Newer
Older
# # ls_ellipsoid and polyToParams3D functions are taken and adapted from
# http://www.juddzone.com/ALGORITHMS/least_squares_3D_ellipsoid.html
# # FindBestFittingEllipsoid3D and GetConvexHull3D are adapted from a discussion at
# https://stackoverflow.com/questions/58501545/python-fit-3d-ellipsoid-oblate-prolate-to-3d-points
from scipy.spatial import ConvexHull
from numpy.linalg import eig, inv
from brick.general.type import site_h, array_t
from brick.component.soma import soma_t
def ls_ellipsoid(xx: array_t, yy: array_t, zz: array_t) -> array_t: # finds the ellipsoid
"""
least squares fit to a 3D-ellipsoid
Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz = 1
Note that sometimes it is expressed as a solution to
Ax^2 + By^2 + Cz^2 + 2Dxy + 2Exz + 2Fyz + 2Gx + 2Hy + 2Iz = 1
where the last six terms have a factor of 2 in them
This is in anticipation of forming a matrix with the polynomial coefficients.
Those terms with factors of 2 are all off diagonal elements. These contribute
two terms when multiplied out (symmetric) so would need to be divided by two
"""
# change xx from vector of length N to Nx1 matrix so we can use hstack
x = xx[:, np_.newaxis]
y = yy[:, np_.newaxis]
z = zz[:, np_.newaxis]
# Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz = 1
J = np_.hstack((x * x, y * y, z * z, x * y, x * z, y * z, x, y, z))
K = np_.ones_like(x) # column of ones
# np.hstack performs a loop over all samples and creates
# a row in J for each x,y,z sample:
# J[ix,0] = x[ix]*x[ix]
# J[ix,1] = y[ix]*y[ix]
# etc.
JT = J.transpose()
try:
InvJTJ = np_.linalg.inv(JTJ);
ABC = np_.dot(InvJTJ, np_.dot(JT, K))
# Rearrange, move the 1 to the other side
# Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz - 1 = 0
# or
# Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0
# where J = -1
ellipsoid_coef = np_.append(ABC, -1)
return (ellipsoid_coef)
except:
print("Singular matrix, cannot find the best fitting ellipsoid.")
return (np_.zeros(9))
def polyToParams3D(ellipsoid_coef: array_t, printMe: bool = False) -> tuple:
"""
gets 3D parameters of an ellipsoid.
convert the polynomial form of the 3D-ellipsoid to parameters
center, axes, and transformation matrix
vec is the vector whose elements are the polynomial
coefficients A..J
returns (center, axes, rotation matrix)
Algebraic form: X.T * Amat * X --> polynomial form
"""
if printMe:
print('\npolynomial\n', ellipsoid_coef)
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
139
140
141
142
143
144
145
146
147
148
149
if ellipsoid_coef == (np_.zeros(9)):
return "NaN", "NaN", "NaN", "NaN"
else:
Amat = np_.array(
[
[ellipsoid_coef[0], ellipsoid_coef[3] / 2.0, ellipsoid_coef[4] / 2.0, ellipsoid_coef[6] / 2.0],
[ellipsoid_coef[3] / 2.0, ellipsoid_coef[1], ellipsoid_coef[5] / 2.0, ellipsoid_coef[7] / 2.0],
[ellipsoid_coef[4] / 2.0, ellipsoid_coef[5] / 2.0, ellipsoid_coef[2], ellipsoid_coef[8] / 2.0],
[ellipsoid_coef[6] / 2.0, ellipsoid_coef[7] / 2.0, ellipsoid_coef[8] / 2.0, ellipsoid_coef[9]]
])
if printMe:
print('\nAlgebraic form of polynomial\n', Amat)
# See B.Bartoni, Preprint SMU-HEP-10-14 Multi-dimensional Ellipsoidal Fitting
# equation 20 for the following method for finding the center
A3 = Amat[0:3, 0:3]
A3inv = inv(A3)
ofs = ellipsoid_coef[6:9] / 2.0
center = -np_.dot(A3inv, ofs)
if printMe:
print('\nCenter at:', center)
# Center the ellipsoid at the origin
Tofs = np_.eye(4)
Tofs[3, 0:3] = center
R = np_.dot(Tofs, np_.dot(Amat, Tofs.T))
if printMe:
print('\nAlgebraic form translated to center\n', R, '\n')
R3 = R[0:3, 0:3]
s1 = -R[3, 3]
R3S = R3 / s1
(el, ec) = eig(R3S)
el = np_.abs(el)
# Sorting in descending order of the eigenvalues and eigenvectors
sorting_vec = np_.argsort(-el)
el = el[sorting_vec]
ec = ec[:, sorting_vec]
# Calculating cartesian axes
recip = 1.0 / el
axes = np_.sqrt(recip) # axes are in ascending order
if printMe:
print('\nAxes are\n', axes, '\n')
# Calculating the orientation
inverse = inv(ec) # inverse is actually the transpose here
if printMe:
print('\nRotation matrix\n', inverse)
# Calculating spherical axes
# # eigenvector 1
r1 = np_.sqrt(sum(axe**2 for axe in ec[:, 0]))
phi1 = np_.arctan2(ec[1, 0], ec[0, 0])
theta1 = np_.arccos(ec[2, 0] / r1)
spherical_axes1 = (r1, theta1, phi1)
# # eigenvector 2
r2 = np_.sqrt(sum(axe**2 for axe in ec[:, 1]))
phi2 = np_.arctan2(ec[1, 1], ec[0, 1])
theta2 = np_.arccos(ec[2, 1] / r2)
spherical_axes2 = (r2, theta2, phi2)
spherical_axes = (spherical_axes1, spherical_axes2)
if printMe:
print('\nAxes in spherical coord are\n', spherical_axes, '\n')
return center, axes, inverse, spherical_axes
def GetConvexHull3D(soma_sites: site_h) -> tuple:
soma_3D = np_.stack((soma_sites[0], soma_sites[1], soma_sites[2]), axis=-1)
hull_3D = ConvexHull(soma_3D)
volume_hull = hull_3D.volume
len_hull = len(hull_3D.vertices)
hull = np_.zeros((len_hull, 3))
for i in range(len(hull_3D.vertices)):
hull[i] = soma_3D[hull_3D.vertices[i]]
convex_hull = np_.transpose(hull)
return convex_hull, volume_hull
def FindBestFittingEllipsoid3D(soma: soma_t) -> tuple:
"""
Find the best fitting ellipsoid for the data points in 3D based on their convex hull.
Return Tuple[ellipsoid coefficients, ellipsoid center, ellipsoid axes, ellipsoid orientation]
"""
# get convex hull
convex_hull, volume_hull = GetConvexHull3D(soma.sites)
# fit ellipsoid on the convex hull
# # get ellipsoid polynomial coefficients
ellipsoid_coef = ls_ellipsoid(convex_hull[0], convex_hull[1], convex_hull[2])
if ellipsoid_coef == (np_.zeros(9)):
return ellipsoid_coef, "NaN", "NaN", "NaN", "NaN", "NaN", "NaN"
else:
# # get ellipsoid 3D parameters
center, axes, orientation, spherical_coor = polyToParams3D(ellipsoid_coef, False)
return ellipsoid_coef, center, axes, orientation, spherical_coor, convex_hull, volume_hull