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Diffstat (limited to 'src/cython/doc')
| -rw-r--r-- | src/cython/doc/cubical_complex_user.rst | 10 | ||||
| -rw-r--r-- | src/cython/doc/examples.rst | 1 | ||||
| -rw-r--r-- | src/cython/doc/fileformats.rst | 10 | ||||
| -rw-r--r-- | src/cython/doc/installation.rst | 65 | ||||
| -rw-r--r-- | src/cython/doc/nerve_gic_complex_ref.rst | 4 | ||||
| -rw-r--r-- | src/cython/doc/nerve_gic_complex_sum.rst | 2 | ||||
| -rw-r--r-- | src/cython/doc/nerve_gic_complex_user.rst | 4 | ||||
| -rw-r--r-- | src/cython/doc/persistence_graphical_tools_ref.rst | 1 | ||||
| -rw-r--r-- | src/cython/doc/persistence_graphical_tools_sum.inc | 4 | ||||
| -rw-r--r-- | src/cython/doc/persistence_graphical_tools_user.rst | 37 | ||||
| -rw-r--r-- | src/cython/doc/persistent_cohomology_user.rst | 14 | ||||
| -rw-r--r-- | src/cython/doc/rips_complex_sum.inc | 6 | ||||
| -rw-r--r-- | src/cython/doc/rips_complex_user.rst | 67 | ||||
| -rw-r--r-- | src/cython/doc/tangential_complex_user.rst | 17 |
14 files changed, 171 insertions, 71 deletions
diff --git a/src/cython/doc/cubical_complex_user.rst b/src/cython/doc/cubical_complex_user.rst index 320bd79b..19120360 100644 --- a/src/cython/doc/cubical_complex_user.rst +++ b/src/cython/doc/cubical_complex_user.rst @@ -83,9 +83,15 @@ 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 +Currently one input from a text file is used. It uses a format inspired from the Perseus software `Perseus software <http://www.sas.upenn.edu/~vnanda/perseus/>`_ by Vidit Nanda. -The file format is described here: :doc:`Perseus <fileformats>`. + +.. note:: + While Perseus assume the filtration of all maximal cubes to be non-negative, over here we do not enforce this and + we allow any filtration values. As a consequence one cannot use ``-1``'s to indicate missing cubes. If you have + missing cubes in your complex, please set their filtration to :math:`+\infty` (aka. ``inf`` in the file). + +The file format is described in details in :ref:`Perseus file format` file format section. .. testcode:: diff --git a/src/cython/doc/examples.rst b/src/cython/doc/examples.rst index 1f02f8a2..edbc2f72 100644 --- a/src/cython/doc/examples.rst +++ b/src/cython/doc/examples.rst @@ -22,6 +22,7 @@ Examples * :download:`rips_complex_diagram_persistence_from_off_file_example.py <../example/rips_complex_diagram_persistence_from_off_file_example.py>` * :download:`rips_complex_diagram_persistence_from_distance_matrix_file_example.py <../example/rips_complex_diagram_persistence_from_distance_matrix_file_example.py>` * :download:`rips_persistence_diagram.py <../example/rips_persistence_diagram.py>` + * :download:`sparse_rips_persistence_diagram.py <../example/sparse_rips_persistence_diagram.py>` * :download:`random_cubical_complex_persistence_example.py <../example/random_cubical_complex_persistence_example.py>` * :download:`coordinate_graph_induced_complex.py <../example/coordinate_graph_induced_complex.py>` * :download:`functional_graph_induced_complex.py <../example/functional_graph_induced_complex.py>` diff --git a/src/cython/doc/fileformats.rst b/src/cython/doc/fileformats.rst index ff20f26e..e205cc8b 100644 --- a/src/cython/doc/fileformats.rst +++ b/src/cython/doc/fileformats.rst @@ -51,10 +51,12 @@ Here is a simple sample file in the 3D case:: 1. 1. 1. +.. _Perseus file format: + Perseus ******* -This file format is the format used by the +This file format is a format inspired from the `Perseus software <http://www.sas.upenn.edu/~vnanda/perseus/>`_ by Vidit Nanda. 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 @@ -88,3 +90,9 @@ Indicate that we have imposed periodic boundary conditions in the direction x, but not in the direction y. Other sample files can be found in the `data/bitmap` folder. + +.. note:: + Unlike in Perseus format the filtration on the maximal cubes can be any + double precision number. Consequently one cannot mark the cubes that are + not present with ``-1``'s. To do that please set their filtration value to + :math:`+\infty` (aka. ``inf`` in the file).
