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ellipsoid.py
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ellipsoid.py
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import numpy as np
from numpy import linalg
from scipy import optimize
from random import random
from geom import Ellipse, Point, Vector2, Circle
class EnclosingEllipsoid:
'''
reference:
https://github.com/minillinim/ellipsoid
'''
def getMinVolEllipse(self, P=None, tolerance=0.0001):
""" 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)
V1 = Vector2(rotation[0][0], rotation[0][1])
V2 = Vector2(rotation[1][0], rotation[1][1])
return Ellipse(Point(center), radii, [V1, V2])
class LSqEllipse:
'''
reference : https://github.com/bdhammel/least-squares-ellipse-fitting
'''
def fit(self, data):
x, y = np.asarray(data, dtype=float)
cx, cy = x.mean(), y.mean()
x = x - cx
y = y - cy
#Quadratic part of design matrix [eqn. 15] from (*)
D1 = np.mat(np.vstack([x**2, x*y, y**2])).T
#Linear part of design matrix [eqn. 16] from (*)
D2 = np.mat(np.vstack([x, y, np.ones(len(x))])).T
#forming scatter matrix [eqn. 17] from (*)
S1 = D1.T*D1
S2 = D1.T*D2
S3 = D2.T*D2
#Constraint matrix [eqn. 18]
C1 = np.mat('0. 0. 2.; 0. -1. 0.; 2. 0. 0.')
#Reduced scatter matrix [eqn. 29]
M=C1.I*(S1-S2*S3.I*S2.T)
#M*|a b c >=l|a b c >. Find eigenvalues and eigenvectors from this equation [eqn. 28]
eval, evec = np.linalg.eig(M)
# eigenvector must meet constraint 4ac - b^2 to be valid.
cond = 4*np.multiply(evec[0, :], evec[2, :]) - np.power(evec[1, :], 2)
a1 = evec[:, np.nonzero(cond.A > 0)[1]]
#|d f g> = -S3^(-1)*S2^(T)*|a b c> [eqn. 24]
a2 = -S3.I*S2.T*a1
# eigenvectors |a b c d f g>
self.coef = np.vstack([a1, a2])
#eigenvectors are the coefficients of an ellipse in general form
#a*x^2 + 2*b*x*y + c*y^2 + 2*d*x + 2*f*y + g = 0 [eqn. 15) from (**) or (***)
a = self.coef[0,0]
b = self.coef[1,0]/2.
c = self.coef[2,0]
d = self.coef[3,0]/2.
f = self.coef[4,0]/2.
g = self.coef[5,0]
#finding center of ellipse [eqn.19 and 20] from (**)
x0 = (c*d-b*f)/(b**2.-a*c)
y0 = (a*f-b*d)/(b**2.-a*c)
#Find the semi-axes lengths [eqn. 21 and 22] from (**)
numerator = 2*(a*f*f+c*d*d+g*b*b-2*b*d*f-a*c*g)
denominator1 = (b*b-a*c)*( (c-a)*np.sqrt(1+4*b*b/((a-c)*(a-c)))-(c+a))
denominator2 = (b*b-a*c)*( (a-c)*np.sqrt(1+4*b*b/((a-c)*(a-c)))-(c+a))
width = np.sqrt(numerator/denominator1)
height = np.sqrt(numerator/denominator2)
# angle of counterclockwise rotation of major-axis of ellipse to x-axis [eqn. 23] from (**)
# or [eqn. 26] from (***).
phi = .5*np.arctan((2.*b)/(a-c))
V1 = Vector2(1, np.tan(phi)).normalize()
V2 = Vector2(-V1.y, V1.x)
x0 += cx
y0 += cy
return Ellipse(Point(x0, y0),[width, height],[V1, V2])
class LSqCircle:
'''
reference :
[1]https://scipy-cookbook.readthedocs.io/items/Least_Squares_Circle.html
[2]https://gist.github.com/lorenzoriano/6799568
'''
def fit(self, data):
x_m = np.mean(data[0])
y_m = np.mean(data[1])
center_estimate = x_m, y_m
center, ier = optimize.leastsq(self._f, center_estimate, args=(data[0],data[1]))
xc, yc = center
Ri = self._calc_R(data[0], data[1], *center)
R = Ri.mean()
residu = np.sum((Ri - R)**2)
return Circle(Point(xc, yc),R)
def _calc_R(self, x,y, xc, yc):
""" calculate the distance of each 2D points from the center (xc, yc) """
return np.sqrt((x-xc)**2 + (y-yc)**2)
def _f(self, c, x, y):
""" calculate the algebraic distance between the data points and the mean circle centered at c=(xc, yc) """
Ri = self._calc_R(x, y, *c)
return Ri - Ri.mean()
ECELL = EnclosingEllipsoid()
LSELL = LSqEllipse()
LSCIR = LSqCircle()