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----
-layout: page
-title: "Alpha complex"
-meta_title: "Alpha complex"
-teaser: ""
-permalink: /alphacomplex/
----
-{::comment}
-Leave the lines above as it is required by the web site generator 'Jekyll'
-{:/comment}
-
-
-## alpha_complex_persistence ##
-
-This program computes the persistent homology with coefficient field Z/pZ of the dD alpha complex built from a dD point cloud.
-The output diagram contains one bar per line, written with the convention:
-
-```
- p dim birth death
-```
-
-where `dim` is the dimension of the homological feature, `birth` and `death` are respectively the birth and death of the feature,
-and `p` is the characteristic of the field *Z/pZ* used for homology coefficients (`p` must be a prime number).
-
-**Usage**
-
-```
- alpha_complex_persistence [options] <input OFF file>
-```
-
-where
-`<input OFF file>` is the path to the input point cloud in [nOFF ASCII format](http://www.geomview.org/docs/html/OFF.html).
-
-**Allowed options**
-
-* `-h [ --help ]` Produce help message
-* `-o [ --output-file ]` Name of file in which the persistence diagram is written. Default print in standard output.
-* `-r [ --max-alpha-square-value ]` (default = inf) Maximal alpha square value for the Alpha complex construction.
-* `-p [ --field-charac ]` (default = 11) Characteristic p of the coefficient field Z/pZ for computing homology.
-* `-m [ --min-persistence ]` (default = 0) Minimal lifetime of homology feature to be recorded. Enter a negative value to see zero length intervals.
-
-**Example**
-
-```
- alpha_complex_persistence -r 32 -p 2 -m 0.45 ../../data/points/tore3D_300.off
-```
-
-N.B.:
-
-* Filtration values are alpha square values.
-
-
-## alpha_complex_3d_persistence ##
-This program computes the persistent homology with coefficient field Z/pZ of the 3D alpha complex built from a 3D point cloud. The output diagram contains one bar per line, written with the convention:
-
-```
-p dim birth death
-```
-
-where `dim` is the dimension of the homological feature, `birth` and `death` are respectively the birth and death of the feature, and `p` is the characteristic of the field *Z/pZ* used for homology coefficients (`p` must be a prime number).
-
-**Usage**
-
-```
- alpha_complex_3d_persistence [options] <input OFF file>
-```
-
-where `<input OFF file>` is the path to the input point cloud in [nOFF ASCII format](http://www.geomview.org/docs/html/OFF.html).
-
-**Allowed options**
-
-* `-h [ --help ]` Produce help message
-* `-o [ --output-file ]` Name of file in which the persistence diagram is written. Default print in standard output.
-* `-p [ --field-charac ]` (default=11) Characteristic p of the coefficient field Z/pZ for computing homology.
-* `-m [ --min-persistence ]` (default = 0) Minimal lifetime of homology feature to be recorded. Enter a negative value to see zero length intervals.
-
-**Example**
-
-```
-alpha_complex_3d_persistence ../../data/points/tore3D_300.off -p 2 -m 0.45
-```
-
-N.B.:
-
-* `alpha_complex_3d_persistence` only accepts OFF files in dimension 3.
-* Filtration values are alpha square values.
-
-
-## exact_alpha_complex_3d_persistence ##
-
-Same as `alpha_complex_3d_persistence`, but using exact computation.
-It is slower, but it is necessary when points are on a grid for instance.
-
-
-
-## weighted_alpha_complex_3d_persistence ##
-
-Same as `alpha_complex_3d_persistence`, but using weighted points.
-
-**Usage**
-
-```
- weighted_alpha_complex_3d_persistence [options] <input OFF file> <weights input file>
-```
-
-where
-
-* `<input OFF file>` is the path to the input point cloud in [nOFF ASCII format](http://www.geomview.org/docs/html/OFF.html).
-* `<input weights file>` is the path to the file containing the weights of the points (one value per line).
-
-**Allowed options**
-
-* `-h [ --help ]` Produce help message
-* `-o [ --output-file ]` Name of file in which the persistence diagram is written. Default print in standard output.
-* `-p [ --field-charac ]` (default=11) Characteristic p of the coefficient field Z/pZ for computing homology.
-* `-m [ --min-persistence ]` (default = 0) Minimal lifetime of homology feature to be recorded. Enter a negative value to see zero length intervals.
-
-**Example**
-
-```
- weighted_alpha_complex_3d_persistence ../../data/points/tore3D_300.off ../../data/points/tore3D_300.weights -p 2 -m 0.45
-```
-
-
-N.B.:
-
-* Weights values are explained on CGAL [Alpha shape](https://doc.cgal.org/latest/Alpha_shapes_3/index.html#title0)
-and [Regular triangulation](https://doc.cgal.org/latest/Triangulation_3/index.html#Triangulation3secclassRegulartriangulation) documentation.
-* Filtration values are alpha square values.
-
-
-## periodic_alpha_complex_3d_persistence ##
-Same as `alpha_complex_3d_persistence`, but using periodic alpha shape 3d.
-Refer to the [CGAL's 3D Periodic Triangulations User Manual](https://doc.cgal.org/latest/Periodic_3_triangulation_3/index.html) for more details.
-
-**Usage**
-
-```
- periodic_alpha_complex_3d_persistence [options] <input OFF file> <cuboid file>
-```
-
-where
-
-* `<input OFF file>` is the path to the input point cloud in [nOFF ASCII format](http://www.geomview.org/docs/html/OFF.html).
-* `<cuboid file>` is the path to the file describing the periodic domain. It must be in the format described
-[here](/doc/latest/fileformats.html#FileFormatsIsoCuboid).
-
-**Allowed options**
-
-* `-h [ --help ]` Produce help message
-* `-o [ --output-file ]` Name of file in which the persistence diagram is written. Default print in standard output.
-* `-p [ --field-charac ]` (default=11) Characteristic p of the coefficient field Z/pZ for computing homology.
-* `-m [ --min-persistence ]` (default = 0) Minimal lifetime of homology feature to be recorded. Enter a negative value to see zero length intervals
-
-
-**Example**
-
-```
-periodic_alpha_complex_3d_persistence ../../data/points/grid_10_10_10_in_0_1.off ../../data/points/iso_cuboid_3_in_0_1.txt -p 3 -m 1.0
-```
-
-N.B.:
-
-* Cuboid file must be in the format described [here](/doc/latest/fileformats.html#FileFormatsIsoCuboid).
-* Filtration values are alpha square values.