\ No newline at end of file diff --git a/src/cython/doc/installation.rst b/src/cython/doc/installation.rst index 43576ec9..855dea44 100644 --- a/src/cython/doc/installation.rst +++ b/src/cython/doc/installation.rst @@ -7,24 +7,23 @@ Installation Compiling ********* -The library uses c++11 and requires `Boost <https://www.boost.org/>`_ ≥ 1.48.0 -and `CMake <https://www.cmake.org/>`_ ≥ 3.1. +The library uses c++11 and requires `Boost <https://www.boost.org/>`_ ≥ 1.56.0, +`CMake <https://www.cmake.org/>`_ ≥ 3.1 to generate makefiles, and +`Cython <https://www.cython.org/>`_ to compile the GUDHI Python module. 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 +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 -=================== +GUDHI Python module compilation +=============================== -To build the GUDHI cython module, run the following commands in a terminal: +To build the GUDHI Python module, run the following commands in a terminal: .. code-block:: bash @@ -32,7 +31,28 @@ To build the GUDHI cython module, run the following commands in a terminal: mkdir build cd build/ cmake .. - make cython + cd cython + make + +GUDHI Python module installation +================================ + +Once the compilation succeeds, one can add the GUDHI Python module path to the +PYTHONPATH: + +.. code-block:: bash + + # For windows, you have to set PYTHONPATH environment variable + export PYTHONPATH='$PYTHONPATH:/path-to-gudhi/build/cython' + +Or install it definitely in your Python packages folder: + +.. code-block:: bash + + cd /path-to-gudhi/build/cython + # May require sudo or administrator privileges + make install + Test suites =========== @@ -45,7 +65,7 @@ following command in a terminal: cd /path-to-gudhi/build/cython # For windows, you have to set PYTHONPATH environment variable export PYTHONPATH='$PYTHONPATH:/path-to-gudhi/build/cython' - ctest -R py_test + make test Debugging issues ================ @@ -54,7 +74,7 @@ If tests fail, please check your PYTHONPATH and try to :code:`import gudhi` and check the errors. The problem can come from a third-party library bad link or installation. -If :code:`import gudhi` succeeds, please have a look to debug informations: +If :code:`import gudhi` succeeds, please have a look to debug information: .. code-block:: python @@ -105,13 +125,17 @@ A complete configuration would be : 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: +To build the documentation, `sphinx-doc <http://www.sphinx-doc.org>`_ and +`sphinxcontrib-bibtex <https://sphinxcontrib-bibtex.readthedocs.io>`_ are +required. As the documentation is auto-tested, `CGAL`_, `Eigen3`_, +`Matplotlib`_, `NumPy`_ and `SciPy`_ are also mandatory to build the +documentation. + +Run the following commands in a terminal: .. code-block:: bash + cd /path-to-gudhi/build/cython make sphinx Optional third-party library @@ -127,7 +151,7 @@ The :doc:`Alpha complex </alpha_complex_user>`, C++ library which provides easy access to efficient and reliable geometric algorithms. -Having CGAL, the Computational Geometry Algorithms Library, version 4.7.0 or +Having CGAL, the Computational Geometry Algorithms Library, version 4.7.0 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>`_. @@ -195,7 +219,7 @@ The following examples require the `Matplotlib <http://matplotlib.org>`_: * :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 +NumPy ===== The :doc:`persistence graphical tools </persistence_graphical_tools_user>` @@ -216,6 +240,13 @@ The following examples require the `NumPy <http://numpy.org>`_: * :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>` +SciPy +===== + +The :doc:`persistence graphical tools </persistence_graphical_tools_user>` +module requires `SciPy <http://scipy.org>`_, a Python-based ecosystem of +open-source software for mathematics, science, and engineering. + Threading Building Blocks ========================= diff --git a/src/cython/doc/nerve_gic_complex_ref.rst b/src/cython/doc/nerve_gic_complex_ref.rst index e24e01fc..abde2e8c 100644 --- a/src/cython/doc/nerve_gic_complex_ref.rst +++ b/src/cython/doc/nerve_gic_complex_ref.rst @@ -1,3 +1,7 @@ +:orphan: + +.. To get rid of WARNING: document isn't included in any toctree + ================================ Cover complexes reference manual ================================ diff --git a/src/cython/doc/nerve_gic_complex_sum.rst b/src/cython/doc/nerve_gic_complex_sum.rst index 72782c7a..523c119f 100644 --- a/src/cython/doc/nerve_gic_complex_sum.rst +++ b/src/cython/doc/nerve_gic_complex_sum.rst @@ -1,5 +1,5 @@ ================================================================= =================================== =================================== -:Author: Mathieu Carrière :Introduced in: GUDHI 2.1.0 :Copyright: GPL v3 +:Author: Mathieu Carrière :Introduced in: GUDHI 2.3.0 :Copyright: GPL v3 :Requires: CGAL :math:`\geq` 4.8.1 ================================================================= =================================== =================================== diff --git a/src/cython/doc/nerve_gic_complex_user.rst b/src/cython/doc/nerve_gic_complex_user.rst index d774827e..44f30e1a 100644 --- a/src/cython/doc/nerve_gic_complex_user.rst +++ b/src/cython/doc/nerve_gic_complex_user.rst @@ -1,3 +1,7 @@ +:orphan: + +.. To get rid of WARNING: document isn't included in any toctree + Cover complexes user manual =========================== Definition diff --git a/src/cython/doc/persistence_graphical_tools_ref.rst b/src/cython/doc/persistence_graphical_tools_ref.rst index a2c6bcef..54aff4bc 100644 --- a/src/cython/doc/persistence_graphical_tools_ref.rst +++ b/src/cython/doc/persistence_graphical_tools_ref.rst @@ -9,3 +9,4 @@ Persistence graphical tools reference manual .. autofunction:: gudhi.__min_birth_max_death .. autofunction:: gudhi.plot_persistence_barcode .. autofunction:: gudhi.plot_persistence_diagram +.. autofunction:: gudhi.plot_persistence_density diff --git a/src/cython/doc/persistence_graphical_tools_sum.inc b/src/cython/doc/persistence_graphical_tools_sum.inc index d602daa7..5577cf99 100644 --- a/src/cython/doc/persistence_graphical_tools_sum.inc +++ b/src/cython/doc/persistence_graphical_tools_sum.inc @@ -1,11 +1,11 @@ ================================================================= =================================== =================================== :Author: Vincent Rouvreau :Introduced in: GUDHI 2.0.0 :Copyright: GPL v3 -:Requires: Matplotlib Numpy +:Requires: matplotlib numpy scipy ================================================================= =================================== =================================== +-----------------------------------------------------------------+-----------------------------------------------------------------------+ | .. 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. | +| img/graphical_tools_representation.png | the user to build easily persistence barcode, diagram or density. | | | | +-----------------------------------------------------------------+-----------------------------------------------------------------------+ | :doc:`persistence_graphical_tools_user` | :doc:`persistence_graphical_tools_ref` | diff --git a/src/cython/doc/persistence_graphical_tools_user.rst b/src/cython/doc/persistence_graphical_tools_user.rst index 292915eb..b2124fdd 100644 --- a/src/cython/doc/persistence_graphical_tools_user.rst +++ b/src/cython/doc/persistence_graphical_tools_user.rst @@ -12,6 +12,9 @@ Definition Show persistence as a barcode ----------------------------- +.. note:: + this function requires matplotlib and numpy to be available + This function can display the persistence result as a barcode: .. plot:: @@ -19,16 +22,22 @@ This function can display the persistence result as a barcode: import gudhi - perseus_file = gudhi.