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# Alpha complex #
## 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.
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