally, it is asked to display information about the simplicial complex. + +.. testcode:: + + import gudhi + rips_complex = gudhi.RipsComplex(points=[[1, 1], [7, 0], [4, 6], [9, 6], [0, 14], [2, 19], [9, 17]], + max_edge_length=12.0) + + simplex_tree = rips_complex.create_simplex_tree(max_dimension=1) + result_str = 'Rips 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)) + +When launching (Rips maximal distance between 2 points is 12.0, is expanded +until dimension 1 - one skeleton graph in other words), the output is: + +.. testoutput:: + + Rips complex is of dimension 1 - 18 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] -> 5.00 + [4, 5] -> 5.39 + [0, 2] -> 5.83 + [0, 1] -> 6.08 + [1, 3] -> 6.32 + [1, 2] -> 6.71 + [5, 6] -> 7.28 + [2, 4] -> 8.94 + [0, 3] -> 9.43 + [4, 6] -> 9.49 + [3, 6] -> 11.00 + +Example from OFF file +^^^^^^^^^^^^^^^^^^^^^ + +This example builds the :doc:`Rips_complex <rips_complex_ref>` from the given +points in an OFF file, and max_edge_length value. +Then it creates a :doc:`Simplex_tree <simplex_tree_ref>` with it. + +Finally, it is asked to display information about the Rips complex. + + +.. testcode:: + + import gudhi + rips_complex = gudhi.RipsComplex(off_file='alphacomplexdoc.off', max_edge_length=12.0) + simplex_tree = rips_complex.create_simplex_tree(max_dimension=1) + result_str = 'Rips 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:: + + Rips complex is of dimension 1 - 18 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] -> 5.00 + [4, 5] -> 5.39 + [0, 2] -> 5.83 + [0, 1] -> 6.08 + [1, 3] -> 6.32 + [1, 2] -> 6.71 + [5, 6] -> 7.28 + [2, 4] -> 8.94 + [0, 3] -> 9.43 + [4, 6] -> 9.49 + [3, 6] -> 11.00 + +Distance matrix +--------------- + +Example from a distance matrix +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +This example builds the one skeleton graph from the given distance matrix, and max_edge_length value. +Then it creates a :doc:`Simplex_tree <simplex_tree_ref>` with it. + +Finally, it is asked to display information about the simplicial complex. + +.. testcode:: + + import gudhi + rips_complex = gudhi.RipsComplex(distance_matrix=[[], + [6.0827625303], + [5.8309518948, 6.7082039325], + [9.4339811321, 6.3245553203, 5], + [13.0384048104, 15.6524758425, 8.94427191, 12.0415945788], + [18.0277563773, 19.6468827044, 13.152946438, 14.7648230602, 5.3851648071], + [17.88854382, 17.1172427686, 12.0830459736, 11, 9.4868329805, 7.2801098893]], + max_edge_length=12.0) + + simplex_tree = rips_complex.create_simplex_tree(max_dimension=1) + result_str = 'Rips 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)) + +When launching (Rips maximal distance between 2 points is 12.0, is expanded +until dimension 1 - one skeleton graph in other words), the output is: + +.. testoutput:: + + Rips complex is of dimension 1 - 18 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] -> 5.00 + [4, 5] -> 5.39 + [0, 2] -> 5.83 + [0, 1] -> 6.08 + [1, 3] -> 6.32 + [1, 2] -> 6.71 + [5, 6] -> 7.28 + [2, 4] -> 8.94 + [0, 3] -> 9.43 + [4, 6] -> 9.49 + [3, 6] -> 11.00 + +Example from csv file +^^^^^^^^^^^^^^^^^^^^^ + +This example builds the :doc:`Rips_complex <rips_complex_ref>` from the given +points in an OFF file, and max_edge_length value. +Then it creates a :doc:`Simplex_tree <simplex_tree_ref>` with it. + +Finally, it is asked to display information about the Rips complex. + + +.. testcode:: + + import gudhi + rips_complex = gudhi.RipsComplex(csv_file='full_square_distance_matrix.csv', max_edge_length=12.0) + simplex_tree = rips_complex.create_simplex_tree(max_dimension=1) + result_str = 'Rips 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:: + + Rips complex is of dimension 1 - 18 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] -> 5.00 + [4, 5] -> 5.39 + [0, 2] -> 5.83 + [0, 1] -> 6.08 + [1, 3] -> 6.32 + [1, 2] -> 6.71 + [5, 6] -> 7.28 + [2, 4] -> 8.94 + [0, 3] -> 9.43 + [4, 6] -> 9.49 + [3, 6] -> 11.00 diff --git a/cython/doc/simplex_tree_ref.rst b/cython/doc/simplex_tree_ref.rst new file mode 100644 index 00000000..6d196843 --- /dev/null +++ b/cython/doc/simplex_tree_ref.rst @@ -0,0 +1,10 @@ +============================= +Simplex tree reference manual +============================= + +.. autoclass:: gudhi.SimplexTree + :members: + :undoc-members: + :show-inheritance: + + .. automethod:: gudhi.SimplexTree.