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:
[1]:
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:
[2]:
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
/home/docs/checkouts/readthedocs.org/user_builds/clifford/conda/v1.2.0/lib/python3.7/site-packages/pyganja/__init__.py:2: UserWarning: Failed to import cef_gui, cef functions will be unavailable
from .script_api import *
We can use this function to create a cluster and then we can visualise this cluster with pyganja.
[3]:
from pyganja import *
clustered_circles = generate_random_object_cluster(10, random_circle)
sc = GanjaScene()
for c in clustered_circles:
sc.add_object(c, rgb2hex([255,0,0]))
draw(sc, scale=0.05)
This cluster generation function appears in clifford tools by default and it can be imported as follows:
[4]:
from clifford.tools.g3c import generate_random_object_cluster
Now that we can generate individual clusters we would like to generate many:
[5]:
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:
[6]:
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:
sc.add_object(c, rgb2hex([255,0,0]))
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.
[7]:
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!
[8]:
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)
sc.add_object(this_obj, rgb2hex(cluster_color))
for c in centroids:
sc.add_object(c, Color.BLACK)
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)
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-8-a9efef63933e> in <module>
30 sc = visualise_n_clusters(all_objects, centroids, labels,
31 color_1=np.array([255, 0, 0]),
---> 32 color_2=np.array([0, 255, 0]))
33 draw(sc, scale=0.05)
<ipython-input-8-a9efef63933e> in visualise_n_clusters(all_objects, centroids, labels, color_1, color_2)
11 alpha = alpha_list[labels[ind]]
12 cluster_color = (alpha * color_1 + (1 - alpha) * color_2)
---> 13 sc.add_object(this_obj, rgb2hex(cluster_color))
14
15 for c in centroids:
~/checkouts/readthedocs.org/user_builds/clifford/conda/v1.2.0/lib/python3.7/site-packages/pyganja/color.py in rgb2hex(x)
7 val = 0
8 for xi in x:
----> 9 val = (val << 8) | _entry_as_uint8(xi)
10 return val
11 else:
~/checkouts/readthedocs.org/user_builds/clifford/conda/v1.2.0/lib/python3.7/site-packages/pyganja/color.py in _entry_as_uint8(v)
22 return operator.index(v)
23 else:
---> 24 raise TypeError("Can't convert {!r} to a color component".format(v))
25
26
TypeError: Can't convert 127.5 to a color component