# Example 2 Clustering Geometric Objects¶

In this example we will look at a few of the tools provided by the clifford package for (4,1) conformal geometric algebra (CGA) and see how we can use them in a practical setting to cluster geometric objects via the simple K-means clustering algorithm provided in clifford.tools

As before the first step in using the package for CGA is to generate and import the algebra:

:

from clifford.g3c import *
print('e1*e1 ', e1*e1)
print('e2*e2 ', e2*e2)
print('e3*e3 ', e3*e3)
print('e4*e4 ', e4*e4)
print('e5*e5 ', e5*e5)

e1*e1  1
e2*e2  1
e3*e3  1
e4*e4  1
e5*e5  -1


The tools submodule of the clifford package contains a wide array of algorithms and tools that can be useful for manipulating objects in CGA. In this case we will be generating a large number of objects and then segmenting them into clusters.

We first need an algorithm for generating a cluster of objects in space. We will construct this cluster by generating a random object and then repeatedly disturbing this object by some small fixed amount and storing the result:

:

from clifford.tools.g3c import *
import numpy as np

def generate_random_object_cluster(n_objects, object_generator, max_cluster_trans=1.0, max_cluster_rot=np.pi/8):
""" Creates a cluster of random objects """
ref_obj = object_generator()
cluster_objects = []
for i in range(n_objects):
r = random_rotation_translation_rotor(maximum_translation=max_cluster_trans, maximum_angle=max_cluster_rot)
new_obj = apply_rotor(ref_obj, r)
cluster_objects.append(new_obj)
return cluster_objects

Failed to import cef_gui, cef functions will be unavailable


We can use this function to create a cluster and then we can visualise this cluster with pyganja.

:

from pyganja import *
clustered_circles = generate_random_object_cluster(10, random_circle)
sc = GanjaScene()
for c in clustered_circles:
draw(sc, scale=0.05)


This cluster generation function appears in clifford tools by default and it can be imported as follows:

:

from clifford.tools.g3c import generate_random_object_cluster


Now that we can generate individual clusters we would like to generate many:

:

def generate_n_clusters( object_generator, n_clusters, n_objects_per_cluster ):
object_clusters = []
for i in range(n_clusters):
cluster_objects = generate_random_object_cluster(n_objects_per_cluster, object_generator,
max_cluster_trans=0.5, max_cluster_rot=np.pi / 16)
object_clusters.append(cluster_objects)
all_objects = [item for sublist in object_clusters for item in sublist]
return all_objects, object_clusters


Again this function appears by default in clifford tools and we can easily visualise the result:

:

from clifford.tools.g3c import generate_n_clusters

all_objects, object_clusters = generate_n_clusters(random_circle, 2, 5)
sc = GanjaScene()
for c in all_objects:
draw(sc, scale=0.05)


Given that we can now generate multiple clusters of objects we can test algorithms for segmenting them.

The function run_n_clusters below generates a lot of objects distributed into n clusters and then attempts to segment the objects to recover the clusters.

:

from clifford.tools.g3c.object_clustering import n_clusters_objects
import time

def run_n_clusters( object_generator, n_clusters, n_objects_per_cluster, n_shotgunning):
all_objects, object_clusters = generate_n_clusters( object_generator, n_clusters, n_objects_per_cluster )
[new_labels, centroids, start_labels, start_centroids] = n_clusters_objects(n_clusters, all_objects,
initial_centroids=None,
n_shotgunning=n_shotgunning,
averaging_method='unweighted')
return all_objects, new_labels, centroids


Lets try it!

:

def visualise_n_clusters(all_objects, centroids, labels,
color_1=np.array([255, 0, 0]),
color_2=np.array([0, 255, 0])):
"""
Utility method for visualising several clusters and their respective centroids
using pyganja
"""
alpha_list = np.linspace(0, 1, num=len(centroids))
sc = GanjaScene()
for ind, this_obj in enumerate(all_objects):
alpha = alpha_list[labels[ind]]
cluster_color = (alpha * color_1 + (1 - alpha) * color_2)

for c in centroids:

return sc

object_generator = random_circle

n_clusters = 3
n_objects_per_cluster = 10
n_shotgunning = 60
all_objects, labels, centroids = run_n_clusters(object_generator, n_clusters,
n_objects_per_cluster, n_shotgunning)

sc = visualise_n_clusters(all_objects, centroids, labels,
color_1=np.array([255, 0, 0]),
color_2=np.array([0, 255, 0]))
draw(sc, scale=0.05)

/home/docs/checkouts/readthedocs.org/user_builds/clifford/conda/v1.1.1/lib/python3.7/site-packages/numba/ir_utils.py:1969: NumbaPendingDeprecationWarning:
Encountered the use of a type that is scheduled for deprecation: type 'reflected list' found for argument 'gradeList' of function 'grade_obj_func'.