__init__ diff --git a/cython/doc/simplex_tree_sum.rst b/cython/doc/simplex_tree_sum.rst new file mode 100644 index 00000000..3174fb62 --- /dev/null +++ b/cython/doc/simplex_tree_sum.rst @@ -0,0 +1,14 @@ +================================================================= =================================== =================================== +:Author: Clément Maria :Introduced in: GUDHI 2.0.0 :Copyright: GPL v3 +================================================================= =================================== =================================== + ++----------------------------------------------------------------+------------------------------------------------------------------------+ +| .. figure:: | The simplex tree is an efficient and flexible data structure for | +| img/Simplex_tree_representation.png | representing general (filtered) simplicial complexes. | +| :alt: Simplex tree representation | | +| :figclass: align-center | The data structure is described in | +| | :cite:`boissonnatmariasimplextreealgorithmica` | +| Simplex tree representation | | ++----------------------------------------------------------------+------------------------------------------------------------------------+ +| :doc:`simplex_tree_user` | :doc:`simplex_tree_ref` | ++----------------------------------------------------------------+------------------------------------------------------------------------+ diff --git a/cython/doc/simplex_tree_user.rst b/cython/doc/simplex_tree_user.rst new file mode 100644 index 00000000..4b1dde19 --- /dev/null +++ b/cython/doc/simplex_tree_user.rst @@ -0,0 +1,68 @@ +Simplex tree user manual +======================== +Definition +---------- + +.. include:: simplex_tree_sum.rst + +A simplicial complex :math:`\mathbf{K}` on a set of vertices :math:`V = \{1, \cdots ,|V|\}` is a collection of +simplices :math:`\{\sigma\}`, :math:`\sigma \subseteq V` such that +:math:`\tau \subseteq \sigma \in \mathbf{K} \rightarrow \tau \in \mathbf{K}`. The dimension :math:`n=|\sigma|-1` of +:math:`\sigma` is its number of elements minus `1`. + +A filtration of a simplicial complex is a function :math:`f:\mathbf{K} \rightarrow \mathbb{R}` satisfying +:math:`f(\tau)\leq f(\sigma)` whenever :math:`\tau \subseteq \sigma`. Ordering the simplices by increasing filtration +values (breaking ties so as a simplex appears after its subsimplices of same filtration value) provides an indexing +scheme. + + +Implementation +-------------- + +There are two implementation of complexes. The first on is the Simplex_tree data structure. +The simplex tree is an efficient and flexible data structure for representing general (filtered) simplicial complexes. +The data structure is described in :cite`boissonnatmariasimplextreealgorithmica`. + +The second one is the Hasse_complex. The Hasse complex is a data structure representing explicitly all co-dimension 1 +incidence relations in a complex. It is consequently faster when accessing the boundary of a simplex, but is less +compact and harder to construct from scratch. + +Example +------- + +.. testcode:: + + import gudhi + st = gudhi.SimplexTree() + if st.insert([0, 1]): + print("[0, 1] inserted") + if st.insert([0, 1, 2], filtration=4.0): + print("[0, 1, 2] inserted") + if st.find([0, 1]): + print("[0, 1] found") + result_str = 'num_vertices=' + repr(st.num_vertices()) + print(result_str) + result_str = 'num_simplices=' + repr(st.num_simplices()) + print(result_str) + print("skeleton(2) =") + for sk_value in st.get_skeleton(2): + print(sk_value) + + +The output is: + +.. testoutput:: + + [0, 1] inserted + [0, 1, 2] inserted + [0, 1] found + num_vertices=3 + num_simplices=7 + skeleton(2) = + ([0, 1, 2], 4.0) + ([0, 1], 0.0) + ([0, 2], 4.0) + ([0], 0.0) + ([1, 2], 4.0) + ([1], 0.0) + ([2], 4.0) diff --git a/cython/doc/strong_witness_complex_ref.rst b/cython/doc/strong_witness_complex_ref.rst new file mode 100644 index 00000000..4ed4fe46 --- /dev/null +++ b/cython/doc/strong_witness_complex_ref.rst @@ -0,0 +1,10 @@ +======================================= +Strong witness complex reference manual +======================================= + +.. autoclass:: gudhi.StrongWitnessComplex + :members: + :undoc-members: + :show-inheritance: + + .. automethod:: gudhi.StrongWitnessComplex.