import typing as ty
from dataclasses import dataclass
from enum import Enum
import numba as nb
import numpy as np
[docs]@nb.njit
def nb_mean_axis_0(array: np.ndarray) -> np.ndarray:
"""
Same as np.mean(array, axis=0) but njitted
"""
mean_array = np.zeros(array.shape[1])
for i in range(array.shape[1]):
mean_array[i] = np.mean(array[:, i])
return mean_array
[docs]@dataclass
class MomentInfo:
moment_function: ty.Callable[[int, int, int, np.ndarray, np.ndarray], float]
mu_arguments: ty.List[ty.Tuple[int, int, int]]
[docs]@nb.njit(cache=False)
def mu(p, q, r, coords, centroid):
"""
Central moment
"""
return np.sum(
((coords[:, 0] - centroid[0]) ** p)
* ((coords[:, 1] - centroid[1]) ** q)
* ((coords[:, 2] - centroid[2]) ** r)
)
[docs]@nb.njit
def O_3(mu_200, mu_020, mu_002):
return mu_200 + mu_020 + mu_002
[docs]@nb.njit
def O_4(mu_200, mu_020, mu_002, mu_110, mu_101, mu_011):
return (
mu_200 * mu_020 * mu_002
+ 2 * mu_110 * mu_101 * mu_011
- mu_002 * mu_110 ** 2
- mu_020 * mu_101 ** 2
- mu_200 * mu_011 ** 2
)
[docs]@nb.njit
def O_5(mu_200, mu_020, mu_002, mu_110, mu_101, mu_011):
return (
mu_200 * mu_020
+ mu_200 * mu_002
+ mu_020 * mu_002
- mu_110 ** 2
- mu_101 ** 2
- mu_011 ** 2
)
[docs]@nb.njit
def F(
mu_201, mu_021, mu_210, mu_300, mu_111, mu_012, mu_003, mu_030, mu_102, mu_120,
):
return (
mu_003 ** 2
+ 6 * mu_012 ** 2
+ 6 * mu_021 ** 2
+ mu_030 ** 2
+ 6 * mu_102 ** 2
+ 15 * mu_111 ** 2
- 3 * mu_102 * mu_120
+ 6 * mu_120 ** 2
- 3 * mu_021 * mu_201
+ 6 * mu_201 ** 2
- 3 * mu_003 * (mu_021 + mu_201)
- 3 * mu_030 * mu_210
+ 6 * mu_210 ** 2
- 3 * mu_012 * (mu_030 + mu_210)
- 3 * mu_102 * mu_300
- 3 * mu_120 * mu_300
+ mu_300 ** 2
)
[docs]@nb.njit
def phi_2(mu_020, mu_011, mu_110, mu_200, mu_002, mu_101):
return (
mu_200 ** 2
+ mu_020 ** 2
+ mu_002 ** 2
+ 2 * mu_110 ** 2
+ 2 * mu_101 ** 2
+ 2 * mu_011 ** 2
)
[docs]@nb.njit
def phi_3(mu_020, mu_011, mu_110, mu_200, mu_002, mu_101):
return (
mu_200 ** 3
+ 3 * mu_200 * mu_110 ** 2
+ 3 * mu_200 * mu_101 ** 2
+ 3 * mu_110 ** 2 * mu_020
+ 3 * mu_101 ** 2 * mu_002
+ mu_020 ** 3
+ 3 * mu_020 * mu_011 ** 2
+ 3 * mu_011 ** 2 * mu_002
+ mu_002 ** 3
+ 6 * mu_110 * mu_101 * mu_011
)
[docs]@nb.njit
def phi_4(
mu_030, mu_021, mu_120, mu_003, mu_111, mu_201, mu_102, mu_210, mu_012, mu_300,