__root_source_dir__ + '/data/bitmap/3d_torus.txt' - periodic_cc = gudhi.PeriodicCubicalComplex(perseus_file=perseus_file) - diag = periodic_cc.persistence() - print("diag = ", diag) - plt = gudhi.plot_persistence_barcode(diag) - plt.show() + off_file = gudhi.__root_source_dir__ + '/data/points/tore3D_300.off' + point_cloud = gudhi.read_off(off_file=off_file) + + rips_complex = gudhi.RipsComplex(points=point_cloud, max_edge_length=0.7) + simplex_tree = rips_complex.create_simplex_tree(max_dimension=3) + diag = simplex_tree.persistence(min_persistence=0.4) + + plot = gudhi.plot_persistence_barcode(diag) + plot.show() Show persistence as a diagram ----------------------------- +.. note:: + this function requires matplotlib and numpy to be available + This function can display the persistence result as a diagram: .. plot:: @@ -43,6 +52,12 @@ This function can display the persistence result as a diagram: legend=True) plt.show() +Persistence density +------------------- + +.. note:: + this function requires matplotlib, numpy and scipy to be available + If you want more information on a specific dimension, for instance: .. plot:: @@ -50,13 +65,9 @@ If you want more information on a specific dimension, for instance: import gudhi + # rips_on_tore3D_1307.pers obtained from write_persistence_diagram method persistence_file=gudhi.__root_source_dir__ + \ '/data/persistence_diagram/rips_on_tore3D_1307.pers' - diag = \ - gudhi.read_persistence_intervals_grouped_by_dimension(persistence_file=\ - persistence_file) - dim = 1 - # Display all points with some transparency - plt = gudhi.plot_persistence_diagram([(dim,interval) for interval in diag[dim]], - max_plots=0, alpha=0.1) + plt = gudhi.plot_persistence_density(persistence_file=persistence_file, + max_intervals=0, dimension=1, legend=True) plt.show() diff --git a/src/cython/doc/persistent_cohomology_user.rst b/src/cython/doc/persistent_cohomology_user.rst index ce7fc685..de83cda1 100644 --- a/src/cython/doc/persistent_cohomology_user.rst +++ b/src/cython/doc/persistent_cohomology_user.rst @@ -10,12 +10,14 @@ 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` | -+---------------------------------------------+----------------------------------------------------------------------+ ++-----------------------------------------------------------------+-----------------------------------------------------------------------+ +| :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` | ++-----------------------------------------------------------------+-----------------------------------------------------------------------+ Computation of persistent cohomology using the algorithm of :cite:`DBLP:journals/dcg/SilvaMV11` and diff --git a/src/cython/doc/rips_complex_sum.inc b/src/cython/doc/rips_complex_sum.inc index 5616bfa9..ea26769a 100644 --- a/src/cython/doc/rips_complex_sum.inc +++ b/src/cython/doc/rips_complex_sum.inc @@ -1,6 +1,6 @@ -================================================================= =================================== =================================== -:Author: Clément Maria, Pawel Dlotko, Vincent Rouvreau :Introduced in: GUDHI 2.0.0 :Copyright: GPL v3 -================================================================= =================================== =================================== +===================================================================== =========================== =================================== +:Author: Clément Maria, Pawel Dlotko, Vincent Rouvreau, Marc Glisse :Introduced in: GUDHI 2.0.0 :Copyright: GPL v3 +===================================================================== =========================== =================================== +----------------------------------------------------------------+------------------------------------------------------------------------+ | .. figure:: | Rips