__init__ diff --git a/cython/doc/tangential_complex_ref.rst b/cython/doc/tangential_complex_ref.rst new file mode 100644 index 00000000..35589475 --- /dev/null +++ b/cython/doc/tangential_complex_ref.rst @@ -0,0 +1,10 @@ +=================================== +Tangential complex reference manual +=================================== + +.. autoclass:: gudhi.TangentialComplex + :members: + :undoc-members: + :show-inheritance: + + .. automethod:: gudhi.TangentialComplex.__init__ diff --git a/cython/doc/tangential_complex_sum.rst b/cython/doc/tangential_complex_sum.rst new file mode 100644 index 00000000..2b05bc10 --- /dev/null +++ b/cython/doc/tangential_complex_sum.rst @@ -0,0 +1,15 @@ +================================================================= =================================== =================================== +:Author: Clément Jamin :Introduced in: GUDHI 2.0.0 :Copyright: GPL v3 +:Requires: CGAL :math:`\geq` 4.8.0 Eigen3 +================================================================= =================================== =================================== + ++----------------------------------------------------------------+------------------------------------------------------------------------+ +| .. figure:: | A Tangential Delaunay complex is a simplicial complex designed to | +| img/tc_examples.png | reconstruct a :math:`k`-dimensional manifold embedded in :math:`d`- | +| :figclass: align-center | dimensional Euclidean space. The input is a point sample coming from | +| | an unknown manifold. The running time depends only linearly on the | +| **Tangential complex representation** | extrinsic dimension :math:`d` and exponentially on the intrinsic | +| | dimension :math:`k`. | ++----------------------------------------------------------------+------------------------------------------------------------------------+ +| :doc:`tangential_complex_user` | :doc:`tangential_complex_ref` | ++----------------------------------------------------------------+------------------------------------------------------------------------+ diff --git a/cython/doc/tangential_complex_user.rst b/cython/doc/tangential_complex_user.rst new file mode 100644 index 00000000..03f9fea6 --- /dev/null +++ b/cython/doc/tangential_complex_user.rst @@ -0,0 +1,195 @@ +Tangential complex user manual +============================== +.. include:: tangential_complex_sum.rst + +Definition +---------- + +A Tangential Delaunay complex is a simplicial complex designed to reconstruct a +:math:`k`-dimensional smooth manifold embedded in :math:`d`-dimensional +Euclidean space. The input is a point sample coming from an unknown manifold, +which means that the points lie close to a structure of "small" intrinsic +dimension. The running time depends only linearly on the extrinsic dimension +:math:`d` and exponentially on the intrinsic dimension :math:`k`. + +An extensive description of the Tangential complex can be found in +:cite:`tangentialcomplex2014`. + +What is a Tangential Complex? +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +Let us start with the description of the Tangential complex of a simple +example, with :math:`k = 1` and :math:`d = 2`. The input data is 4 points +:math:`P` located on a curve embedded in 2D. + +.. figure:: img/tc_example_01.png + :alt: The input + :figclass: align-center + + The input + +For each point :math:`p`, estimate its tangent subspace :math:`T_p` (e.g. +using PCA). + +.. figure:: img/tc_example_02.png + :alt: The estimated normals + :figclass: align-center + + The estimated normals + + +Let us add the Voronoi diagram of the points in orange. For each point +:math:`p`, construct its star in the Delaunay triangulation of :math:`P` +restricted to :math:`T_p`. + +.. figure:: img/tc_example_03.png + :alt: The Voronoi diagram + :figclass: align-center + + The Voronoi diagram + +The Tangential Delaunay complex is the union of those stars. + +In practice, neither the ambient Voronoi diagram nor the ambient Delaunay +triangulation is computed. Instead, local :math:`k`-dimensional regular +triangulations are computed with a limited number of points as we only need the +star of each point. More details can be found in :cite:`tangentialcomplex2014`. + +Inconsistencies +^^^^^^^^^^^^^^^ +Inconsistencies between the stars can occur. An inconsistency occurs when a +simplex is not in the star of all its vertices. + +Let us take the same example. + +.. figure:: img/tc_example_07_before.png + :alt: Before + :figclass: align-center + + Before + +Let us slightly move the tangent subspace :math:`T_q` + +.. figure:: img/tc_example_07_after.png + :alt: After + :figclass: align-center + + After + +Now, the star of :math:`Q` contains :math:`QP`, but the star of :math:`P` does +not contain :math:`QP`. We have an inconsistency. + +.. figure:: img/tc_example_08.png + :alt: After + :figclass: align-center + + After + +One way to solve inconsistencies is to randomly perturb the positions of the +points involved in an inconsistency. In the current implementation, this +perturbation is done in the tangent subspace of each point. The maximum +perturbation radius is given as a parameter to the constructor. + +In most cases, we recommend to provide a point set where the minimum distance +between any two points is not too small. This can be achieved using the +functions provided by the Subsampling module. Then, a good value to start with +for the maximum perturbation radius would be around half the minimum distance +between any two points. The Example with perturbation below shows an example of +such a process. + +In most cases, this process is able to dramatically reduce the number of +inconsistencies, but is not guaranteed to succeed. + +Output +^^^^^^ +The result of the computation is exported as a Simplex_tree. It is the union of +the stars of all the input points. A vertex in the Simplex Tree is the index of +the point in the range provided by the user. The point corresponding to a +vertex can also be obtained through the Tangential_complex::get_point function. +Note that even if the positions of the points are perturbed, their original +positions are kept (e.g. Tangential_complex::get_point returns the original +position of the point). + +The result can be obtained after the computation of the Tangential complex +itself and/or after the perturbation process. + + +Simple example +-------------- + +This example builds the Tangential complex of point set read in an OFF file. + +.. testcode:: + + import gudhi + tc = gudhi.TangentialComplex(off_file='alphacomplexdoc.off') + result_str = 'Tangential contains ' + repr(tc.num_simplices()) + \ + ' simplices - ' + repr(tc.num_vertices()) + ' vertices.' + print(result_str) + + st = tc.create_simplex_tree() + result_str = 'Simplex tree is of dimension ' + repr(st.dimension()) + \ + ' - ' + repr(st.num_simplices()) + ' simplices - ' + \ + repr(st.num_vertices()) + ' vertices.' + print(result_str) + for filtered_value in st.get_filtration(): + print(filtered_value[0]) + +The output is: + +.. testoutput:: + + Tangential contains 12 simplices - 7 vertices. + Simplex tree is of dimension 1 - 15 simplices - 7 vertices. + [0] + [1] + [0, 1] + [2] + [0, 2] + [1, 2] + [3] + [1, 3] + [4] + [2, 4] + [5] + [4, 5] + [6] + [3, 6] + [5, 6] + + +Example with perturbation +------------------------- + +This example builds the Tangential complex of a point set, then tries to solve +inconsistencies by perturbing the positions of points involved in inconsistent +simplices. + +.. testcode:: + + import gudhi + tc = gudhi.TangentialComplex(points=[[0.0, 0.0], [1.0, 0.0], [0.0, 1.0], [1.0, 1.0]]) + result_str = 'Tangential contains ' + repr(tc.num_vertices()) + ' vertices.' + print(result_str) + + if tc.num_inconsistent_simplices() > 0: + print('Tangential contains inconsistencies.') + + tc.fix_inconsistencies_using_perturbation(10, 60) + if tc.num_inconsistent_simplices() == 