):
return (
mu_300 ** 2
+ mu_030 ** 2
+ mu_003 ** 2
+ 3 * mu_210 ** 2
+ 3 * mu_201 ** 2
+ 3 * mu_120 ** 2
+ 3 * mu_102 ** 2
+ 3 * mu_021 ** 2
+ 3 * mu_012 ** 2
+ 6 * mu_111 ** 2
)
[docs]@nb.njit
def phi_5(mu_030, mu_021, mu_120, mu_003, mu_201, mu_102, mu_210, mu_012, mu_300):
return (
mu_300 ** 2
+ 2 * mu_300 * mu_120
+ 2 * mu_300 * mu_102
+ 2 * mu_210 * mu_030
+ 2 * mu_201 * mu_003
+ mu_030 ** 2
+ 2 * mu_030 * mu_012
+ 2 * mu_021 * mu_003
+ mu_003 ** 2
+ mu_210 ** 2
+ 2 * mu_210 * mu_012
+ mu_201 ** 2
+ 2 * mu_201 * mu_021
+ mu_120 ** 2
+ 2 * mu_120 * mu_102
+ mu_102 ** 2
+ mu_021 ** 2
+ mu_012 ** 2
)
[docs]@nb.njit
def phi_6(
mu_030, mu_021, mu_120, mu_003, mu_111, mu_201, mu_102, mu_210, mu_012, mu_300,
):
return (
1 * mu_300 ** 4
+ 6 * mu_300 ** 2 * mu_210 ** 2
+ 6 * mu_300 ** 2 * mu_201 ** 2
+ 2 * mu_300 ** 2 * mu_120 ** 2
+ 4 * mu_300 ** 2 * mu_111 ** 2
+ 2 * mu_300 ** 2 * mu_102 ** 2
+ 8 * mu_300 * mu_210 ** 2 * mu_120
+ 16 * mu_300 * mu_210 * mu_201 * mu_111
+ 4 * mu_300 * mu_210 * mu_120 * mu_030
+ 8 * mu_300 * mu_210 * mu_111 * mu_021
+ 4 * mu_300 * mu_210 * mu_102 * mu_012
+ 8 * mu_300 * mu_201 ** 2 * mu_102
+ 4 * mu_300 * mu_201 * mu_120 * mu_021
+ 8 * mu_300 * mu_201 * mu_111 * mu_012
+ 4 * mu_300 * mu_201 * mu_102 * mu_003
+ 2 * mu_210 ** 2 * mu_030 ** 2
+ 4 * mu_210 * mu_201 * mu_030 * mu_021
+ 4 * mu_210 * mu_201 * mu_012 * mu_003
+ 8 * mu_210 * mu_120 ** 2 * mu_030
+ 8 * mu_210 * mu_111 * mu_102 * mu_003
+ 2 * mu_201 ** 2 * mu_003 ** 2
+ 8 * mu_201 * mu_120 * mu_111 * mu_030
+ 8 * mu_201 * mu_102 ** 2 * mu_003
+ 6 * mu_120 ** 2 * mu_030 ** 2
+ 16 * mu_120 * mu_111 * mu_030 * mu_021
+ 8 * mu_120 * mu_111 * mu_012 * mu_003
+ 4 * mu_120 * mu_102 * mu_030 * mu_012
+ 4 * mu_120 * mu_102 * mu_021 * mu_003
+ 4 * mu_111 ** 2 * mu_030 ** 2
+ 4 * mu_111 ** 2 * mu_003 ** 2
+ 8 * mu_111 * mu_102 * mu_030 * mu_021
+ 16 * mu_111 * mu_102 * mu_012 * mu_003
+ 6 * mu_102 ** 2 * mu_003 ** 2
+ 1 * mu_030 ** 4
+ 6 * mu_030 ** 2 * mu_021 ** 2
+ 2 * mu_030 ** 2 * mu_012 ** 2
+ 8 * mu_030 * mu_021 ** 2 * mu_012
+ 4 * mu_030 * mu_021 * mu_012 * mu_003
+ 2 * mu_021 ** 2 * mu_003 ** 2
+ 8 * mu_021 * mu_012 ** 2 * mu_003
+ 6 * mu_012 ** 2 * mu_003 ** 2
+ 1 * mu_003 ** 4
+ 5 * mu_210 ** 4
+ 10 * mu_210 ** 2 * mu_201 ** 2
