# Witness complex

For more details about the witness complex, please read the user manual of the package.

## weak_witness_persistence

This program computes the persistent homology with coefficient field *Z/pZ* of a Weak witness complex defined on a set of input points.
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.

**Usage**

`weak_witness_persistence [options] <OFF input file>`

**Allowed options**

`-h [ --help ]`

Produce help message`-l [ --landmarks ]`

Number of landmarks to choose from the point cloud.`-o [ --output-file ]`

Name of file in which the persistence diagram is written. By default, print in std::cout.`-a [ --max-sq-alpha ]`

(default = inf) Maximal squared relaxation parameter.`-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.`-d [ --cpx-dimension ]`

(default = 2147483647) Maximal dimension of the weak witness complex we want to compute.

**Example**

`weak_witness_persistence data/points/tore3D_1307.off -l 20 -a 0.5 -m 0.006`

N.B.: output is random as the 20 landmarks are chosen randomly.

## strong_witness_persistence

This program computes the persistent homology with coefficient field *Z/pZ* of a Strong witness complex defined on a set of input points.
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.

**Usage**

`strong_witness_persistence [options] <OFF input file>`

**Allowed options**

`-h [ --help ]`

Produce help message`-l [ --landmarks ]`

Number of landmarks to choose from the point cloud.`-o [ --output-file ]`

Name of file in which the persistence diagram is written. By default, print in std::cout.`-a [ --max-sq-alpha ]`

(default = inf) Maximal squared relaxation parameter.`-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.`-d [ --cpx-dimension ]`

(default = 2147483647) Maximal dimension of the weak witness complex we want to compute.

**Example**

`strong_witness_persistence data/points/tore3D_1307.off -l 20 -a 0.5 -m 0.06`

N.B.: output is random as the 20 landmarks are chosen randomly.