0: + print('Inconsistencies has been fixed.') + +The output is: + +.. testoutput:: + + Tangential contains 4 vertices. + Inconsistencies has been fixed. + + +Bibliography +============ + +.. bibliography:: bibliography.bib + :filter: docnames + :style: unsrt diff --git a/cython/doc/todos.rst b/cython/doc/todos.rst new file mode 100644 index 00000000..78972a4c --- /dev/null +++ b/cython/doc/todos.rst @@ -0,0 +1,5 @@ +========== +To be done +========== + +.. todolist:: diff --git a/cython/doc/witness_complex_ref.rst b/cython/doc/witness_complex_ref.rst new file mode 100644 index 00000000..c78760cb --- /dev/null +++ b/cython/doc/witness_complex_ref.rst @@ -0,0 +1,10 @@ +================================ +Witness complex reference manual +================================ + +.. autoclass:: gudhi.WitnessComplex + :members: + :undoc-members: + :show-inheritance: + + .. automethod:: gudhi.WitnessComplex.__init__ diff --git a/cython/doc/witness_complex_sum.rst b/cython/doc/witness_complex_sum.rst new file mode 100644 index 00000000..b65522ba --- /dev/null +++ b/cython/doc/witness_complex_sum.rst @@ -0,0 +1,17 @@ +================================================================= =================================== =================================== +:Author: Siargey Kachanovich :Introduced in: GUDHI 2.0.0 :Copyright: GPL v3 +:Euclidean version requires: CGAL :math:`\geq` 4.6.0 Eigen3 +================================================================= =================================== =================================== + ++-----------------------------------------------------------------+----------------------------------------------------------------------+ +| .. image:: | Witness complex :math:`Wit(W,L)` is a simplicial complex defined on | +| img/Witness_complex_representation.png | two sets of points in :math:`\mathbb{R}^D`. | +| | | +| | The data structure is described in | +| | :cite:`boissonnatmariasimplextreealgorithmica`. | ++-----------------------------------------------------------------+----------------------------------------------------------------------+ +| :doc:`witness_complex_user` | * :doc:`witness_complex_ref` | +| | * :doc:`strong_witness_complex_ref` | +| | * :doc:`euclidean_witness_complex_ref` | +| | * :doc:`euclidean_strong_witness_complex_ref` | ++-----------------------------------------------------------------+----------------------------------------------------------------------+ diff --git a/cython/doc/witness_complex_user.rst b/cython/doc/witness_complex_user.rst new file mode 100644 index 00000000..aa9cbb2c --- /dev/null +++ b/cython/doc/witness_complex_user.rst @@ -0,0 +1,131 @@ +Witness complex user manual +=========================== + +.. include:: witness_complex_sum.rst + +Definitions +----------- + +Witness complex is a simplicial complex defined on two sets of points in :math:`\mathbb{R}^D`: + +- :math:`W` set of **witnesses** and +- :math:`L` set of **landmarks**. + +Even though often the set of landmarks :math:`L` is a subset of the set of witnesses :math:`W`, it is not a requirement +for the current implementation. + +Landmarks are the vertices of the simplicial complex and witnesses help to decide on which simplices are inserted via a +predicate "is witnessed". + +De Silva and Carlsson in their paper :cite:`de2004topological` differentiate **weak witnessing** and +**strong witnessing**: + +- *weak*: :math:`\sigma \subset L` is witnessed by :math:`w \in W` if :math:`\forall l \in \sigma,\ \forall l' \in \mathbf{L \setminus \sigma},\ d(w,l) \leq d(w,l')` +- *strong*: :math:`\sigma \subset L` is witnessed by :math:`w \in W` if :math:`\forall l \in \sigma,\ \forall l' \in \mathbf{L},\ d(w,l) \leq d(w,l')` + +where :math:`d(.,.)