+ 16 * mu_210 ** 2 * mu_120 ** 2
+ 20 * mu_210 ** 2 * mu_111 ** 2
+ 4 * mu_210 ** 2 * mu_102 ** 2
+ 4 * mu_210 ** 2 * mu_021 ** 2
+ 2 * mu_210 ** 2 * mu_012 ** 2
+ 24 * mu_210 * mu_201 * mu_120 * mu_111
+ 24 * mu_210 * mu_201 * mu_111 * mu_102
+ 8 * mu_210 * mu_201 * mu_021 * mu_012
+ 24 * mu_210 * mu_120 * mu_111 * mu_021
+ 8 * mu_210 * mu_120 * mu_102 * mu_012
+ 16 * mu_210 * mu_111 ** 2 * mu_012
+ 5 * mu_201 ** 4
+ 4 * mu_201 ** 2 * mu_120 ** 2
+ 20 * mu_201 ** 2 * mu_111 ** 2
+ 16 * mu_201 ** 2 * mu_102 ** 2
+ 2 * mu_201 ** 2 * mu_021 ** 2
+ 4 * mu_201 ** 2 * mu_012 ** 2
+ 8 * mu_201 * mu_120 * mu_102 * mu_021
+ 16 * mu_201 * mu_111 ** 2 * mu_021
+ 24 * mu_201 * mu_111 * mu_102 * mu_012
+ 5 * mu_120 ** 4
+ 20 * mu_120 ** 2 * mu_111 ** 2
+ 2 * mu_120 ** 2 * mu_102 ** 2
+ 10 * mu_120 ** 2 * mu_021 ** 2
+ 4 * mu_120 ** 2 * mu_012 ** 2
+ 16 * mu_120 * mu_111 ** 2 * mu_102
+ 24 * mu_120 * mu_111 * mu_021 * mu_012
+ 20 * mu_111 ** 2 * mu_102 ** 2
+ 20 * mu_111 ** 2 * mu_021 ** 2
+ 20 * mu_111 ** 2 * mu_012 ** 2
+ 24 * mu_111 * mu_102 * mu_021 * mu_012
+ 5 * mu_102 ** 4
+ 4 * mu_102 ** 2 * mu_021 ** 2
+ 10 * mu_102 ** 2 * mu_012 ** 2
+ 5 * mu_021 ** 4
+ 16 * mu_021 ** 2 * mu_012 ** 2
+ 5 * mu_012 ** 4
+ 12 * mu_111 ** 4
)
[docs]@nb.njit
def phi_7(
mu_030, mu_021, mu_120, mu_003, mu_111, mu_201, mu_102, mu_210, mu_012, mu_300,
):
return (
1 * mu_300 ** 4
+ 1 * mu_300 ** 3 * mu_120
+ 1 * mu_300 ** 3 * mu_102
+ 5 * mu_300 ** 2 * mu_210 ** 2
+ 1 * mu_300 ** 2 * mu_210 * mu_030
+ 1 * mu_300 ** 2 * mu_210 * mu_012
+ 5 * mu_300 ** 2 * mu_201 ** 2
+ 1 * mu_300 ** 2 * mu_201 * mu_021
+ 1 * mu_300 ** 2 * mu_201 * mu_003
+ 1 * mu_300 ** 2 * mu_120 ** 2
+ 2 * mu_300 ** 2 * mu_111 ** 2
+ 1 * mu_300 ** 2 * mu_102 ** 2
+ 11 * mu_300 * mu_210 ** 2 * mu_120
+ 4 * mu_300 * mu_210 ** 2 * mu_102
+ 14 * mu_300 * mu_210 * mu_201 * mu_111
+ 4 * mu_300 * mu_210 * mu_120 * mu_030
+ 2 * mu_300 * mu_210 * mu_120 * mu_012
+ 6 * mu_300 * mu_210 * mu_111 * mu_021
+ 2 * mu_300 * mu_210 * mu_111 * mu_003
+ 2 * mu_300 * mu_210 * mu_102 * mu_012
+ 4 * mu_300 * mu_201 ** 2 * mu_120
+ 11 * mu_300 * mu_201 ** 2 * mu_102
+ 2 * mu_300 * mu_201 * mu_120 * mu_021
+ 2 * mu_300 * mu_201 * mu_111 * mu_030
+ 6 * mu_300 * mu_201 * mu_111 * mu_012