complex is a simplicial complex constructed from a one skeleton | diff --git a/src/cython/doc/rips_complex_user.rst b/src/cython/doc/rips_complex_user.rst index a8c06cf9..1d340dbe 100644 --- a/src/cython/doc/rips_complex_user.rst +++ b/src/cython/doc/rips_complex_user.rst @@ -7,27 +7,27 @@ Rips complex user manual Definition ---------- -======================================================= ===================================== ===================================== -:Authors: Clément Maria, Pawel Dlotko, Vincent Rouvreau :Introduced in: GUDHI 2.0.0 :Copyright: GPL v3 -======================================================= ===================================== ===================================== +==================================================================== ================================ ====================== +:Authors: Clément Maria, Pawel Dlotko, Vincent Rouvreau, Marc Glisse :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 `Rips complex <https://en.wikipedia.org/wiki/Vietoris%E2%80%93Rips_complex>`_ is a simplicial complex that +generalizes proximity (:math:`\varepsilon`-ball) graphs to higher dimensions. The vertices correspond to the input +points, and a simplex is present if and only if its diameter is smaller than some parameter α. Considering all +parameters α defines a filtered simplicial complex, where the filtration value of a simplex is its diameter. +The filtration can be restricted to values α smaller than some threshold, to reduce its size. -The filtration value of each edge is computed from a user-given distance function, or directly from the distance -matrix. +The input discrete metric space can be provided as a point cloud plus a distance function, or as a 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 the graph, :doc:`RipsComplex <rips_complex_ref>` first builds the graph and +inserts it into the data structure. It then expands the simplicial complex (adds the simplices corresponding to cliques) +when required. The expansion can be stopped at dimension `max_dimension`, by default 1. -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). +A vertex name corresponds to the index of the point in the given range (aka. the point cloud). .. figure:: ../../doc/Rips_complex/rips_complex_representation.png @@ -38,8 +38,27 @@ Vertex name correspond to the index of the point in the given range (aka. the po 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. +If the `RipsComplex` interfaces are not detailed enough for your need, please refer to rips_persistence_step_by_step.cpp +C++ example, where the graph construction over the Simplex_tree is more detailed. + +A Rips complex can easily become huge, even if we limit the length of the edges +and the dimension of the simplices. One easy trick, before building a Rips +complex on a point cloud, is to call `sparsify_point_set` which removes points +that are too close to each other. This does not change its persistence diagram +by more than the length used to define "too close". + +A more general technique is to use a sparse approximation of the Rips +introduced by Don Sheehy :cite:`sheehy13linear`. We are using the version +described in :cite:`buchet16efficient` (except that we multiply all filtration +values by 2, to match the usual Rips complex). :cite:`cavanna15geometric` proves +a :math:`\frac{1}{1-\varepsilon}`-interleaving, although in practice the +error is usually smaller. A more intuitive presentation of the idea is +available in :cite:`cavanna15geometric`, and in a video +:cite:`cavanna15visualizing`. Passing an extra argument `sparse=0.3` at the +construction of a `RipsComplex` object asks it to build a sparse Rips with +parameter :math:`\varepsilon=0.3`, while the default `sparse=None` builds the +regular Rips complex. + Point cloud ----------- @@ -47,7 +66,7 @@ Point cloud Example from a point cloud ^^^^^^^^^^^^^^^^^^^^^^^^^^ -This example builds the one skeleton graph from the given points, and max_edge_length value. +This example builds the neighborhood graph from the given points, up to max_edge_length. Then it creates a :doc:`Simplex_tree <simplex_tree_ref>` with it. Finally, it is asked to display information about the simplicial complex. @@ -56,7 +75,7 @@ Finally, it is asked to display information about the simplicial complex. 