` is a distance function. + +Both definitions can be relaxed by a real value :math:`\alpha`: + +- *weak*: :math:`\sigma \subset L` is :math:`\alpha`-witnessed by :math:`w \in W` if :math:`\forall l \in \sigma,\ \forall l' \in \mathbf{L \setminus \sigma},\ d(w,l)^2 \leq d(w,l')^2 + \alpha^2` +- *strong*: :math:`\sigma \subset L` is :math:`\alpha`-witnessed by :math:`w \in W` if :math:`\forall l \in \sigma,\ \forall l' \in \mathbf{L},\ d(w,l)^2 \leq d(w,l')^2 + \alpha^2` + +which leads to definitions of **weak relaxed witness complex** (or just relaxed witness complex for short) and +**strong relaxed witness complex** respectively. + +.. figure:: img/swit.svg + :alt: Strongly witnessed simplex + :figclass: align-center + + Strongly witnessed simplex + + +In particular case of 0-relaxation, weak complex corresponds to **witness complex** introduced in +:cite:`de2004topological`, whereas 0-relaxed strong witness complex consists of just vertices and is not very +interesting. Hence for small relaxation weak version is preferable. +However, to capture the homotopy type (for example using Gudhi::persistent_cohomology::Persistent_cohomology) it is +often necessary to work with higher filtration values. In this case strong relaxed witness complex is faster to compute +and offers similar results. + +Implementation +-------------- + +The two complexes described above are implemented in the corresponding classes + +- :doc:`witness_complex_ref` +- :doc:`strong_witness_complex_ref` +- :doc:`euclidean_witness_complex_ref` +- :doc:`euclidean_strong_witness_complex_ref` + +The construction of the Euclidean versions of complexes follow the same scheme: + +1. Construct a search tree on landmarks. +2. Construct lists of nearest landmarks for each witness. +3. Construct the witness complex for nearest landmark lists. + +In the non-Euclidean classes, the lists of nearest landmarks are supposed to be given as input. + +The constructors take on the steps 1 and 2, while the function 'create_complex' executes the step 3. + +Constructing weak relaxed witness complex from an off file +---------------------------------------------------------- + +Let's start with a simple example, which reads an off point file and computes a weak witness complex. + +.. code-block:: python + + import gudhi + import argparse + + parser = argparse.ArgumentParser(description='EuclideanWitnessComplex creation from ' + 'points read in a OFF file.', + epilog='Example: ' + 'example/witness_complex_diagram_persistence_from_off_file_example.py ' + '-f ../data/points/tore3D_300.off -a 1.0 -n 20 -d 2' + '- Constructs a alpha complex with the ' + 'points from the given OFF file.') + parser.add_argument("-f", "--file", type=str, required=True) + parser.add_argument("-a", "--max_alpha_square", type=float, required=True) + parser.add_argument("-n", "--number_of_landmarks", type=int, required=True) + parser.add_argument("-d", "--limit_dimension", type=int, required=True) + + args = parser.parse_args() + + with open(args.file, 'r') as f: + first_line = f.readline() + if (first_line == 'OFF\n') or (first_line == 'nOFF\n'): + print("#####################################################################") + print("EuclideanWitnessComplex creation from points read in a OFF file") + + witnesses = gudhi.read_off(off_file=args.file) + landmarks = gudhi.pick_n_random_points(points=witnesses, nb_points=args.number_of_landmarks) + + message = "EuclideanWitnessComplex with max_edge_length=" + repr(args.max_alpha_square) + \ + " - Number of landmarks=" + repr(args.number_of_landmarks) + print(message) + + witness_complex = gudhi.EuclideanWitnessComplex(witnesses=witnesses, landmarks=landmarks) + simplex_tree = witness_complex.create_simplex_tree(max_alpha_square=args.max_alpha_square, + limit_dimension=args.limit_dimension) + + message = "Number of simplices=" + repr(simplex_tree.num_simplices()) + print(message) + else: + print(args.file, "is not a valid OFF file") + + f.close() + + +Example2: Computing persistence using strong relaxed witness complex +-------------------------------------------------------------------- + +Here is an example of constructing a strong witness complex filtration and computing persistence on it: + +* :download:`euclidean_strong_witness_complex_diagram_persistence_from_off_file_example.py <../example/periodic_cubical_complex_barcode_persistence_from_perseus_file_example.py>` + +Bibliography +============ + +.. bibliography:: bibliography.bib + :filter: docnames + :style: unsrt |
