Toroidal coordinates on surface of genus two

An example of how, in the presence of more than one prominent circular feature in the data, the toroidal coordinates algorithm returns circle-valued maps that are geometrically less correlated and easier to interpret than the ones returned by running the circular coordinates algorithm on each individual class.

import matplotlib.pyplot as plt
from dreimac import CircularCoords, ToroidalCoords, GeometryExamples, PlotUtils, CircleMapUtils
from persim import plot_diagrams

We start by displaying the data, which consists of a pointcloud in 3D.

X = GeometryExamples.genus_two_surface()

fig = plt.figure(figsize=(4,4))
ax = fig.add_subplot(projection="3d")
ax.scatter(X[:,0], X[:,1], X[:,2], s=0.1)
PlotUtils.set_axes_equal(ax) ; plt.axis("off") ; fig.subplots_adjust(top=1, bottom=0, left=0, right=1)

Compute and display the Rips persistent cohomology.

n_landmarks = 750

tc = ToroidalCoords(X, n_landmarks=n_landmarks)
plot_diagrams(tc._dgms)

We now construct two sets of four circle-valued maps. For the first set, we use the toroidal coordinates with input the four most prominent persistent cohomology classes; for the second, we run the circular coordinates algorithm on each of the four individual classes.

cohomology_classes = [0, 1, 2, 3]
toroidal_coords = tc.get_coordinates(cocycle_idxs=cohomology_classes)

We display the circle-valued representations obtained with toroidal coordinates using two different colorings.

fig = plt.figure(figsize=(9,4))
fig.suptitle("Toroidal coordinates with cyclic colormap", fontsize=20)
for i, coordinate in enumerate(toroidal_coords):
    plt.subplot(1, len(toroidal_coords), i + 1)
    plt.scatter(X[:, 1], X[:, 0], s=20, c=CircleMapUtils.to_sinebow(CircleMapUtils.center(coordinate)))
    plt.title("toroidal\ncoordinate " + str(i+1))
    plt.gca().set_aspect("equal") ; _ = plt.axis("off")

fig = plt.figure(figsize=(9, 4))
fig.suptitle("Toroidal coordinates with levelset colormap", fontsize=20)
for i, coordinate in enumerate(toroidal_coords):
    plt.subplot(1, len(toroidal_coords), i + 1)
    plt.scatter(
        X[:, 1],
        X[:, 0],
        s=2,
        c=CircleMapUtils.levelset_coloring(CircleMapUtils.center(coordinate)),
    )
    plt.title("toroidal\ncoordinate " + str(i + 1))
    plt.gca().set_aspect("equal")
    _ = plt.axis("off")

cc = CircularCoords(X, n_landmarks=n_landmarks)
circular_coords1 = cc.get_coordinates(cocycle_idx=cohomology_classes[0])
circular_coords2 = cc.get_coordinates(cocycle_idx=cohomology_classes[1])
circular_coords3 = cc.get_coordinates(cocycle_idx=cohomology_classes[2])
circular_coords4 = cc.get_coordinates(cocycle_idx=cohomology_classes[3])
circular_coords = [ circular_coords1, circular_coords2, circular_coords3, circular_coords4 ]

We display the circle-valued representations obtained with circular coordinates.

fig = plt.figure(figsize=(9,4))
fig.suptitle("Circular coordinates with cyclic colormap", fontsize=20)
for i, coordinate in enumerate(circular_coords):
    plt.subplot(1, len(toroidal_coords), i + 1)
    plt.scatter(X[:, 1], X[:, 0], s=2, c=CircleMapUtils.to_sinebow(coordinate))
    plt.title("circular\ncoordinate " + str(i+1))
    plt.gca().set_aspect("equal") ; _ = plt.axis("off")

fig = plt.figure(figsize=(9, 4))
fig.suptitle("Circular coordinates with levelset colormap", fontsize=20)
for i, coordinate in enumerate(circular_coords):
    plt.subplot(1, len(toroidal_coords), i + 1)
    plt.scatter(
        X[:, 1],
        X[:, 0],
        s=2,
        c=CircleMapUtils.levelset_coloring(CircleMapUtils.center(coordinate)),
    )
    plt.title("toroidal\ncoordinate " + str(i + 1))
    plt.gca().set_aspect("equal")
    _ = plt.axis("off")