Add weighted Delaunay triangulation

cechmate/filtrations/alpha.py

@@ -4,6 +4,7 @@
 import numpy as np
 import numpy.linalg as linalg
 from scipy import spatial
+from scipy.spatial import ConvexHull
 
 from .base import BaseFiltration
 
@@ -26,16 +27,21 @@ class Alpha(BaseFiltration):
 
  """
 
- def build(self, X):
+ def build(self, X, weights=[]):
  """
  Do the Alpha filtration of a Euclidean point set (requires scipy)
 
  Parameters
  ===========
  X: Nxd array
  Array of N Euclidean vectors in d dimensions
+ weights : Nx1 array
+ Array of weights (default is non-weighted)
+
  """
-
+ self.weights = weights
+
+
  if X.shape[0] < X.shape[1]:
  warnings.warn(
  "The input point cloud has more columns than rows; "
@@ -47,14 +53,17 @@ def build(self, X):
 
  ## Step 1: Figure out the filtration
  if self.verbose:
- print("Doing spatial.Delaunay triangulation...")
+ print("Doing Delaunay triangulation...")
  tic = time.time()
-
- delaunay_faces = spatial.Delaunay(X).simplices
+
+ if not np.shape(self.weights)[0]==0:
+ delaunay_faces = self._weighted_delaunay(X)
+ else:
+ delaunay_faces = spatial.Delaunay(X).simplices
 
  if self.verbose:
  print(
- "Finished spatial.Delaunay triangulation (Elapsed Time %.3g)"
+ "Finished Delaunay triangulation (Elapsed Time %.3g)"
  % (time.time() - tic)
  )
  print("Building alpha filtration...")
@@ -169,3 +178,46 @@ def _get_circumcenter(self, X):
  x = x.dot(V.T) + muV
  return (x, rSqr)
  return (np.inf, np.inf) # SC2 (Points not in general position)
+
+ def _weighted_delaunay(self, X):
+ """
+ Computes a weighted delaunay triangulation using the duality of delaunay triangulation with a convex hull of a dimension higher
+ This function is triggered if a weight function is input to build the Alpha filtration 
+ and replaces the delaunay triangulation of scipy.
+
+ Returns
+ -------
+ Triangular faces from a weighted delaunay triangulation in the form of a numpy array. 
+ Similar to the output from scipy.spatial.Delaunay
+
+ """
+ num, dim = np.shape(X)
+
+ lifted = np.zeros((num,dim+1))
+
+ for i in range(num):
+ p = X[i,:]
+ lifted[i,:] = np.append(p,np.sum(p**2) - self.weights[i]**2)
+ pinf = np.append(np.zeros((1,dim)),1e10); 
+ lifted = np.vstack((lifted, pinf))
+ hull = ConvexHull(lifted)
+ delaunay = []
+ for simplex in hull.simplices:
+ if num not in simplex:
+ delaunay.append(simplex.tolist())
+
+ return np.asarray(delaunay)
+
+
+
+
+
+
+
+
+
+
+
+
+
+

docs/notebooks/BasicUsage.ipynb

@@ -331,6 +331,27 @@
  "print(\"H1:\\n\", dgms[1])"
  ]
  },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Weighted Alpha filtrations\n",
+ "He we showcase an usage of Alpha filtrations when the points have some weights. \n",
+ "This is done by computing a weighted delaunay triangulation and changing the circum radius accordingly for Alpha filtratiion computation."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "num_points = 10\n",
+ "points = np.random.rand(num_points, 2)\n",
+ "weights = np.random.rand(num_points,1)\n",
+ "filtration = alpha.build(points, weights=weights)"
+ ]
+ },
  {
  "cell_type": "code",
  "execution_count": null,
@@ -355,7 +376,7 @@
  "name": "python",
  "nbconvert_exporter": "python",
  "pygments_lexer": "ipython3",
- "version": "3.7.1"
+ "version": "3.7.3"
  }
  },
  "nbformat": 4,