# Čech complex #

## cech_persistence ##
This program computes the persistent homology with coefficient field *Z/pZ* of
a Čech complex defined on a set of input points, using Euclidean distance. 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**

`cech_persistence [options] <OFF input file>`

**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-edge-length ]` (default = inf) Maximal length of an edge for the Čech complex construction.
* `-d [ --cpx-dimension ]` (default = 1) Maximal dimension of the Čech complex we want to compute.
* `-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.

Beware: this program may use a lot of RAM and take a lot of time if `max-edge-length` is set to a large value.

**Example 1 with Z/2Z coefficients**

`cech_persistence ../../data/points/tore3D_1307.off -r 0.25 -m 0.5 -d 3 -p 2`

**Example 2 with Z/3Z coefficients**

`cech_persistence ../../data/points/tore3D_1307.off -r 0.25 -m 0.5 -d 3 -p 3`