/
ellipsoid.py
138 lines (109 loc) · 4.48 KB
/
ellipsoid.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
#!/usr/bin/python
from __future__ import division
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
import sys
import numpy as np
from numpy import linalg
from random import random
class EllipsoidTool:
"""Some stuff for playing with ellipsoids"""
def __init__(self): pass
def getMinVolEllipse(self, P=None, tolerance=0.01):
""" Find the minimum volume ellipsoid which holds all the points
Based on work by Nima Moshtagh
http://www.mathworks.com/matlabcentral/fileexchange/9542
and also by looking at:
http://cctbx.sourceforge.net/current/python/scitbx.math.minimum_covering_ellipsoid.html
Which is based on the first reference anyway!
Here, P is a numpy array of N dimensional points like this:
P = [[x,y,z,...], <-- one point per line
[x,y,z,...],
[x,y,z,...]]
Returns:
(center, radii, rotation)
"""
(N, d) = np.shape(P)
d = float(d)
# Q will be our working array
Q = np.vstack([np.copy(P.T), np.ones(N)])
QT = Q.T
# initializations
err = 1.0 + tolerance
u = (1.0 / N) * np.ones(N)
# Khachiyan Algorithm
while err > tolerance:
V = np.dot(Q, np.dot(np.diag(u), QT))
M = np.diag(np.dot(QT , np.dot(linalg.inv(V), Q))) # M the diagonal vector of an NxN matrix
j = np.argmax(M)
maximum = M[j]
step_size = (maximum - d - 1.0) / ((d + 1.0) * (maximum - 1.0))
new_u = (1.0 - step_size) * u
new_u[j] += step_size
err = np.linalg.norm(new_u - u)
u = new_u
# center of the ellipse
center = np.dot(P.T, u)
# the A matrix for the ellipse
A = linalg.inv(
np.dot(P.T, np.dot(np.diag(u), P)) -
np.array([[a * b for b in center] for a in center])
) / d
# Get the values we'd like to return
U, s, rotation = linalg.svd(A)
radii = 1.0/np.sqrt(s)
return (center, radii, rotation)
def getEllipsoidVolume(self, radii):
"""Calculate the volume of the blob"""
return 4./3.*np.pi*radii[0]*radii[1]*radii[2]
def plotEllipsoid(self, center, radii, rotation, ax=None, plotAxes=False, cageColor='b', cageAlpha=0.2):
"""Plot an ellipsoid"""
make_ax = ax == None
if make_ax:
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
u = np.linspace(0.0, 2.0 * np.pi, 100)
v = np.linspace(0.0, np.pi, 100)
# cartesian coordinates that correspond to the spherical angles:
x = radii[0] * np.outer(np.cos(u), np.sin(v))
y = radii[1] * np.outer(np.sin(u), np.sin(v))
z = radii[2] * np.outer(np.ones_like(u), np.cos(v))
# rotate accordingly
for i in range(len(x)):
for j in range(len(x)):
[x[i,j],y[i,j],z[i,j]] = np.dot([x[i,j],y[i,j],z[i,j]], rotation) + center
if plotAxes:
# make some purdy axes
axes = np.array([[radii[0],0.0,0.0],
[0.0,radii[1],0.0],
[0.0,0.0,radii[2]]])
# rotate accordingly
for i in range(len(axes)):
axes[i] = np.dot(axes[i], rotation)
# plot axes
for p in axes:
X3 = np.linspace(-p[0], p[0], 100) + center[0]
Y3 = np.linspace(-p[1], p[1], 100) + center[1]
Z3 = np.linspace(-p[2], p[2], 100) + center[2]
ax.plot(X3, Y3, Z3, color=cageColor)
# plot ellipsoid
ax.plot_wireframe(x, y, z, rstride=4, cstride=4, color=cageColor, alpha=cageAlpha)
if make_ax:
plt.show()
plt.close(fig)
del fig
if __name__ == "__main__":
# make 100 random points
P = np.reshape([random()*100 for i in range(300)],(100,3))
# find the ellipsoid
ET = EllipsoidTool()
(center, radii, rotation) = ET.getMinVolEllipse(P, .01)
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
# plot points
ax.scatter(P[:,0], P[:,1], P[:,2], color='g', marker='*', s=100)
# plot ellipsoid
ET.plotEllipsoid(center, radii, rotation, ax=ax, plotAxes=True)
plt.show()
plt.close(fig)
del fig