+ 2 * mu_300 * mu_201 * mu_102 * mu_021
+ 4 * mu_300 * mu_201 * mu_102 * mu_003
+ 3 * mu_300 * mu_120 ** 3
+ 1 * mu_300 * mu_120 ** 2 * mu_102
+ 8 * mu_300 * mu_120 * mu_111 ** 2
+ 1 * mu_300 * mu_120 * mu_102 ** 2
+ 1 * mu_300 * mu_120 * mu_030 ** 2
+ 2 * mu_300 * mu_120 * mu_021 ** 2
+ 1 * mu_300 * mu_120 * mu_012 ** 2
+ 8 * mu_300 * mu_111 ** 2 * mu_102
+ 2 * mu_300 * mu_111 * mu_030 * mu_021
+ 4 * mu_300 * mu_111 * mu_021 * mu_012
+ 2 * mu_300 * mu_111 * mu_012 * mu_003
+ 3 * mu_300 * mu_102 ** 3
+ 1 * mu_300 * mu_102 * mu_021 ** 2
+ 2 * mu_300 * mu_102 * mu_012 ** 2
+ 1 * mu_300 * mu_102 * mu_003 ** 2
+ 3 * mu_210 ** 3 * mu_030
+ 2 * mu_210 ** 2 * mu_201 * mu_003
+ 1 * mu_210 ** 2 * mu_030 ** 2
+ 1 * mu_210 ** 2 * mu_030 * mu_012
+ 1 * mu_210 ** 2 * mu_021 * mu_003
+ 2 * mu_210 * mu_201 ** 2 * mu_030
+ 2 * mu_210 * mu_201 * mu_030 * mu_021
+ 2 * mu_210 * mu_201 * mu_012 * mu_003
+ 11 * mu_210 * mu_120 ** 2 * mu_030
+ 4 * mu_210 * mu_120 * mu_111 * mu_003
+ 2 * mu_210 * mu_120 * mu_102 * mu_030
+ 8 * mu_210 * mu_111 ** 2 * mu_030
+ 6 * mu_210 * mu_111 * mu_102 * mu_003
+ 1 * mu_210 * mu_102 ** 2 * mu_030
+ 1 * mu_210 * mu_030 ** 3
+ 4 * mu_210 * mu_030 * mu_021 ** 2
+ 1 * mu_210 * mu_030 * mu_012 ** 2
+ 2 * mu_210 * mu_021 * mu_012 * mu_003
+ 1 * mu_210 * mu_012 * mu_003 ** 2
+ 3 * mu_201 ** 3 * mu_003
+ 1 * mu_201 ** 2 * mu_030 * mu_012
+ 1 * mu_201 ** 2 * mu_021 * mu_003
+ 1 * mu_201 ** 2 * mu_003 ** 2
+ 1 * mu_201 * mu_120 ** 2 * mu_003
+ 6 * mu_201 * mu_120 * mu_111 * mu_030
+ 2 * mu_201 * mu_120 * mu_102 * mu_003
+ 8 * mu_201 * mu_111 ** 2 * mu_003
+ 4 * mu_201 * mu_111 * mu_102 * mu_030
+ 11 * mu_201 * mu_102 ** 2 * mu_003
+ 1 * mu_201 * mu_030 ** 2 * mu_021
+ 2 * mu_201 * mu_030 * mu_021 * mu_012
+ 1 * mu_201 * mu_021 ** 2 * mu_003
+ 4 * mu_201 * mu_012 ** 2 * mu_003
+ 1 * mu_201 * mu_003 ** 3
+ 5 * mu_120 ** 2 * mu_030 ** 2
+ 4 * mu_120 ** 2 * mu_030 * mu_012
+ 2 * mu_120 ** 2 * mu_021 * mu_003
+ 14 * mu_120 * mu_111 * mu_030 * mu_021
+ 2 * mu_120 * mu_111 * mu_030 * mu_003
+ 6 * mu_120 * mu_111 * mu_012 * mu_003
+ 1 * mu_120 * mu_102 * mu_030 ** 2
+ 2 * mu_120 * mu_102 * mu_030 * mu_012
+ 2 * mu_120 * mu_102 * mu_021 * mu_003
+ 1 * mu_120 * mu_102 * mu_003 ** 2
+ 2 * mu_111 ** 2 * mu_030 ** 2