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) + 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()) + ' - ' + \ @@ -92,10 +111,20 @@ until dimension 1 - one skeleton graph in other words), the output is: [4, 6] -> 9.49 [3, 6] -> 11.00 +Notice that if we use + +.. code-block:: python + + rips_complex = gudhi.RipsComplex(points=[[1, 1], [7, 0], [4, 6], [9, 6], [0, 14], [2, 19], [9, 17]], + max_edge_length=12.0, sparse=2) + +asking for a very sparse version (theory only gives some guarantee on the meaning of the output if `sparse<1`), +2 to 5 edges disappear, depending on the random vertex used to start the sparsification. + Example from OFF file ^^^^^^^^^^^^^^^^^^^^^ -This example builds the :doc:`Rips_complex <rips_complex_ref>` from the given +This example builds the :doc:`RipsComplex <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. @@ -200,7 +229,7 @@ until dimension 1 - one skeleton graph in other words), the output is: Example from csv file ^^^^^^^^^^^^^^^^^^^^^ -This example builds the :doc:`Rips_complex <rips_complex_ref>` from the given +This example builds the :doc:`RipsComplex <rips_complex_ref>` from the given distance matrix in a csv file, and max_edge_length value. Then it creates a :doc:`Simplex_tree <simplex_tree_ref>` with it. diff --git a/src/cython/doc/tangential_complex_user.rst b/src/cython/doc/tangential_complex_user.rst index 5ce69e86..ebfe1e29 100644 --- a/src/cython/doc/tangential_complex_user.rst +++ b/src/cython/doc/tangential_complex_user.rst @@ -23,8 +23,10 @@ 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. +example, with :math:`k = 1` and :math:`d = 2`. The point set +:math:`\mathscr P` is located on a closed curve embedded in 2D. +Only 4 points will be displayed (more are required for PCA) to simplify the +figures. .. figure:: ../../doc/Tangential_complex/tc_example_01.png :alt: The input @@ -32,8 +34,7 @@ example, with :math:`k = 1` and :math:`d = 2`. The input data is 4 points The input -For each point :math:`p`, estimate its tangent subspace :math:`T_p` (e.g. -using PCA). +For each point :math:`P`, estimate its tangent subspace :math:`T_P` using PCA. .. figure:: ../../doc/Tangential_complex/tc_example_02.png :alt: The estimated normals @@ -43,8 +44,8 @@ using PCA). 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`. +:math:`P`, construct its star in the Delaunay triangulation of +:math:`\mathscr P` restricted to :math:`T_P`. .. figure:: ../../doc/Tangential_complex/tc_example_03.png :alt: The Voronoi diagram @@ -72,7 +73,7 @@ Let us take the same example. Before -Let us slightly move the tangent subspace :math:`T_q` +Let us slightly move the tangent subspace :math:`T_Q` .. figure:: ../../doc/Tangential_complex/tc_example_07_after.png :alt: After @@ -128,6 +129,7 @@ This example builds the Tangential complex of point set read in an OFF file. import gudhi tc = gudhi.TangentialComplex(intrisic_dim = 1, off_file=gudhi.__root_source_dir__ + '/data/points/alphacomplexdoc.off') + tc.compute_tangential_complex() result_str = 'Tangential contains ' + repr(tc.num_simplices()) + \ ' simplices - ' + repr(tc.num_vertices()) + ' vertices.' print(result_str) @@ -175,6 +177,7 @@ simplices. import gudhi tc = gudhi.TangentialComplex(intrisic_dim = 1, points=[[0.0, 0.0], [1.0, 0.0], [0.0, 1.0], [1.0, 1.0]]) + tc.compute_tangential_complex() result_str = 'Tangential contains ' + repr(tc.num_vertices()) + ' vertices.' print(result_str) |