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doc/examples/cohomology.rst

author | Camille Wormser <camille.wormser@normalesup.org> |

Wed, 12 Aug 2009 07:16:35 -0700 | |

branch | dev |

changeset 162 | eec482c29319 |

parent 146 | 4e27f1f7c169 |

permissions | -rw-r--r-- |

Removing unused parameter names in order to avoid warnings
-> no warning anymore in zigzag-persistence

.. _cohomology-parametrization: Parametrizing a point set using circle valued functions ======================================================= The procedure described below is explained in detail in [dSVJ09]_. .. program:: rips-pairwise-cohomology One can use :sfile:`examples/cohomology/rips-pairwise-cohomology.cpp` to compute persistent pairing of the Rips filtration using the persistent cohomology algorithm. It takes as input a file containing a point set in Euclidean space (one per line) as well as the following command-line flags: .. cmdoption:: -p, --prime The prime to use in the computation (defaults to 11). .. cmdoption:: -m, --max-distance Maximum cutoff parameter up to which to compute the complex. .. cmdoption:: -s, --skeleton-dimension Skeleton to compute; persistent pairs output will be this number minus 1 (defaults to 2). .. cmdoption:: -b, --boundary Filename where to output the boundary matrix. .. cmdoption:: -c, --cocycle Prefix of the filenames where to output the 1-dimensional cocycles. .. cmdoption:: -v, --vertices Filename where to output the simplex vertex mapping. .. cmdoption:: -d, --diagram Filename where to output the persistence diagram. For example:: rips-pairwise-cohomology points.txt -m 1 -b points.bdry -c points -v points.vrt -d points.dgm Assuming that at the threshold value of 1 (``-m 1`` above) Rips complex contains 1-dimensional cocycles, they will be output into filenames of the form ``points-0.ccl``, ``points-1.ccl``, etc. Subsequently one can use :sfile:`examples/cohomology/cocycle.py` to assign to each vertex of the input point set a circle-valued function. It takes the boundary matrix, cocycle, and simplex-vertex map as an input (all produced at the previous step):: cocycle.py points.bdry points-0.ccl points.vrt The above command outputs a file ``points-0.val`` which contains values assigned to the input points (the lines match the lines of the input file ``points.txt``, but also contains the indices). Plotting -------- Two auxilliary tools allow one to visualize the values assigned to the points (using Matplotlib_): :sfile:`tools/plot-values/plot.py` and :sfile:`tools/plot-values/scatter.py`:: plot.py points-0.val points.txt scatter.py points-0.val points-1.val .. _Matplotlib: http://matplotlib.sourceforge.net/ Dependency ---------- The Python `LSQR code`_ (ported from the `Stanford MATLAB implementation`_ to Python by `Jeffery Kline`_) included with Dionysus, and used in :sfile:`examples/cohomology/cocycle.py`, requires CVXOPT_. .. _`LSQR code`: http://pages.cs.wisc.edu/~kline/cvxopt/ .. _CVXOPT: http://abel.ee.ucla.edu/cvxopt/ .. _`Stanford MATLAB implementation`: http://www.stanford.edu/group/SOL/software/lsqr.html .. _`Jeffery Kline`: http://pages.cs.wisc.edu/~kline/ .. _rips-pairwise-cohomology: Python cohomology computation ----------------------------- :sfile:`examples/cohomology/rips-pairwise-cohomology.py` gives an example of the same computation performed in Python (but with the output in a different format). After the simplicial complex is computed in a list `simplices`, and the list is sorted with respect to the Rips filtration order, the simplices are inserted into the :class:`CohomologyPersistence` one by one:: # list simplices is created ch = CohomologyPersistence(prime) complex = {} for s in simplices: i,d = ch.add([complex[sb] for sb in s.boundary], (s.dimension(), s.data)) complex[s] = i if d: dimension, birth = d print dimension, birth, s.data # else birth Above dictionary `complex` maintains the map of simplices to indices returned by :meth:`CohomologyPersistence.add`. The pair `(dimension, data)` is used as the birth value. Here `data` is the value associated with the simplex in the Rips filtration. The pair is returned back if a death occurs, and is printed on the standard output. After the for loop finishes, one may output infinite persistence classes with the following for loop:: for ccl in ch: dimension, birth = ccl.birth if dimension >= skeleton: continue print dimension, birth, 'inf' # dimension, simplex data = birth Naturally one may iterate over `ccl` which is of type :class:`Cocycle` and extract more information. For example, this is necessary to get the coefficients that serve as the input for :sfile:`examples/cohomology/cocycle.py`.