+ 8 * mu_111 ** 2 * mu_030 * mu_012
+ 8 * mu_111 ** 2 * mu_021 * mu_003
+ 2 * mu_111 ** 2 * mu_003 ** 2
+ 6 * mu_111 * mu_102 * mu_030 * mu_021
+ 2 * mu_111 * mu_102 * mu_030 * mu_003
+ 14 * mu_111 * mu_102 * mu_012 * mu_003
+ 2 * mu_102 ** 2 * mu_030 * mu_012
+ 4 * mu_102 ** 2 * mu_021 * mu_003
+ 5 * mu_102 ** 2 * mu_003 ** 2
+ 1 * mu_030 ** 4
+ 1 * mu_030 ** 3 * mu_012
+ 5 * mu_030 ** 2 * mu_021 ** 2
+ 1 * mu_030 ** 2 * mu_021 * mu_003
+ 1 * mu_030 ** 2 * mu_012 ** 2
+ 11 * mu_030 * mu_021 ** 2 * mu_012
+ 4 * mu_030 * mu_021 * mu_012 * mu_003
+ 3 * mu_030 * mu_012 ** 3
+ 1 * mu_030 * mu_012 * mu_003 ** 2
+ 3 * mu_021 ** 3 * mu_003
+ 1 * mu_021 ** 2 * mu_003 ** 2
+ 11 * mu_021 * mu_012 ** 2 * mu_003
+ 1 * mu_021 * mu_003 ** 3
+ 5 * mu_012 ** 2 * mu_003 ** 2
+ 1 * mu_003 ** 4
+ 2 * mu_210 ** 4
+ 2 * mu_210 ** 3 * mu_012
+ 4 * mu_210 ** 2 * mu_201 ** 2
+ 5 * mu_210 ** 2 * mu_201 * mu_021
+ 10 * mu_210 ** 2 * mu_120 ** 2
+ 5 * mu_210 ** 2 * mu_120 * mu_102
+ 6 * mu_210 ** 2 * mu_111 ** 2
+ 1 * mu_210 ** 2 * mu_102 ** 2
+ 1 * mu_210 ** 2 * mu_021 ** 2
+ 5 * mu_210 * mu_201 ** 2 * mu_012
+ 18 * mu_210 * mu_201 * mu_120 * mu_111
+ 18 * mu_210 * mu_201 * mu_111 * mu_102
+ 4 * mu_210 * mu_201 * mu_021 * mu_012
+ 5 * mu_210 * mu_120 ** 2 * mu_012
+ 18 * mu_210 * mu_120 * mu_111 * mu_021
+ 4 * mu_210 * mu_120 * mu_102 * mu_012
+ 12 * mu_210 * mu_111 ** 2 * mu_012
+ 12 * mu_210 * mu_111 * mu_102 * mu_021
+ 5 * mu_210 * mu_102 ** 2 * mu_012
+ 5 * mu_210 * mu_021 ** 2 * mu_012
+ 2 * mu_210 * mu_012 ** 3
+ 2 * mu_201 ** 4
+ 2 * mu_201 ** 3 * mu_021
+ 1 * mu_201 ** 2 * mu_120 ** 2
+ 5 * mu_201 ** 2 * mu_120 * mu_102
+ 6 * mu_201 ** 2 * mu_111 ** 2
+ 10 * mu_201 ** 2 * mu_102 ** 2
+ 1 * mu_201 ** 2 * mu_012 ** 2
+ 5 * mu_201 * mu_120 ** 2 * mu_021
+ 12 * mu_201 * mu_120 * mu_111 * mu_012
+ 4 * mu_201 * mu_120 * mu_102 * mu_021
+ 12 * mu_201 * mu_111 ** 2 * mu_021
+ 18 * mu_201 * mu_111 * mu_102 * mu_012
+ 5 * mu_201 * mu_102 ** 2 * mu_021
+ 2 * mu_201 * mu_021 ** 3
+ 5 * mu_201 * mu_021 * mu_012 ** 2
+ 2 * mu_120 ** 4
+ 2 * mu_120 ** 3 * mu_102
+ 6 * mu_120 ** 2 * mu_111 ** 2
+ 4 * mu_120 ** 2 * mu_021 ** 2
+ 1 * mu_120 ** 2 * mu_012 ** 2
+ 12 * mu_120 * mu_111 ** 2 * mu_102
+ 18 * mu_120 * mu_111 * mu_021 * mu_012
+ 2 * mu_120 * mu_102 ** 3
