Example

This page demonstrates how to use RIVET to visualize the topological structure of a point cloud. We consider the point cloud in \(\mathbb{R}^2\) pictured below. We emphasize, however, that RIVET can handle point clouds in any dimension, as well as other data types.

Point cloud

Evidently, this point cloud contains a dense circle of points, as if many of the points were sampled from an annulus. However, we also observe outliers, both inside and outside of the annulus. Note that one-parameter persistence (e.g., using a Rips filtration) cannot easily detect the dense cycle, due to the presence of outliers.

We will associate to this data a function-Rips bifiltration, taking the function on the points to be a density estimate; here, we use a simple ball density estimator, which for each point \(p\) simply counts the number of points within a unit distance of \(p\). The following diagram colors the points by density.

Point cloud with density estimator

Input File

We next prepare an input file for RIVET. This is a text file written in the format described on the Input Data page. The first lines of the text file appear below (the complete file is in the RIVET repository):

points
2
4
[-] density
1.57    2.40    20
1.21    2.70    22
0.79    -2.44   32
...

The first four lines are parameters for RIVET: points tells RIVET that the file contains point-cloud data, 2 indicates that each point is specified by two coordinates, 4 specifies that the maximum length of edges to be constructed in the Rips filtration will be 4, and density is a text label for the horizontal axis in RIVET. The [-] preceding the axis label tells RIVET to filter by density in decreasing order, so that points with the largest density appear first in the bifiltration.

Computation from the GUI

We proceed using the the RIVET GUI. With RIVET installed, we run rivet_GUI. The input data dialog box appears. We select the data file and computation parameters, as shown below. Since we are interested in detecting a cycle, we select homology dimension 1. We must also set parameters for the coarseness of the computation, via the bins selectors. These tell RIVET to round the computed values for density and distance to a specified number of equally-spaced values, which speeds up the computation. We will choose 40 bins in each direction.

RIVET input data dialog box

(Note that setting the bin selectors to zero will cause RIVET not to round the computed values, is more computationally expensive.)

We click Compute, and RIVET computes the augmented arrangement. This may take several minutes, depending on the computing power available.

Visualization

When the Hilbert function and bigraded Betti numbers have been computed, visualizations of these appear in the Line Selection Window on the left side of the RIVET window (see the The RIVET Visualization page for more details). When the computation of the augmented arrangement is complete, a barcode appears in the Line Selection Window and a persistence diagram appears in the Persistence Diagram window in RIVET, as shown below. RIVET is now ready for interactive browsing of barcodes along linear slices through the bipersistence persistence module.

RIVET visualization window

For this data, note that the barcode contains a single long bar when the selected line goes roughly from the lower-left corner to the upper-right corner of the Line Selection Window. This bar corresponds to the dense cycle of points in the point cloud.

Furthermore, note that selecting vertical lines effectively thresholds the points by density. That is, selecting a vertical line with density value \(d\) produces a barcode computed from a Rips filtration on only those points with density value greater than \(d\), as shown below. This effectively reduces the analysis to one-parameter persistence, using a density threshold. The RIVET GUI allows the user to slide the vertical line left and right, thereby displaying the barcodes for different choices of the density threshold.

RIVET visualization window

Computation from the Console

Rather than using the RIVET GUI, one may use the RIVET console application to compute the augmented arrangement and even obtain barcodes. This is done using the command line, as described in Running RIVET. For example, the computation described above can be obtained from rivet_console using the following command, run from the root directory of the RIVET repository:

./rivet_GUI/data/Test_Point_Clouds/circle_300pts_density.txt circle_300_computed.mif -H 1 -X 40 -y 40

This will produce a module invariants file circle_300_computed.mif, which may then be loaded into the RIVET GUI or queried for barcodes on a collection of user-chosen lines. Please see Running RIVET for more details.