Tessellation Graphs¶

Module: pycgsp.graph

Routines for building graphs from tessellations/point sets in \(\mathbb{R}^3\).

cvxhull_graph(R[, cheb_normalized, …])

Build the convex hull graph of a point set in \(\mathbb{R}^3\).

healpix_nngraph(nside[, cheb_normalized, …])

Build the nearest neighbour graph of a HEALPix spherical point set.

cvxhull_graph(R: numpy.ndarray, cheb_normalized: bool = True, compute_differential_operator: bool = True) → Tuple[pygsp.graphs.graph.Graph, float][source]¶

Build the convex hull graph of a point set in \(\mathbb{R}^3\).

The graph edges have exponential-decay weighting.

Definitions of the graph Laplacians:

\[L = I - D^{-1/2} W D^{-1/2},\qquad L_{n} = (2 / \mu_{\max}) L - I\]

Parameters

R (ndarray) – (N,3) Cartesian coordinates of point set with size N. All points must be distinct.
cheb_normalized (bool) – Rescale Laplacian spectrum to [-1, 1].
compute_differential_operator (bool) – Computes the graph gradient.

Returns

G (Graph) – If cheb_normalized = True, G.Ln is created (Chebyshev Laplacian \(L_{n}\) above) If compute_differential_operator = True, G.D is created and contains the gradient.
rho (float) – Scale parameter \(\rho\) corresponding to the average distance of a point on the graph to its nearest neighbors.

Examples

import numpy as np
from pycgsp.graph import cvxhull_graph
from pygsp.plotting import plot_graph
theta, phi = np.linspace(0,np.pi,6, endpoint=False)[1:], np.linspace(0,2*np.pi,9, endpoint=False)
theta, phi = np.meshgrid(theta, phi)
x,y,z = np.cos(phi)*np.sin(theta), np.sin(phi)*np.sin(theta), np.cos(theta)
R = np.stack((x.flatten(), y.flatten(), z.flatten()), axis=-1)
G, _ = cvxhull_graph(R)
plot_graph(G)

(Source code, png, hires.png, pdf)

Warning

In the newest version of PyGSP (> 0.5.1) the convention is changed: Graph.D is the divergence operator and Graph.D.transpose() the gradient (see routine Graph.compute_differential_operator). The code should be adapted when this new version is released.

healpix_nngraph(nside: int, cheb_normalized: bool = True, compute_differential_operator: bool = True) → Tuple[pygsp.graphs.graph.Graph, float][source]¶

Build the nearest neighbour graph of a HEALPix spherical point set.