+ 5 * mu_120 * mu_102 * mu_021 ** 2
+ 5 * mu_120 * mu_102 * mu_012 ** 2
+ 6 * mu_111 ** 2 * mu_102 ** 2
+ 6 * mu_111 ** 2 * mu_021 ** 2
+ 6 * mu_111 ** 2 * mu_012 ** 2
+ 18 * mu_111 * mu_102 * mu_021 * mu_012
+ 2 * mu_102 ** 4
+ 1 * mu_102 ** 2 * mu_021 ** 2
+ 4 * mu_102 ** 2 * mu_012 ** 2
+ 2 * mu_021 ** 4
+ 10 * mu_021 ** 2 * mu_012 ** 2
+ 2 * mu_012 ** 4
)
[docs]@nb.njit
def phi_8(
mu_030, mu_021, mu_120, mu_003, mu_111, mu_201, mu_102, mu_210, mu_012, mu_300,
):
return (
1 * mu_300 ** 4
+ 2 * mu_300 ** 3 * mu_120
+ 2 * mu_300 ** 3 * mu_102
+ 4 * mu_300 ** 2 * mu_210 ** 2
+ 2 * mu_300 ** 2 * mu_210 * mu_030
+ 2 * mu_300 ** 2 * mu_210 * mu_012
+ 4 * mu_300 ** 2 * mu_201 ** 2
+ 2 * mu_300 ** 2 * mu_201 * mu_021
+ 2 * mu_300 ** 2 * mu_201 * mu_003
+ 2 * mu_300 ** 2 * mu_120 ** 2
+ 2 * mu_300 ** 2 * mu_120 * mu_102
+ 2 * mu_300 ** 2 * mu_111 ** 2
+ 2 * mu_300 ** 2 * mu_102 ** 2
+ 10 * mu_300 * mu_210 ** 2 * mu_120
+ 6 * mu_300 * mu_210 ** 2 * mu_102
+ 8 * mu_300 * mu_210 * mu_201 * mu_111
+ 8 * mu_300 * mu_210 * mu_120 * mu_030
+ 6 * mu_300 * mu_210 * mu_120 * mu_012
+ 8 * mu_300 * mu_210 * mu_111 * mu_021
+ 4 * mu_300 * mu_210 * mu_111 * mu_003
+ 2 * mu_300 * mu_210 * mu_102 * mu_030
+ 4 * mu_300 * mu_210 * mu_102 * mu_012
+ 6 * mu_300 * mu_201 ** 2 * mu_120
+ 10 * mu_300 * mu_201 ** 2 * mu_102
+ 4 * mu_300 * mu_201 * mu_120 * mu_021
+ 2 * mu_300 * mu_201 * mu_120 * mu_003
+ 4 * mu_300 * mu_201 * mu_111 * mu_030
+ 8 * mu_300 * mu_201 * mu_111 * mu_012
+ 6 * mu_300 * mu_201 * mu_102 * mu_021
+ 8 * mu_300 * mu_201 * mu_102 * mu_003
+ 2 * mu_300 * mu_120 ** 3
+ 2 * mu_300 * mu_120 ** 2 * mu_102
+ 4 * mu_300 * mu_120 * mu_111 ** 2
+ 2 * mu_300 * mu_120 * mu_102 ** 2
+ 2 * mu_300 * mu_120 * mu_030 ** 2
+ 2 * mu_300 * mu_120 * mu_030 * mu_012
+ 2 * mu_300 * mu_120 * mu_021 ** 2
+ 2 * mu_300 * mu_120 * mu_021 * mu_003
+ 4 * mu_300 * mu_111 ** 2 * mu_102
+ 4 * mu_300 * mu_111 * mu_030 * mu_021
+ 8 * mu_300 * mu_111 * mu_021 * mu_012
+ 4 * mu_300 * mu_111 * mu_012 * mu_003
+ 2 * mu_300 * mu_102 ** 3
+ 2 * mu_300 * mu_102 * mu_030 * mu_012
+ 2 * mu_300 * mu_102 * mu_021 * mu_003
+ 2 * mu_300 * mu_102 * mu_012 ** 2
+ 2 * mu_300 * mu_102 * mu_003 ** 2
+ 2 * mu_210 ** 3 * mu_030
+ 2 * mu_210 ** 2 * mu_201 * mu_003
+ 2 * mu_210 ** 2 * mu_030 ** 2
+ 2 * mu_210 ** 2 * mu_030 * mu_012
+ 2 * mu_210 * mu_201 ** 2 * mu_030
+ 4 * mu_210 * mu_201 * mu_030 * mu_021
+ 2 * mu_210 * mu_201 * mu_030 * mu_003
+ 4 * mu_210 * mu_201 * mu_012 * mu_003
+ 10 * mu_210 * mu_120 ** 2 * mu_030
+ 8 * mu_210 * mu_120 * mu_111 * mu_003
+ 6 * mu_210 * mu_120 * mu_102 * mu_030
+ 4 * mu_210 * mu_111 ** 2 * mu_030
+ 8 * mu_210 * mu_111 * mu_102 * mu_003
+ 2 * mu_210 * mu_030 ** 3
+ 2 * mu_210 * mu_030 ** 2 * mu_012
+ 6 * mu_210 * mu_030 * mu_021 ** 2
+ 2 * mu_210 * mu_030 * mu_021 * mu_003
+ 2 * mu_210 * mu_030 * mu_012 ** 2
+ 6 * mu_210 * mu_021 * mu_012 * mu_003
+ 2 * mu_210 * mu_012 * mu_003 ** 2
+ 2 * mu_201 ** 3 * mu_003
+ 2 * mu_201 ** 2 * mu_021 * mu_003
+ 2 * mu_201 ** 2 * mu_003 ** 2
+ 8 * mu_201 * mu_120 * mu_111 * mu_030
+ 6 * mu_201 * mu_120 * mu_102 * mu_003
+ 4 * mu_201 * mu_111 ** 2 * mu_003
+ 8 * mu_201 * mu_111 * mu_102 * mu_030
+ 10 * mu_201 * mu_102 ** 2 * mu_003
+ 2 * mu_201 * mu_030 ** 2 * mu_021
+ 6 * mu_201 * mu_030 * mu_021 * mu_012
+ 2 * mu_201 * mu_030 * mu_012 * mu_003
+ 2 * mu_201 * mu_021 ** 2 * mu_003
+ 2 * mu_201 * mu_021 * mu_003 ** 2
+ 6 * mu_201 * mu_012 ** 2 * mu_003
+ 2 * mu_201 * mu_003 ** 3
+ 4 * mu_120 ** 2 * mu_030 ** 2
+ 6 * mu_120 ** 2 * mu_030 * mu_012
+ 2 * mu_120 ** 2 * mu_021 * mu_003
+ 8 * mu_120 * mu_111 * mu_030 * mu_021
+ 4 * mu_120 * mu_111 * mu_030 * mu_003
+ 8 * mu_120 * mu_111 * mu_012 * mu_003
+ 2 * mu_120 * mu_102 * mu_030 ** 2
+ 4 * mu_120 * mu_102 * mu_030 * mu_012
+ 4 * mu_120 * mu_102 * mu_021 * mu_003
+ 2 * mu_120 * mu_102 * mu_003 ** 2
+ 2 * mu_111 ** 2 * mu_030 ** 2
+ 4 * mu_111 ** 2 * mu_030 * mu_012
+ 4 * mu_111 ** 2 * mu_021 * mu_003
+ 2 * mu_111 ** 2 * mu_003 ** 2
+ 8 * mu_111 * mu_102 * mu_030 * mu_021
+ 4 * mu_111 * mu_102 * mu_030 * mu_003
+ 8 * mu_111 * mu_102 * mu_012 * mu_003
+ 2 * mu_102 ** 2 * mu_030 * mu_012
+ 6 * mu_102 ** 2 * mu_021 * mu_003
+ 4 * mu_102 ** 2 * mu_003 ** 2
+ 1 * mu_030 ** 4
+ 2 * mu_030 ** 3 * mu_012
+ 4 * mu_030 ** 2 * mu_021 ** 2
+ 2 * mu_030 ** 2 * mu_021 * mu_003
+ 2 * mu_030 ** 2 * mu_012 ** 2
+ 10 * mu_030 * mu_021 ** 2 * mu_012
+ 8 * mu_030 * mu_021 * mu_012 * mu_003
+ 2 * mu_030 * mu_012 ** 3
+ 2 * mu_030 * mu_012 * mu_003 ** 2
+ 2 * mu_021 ** 3 * mu_003
+ 2 * mu_021 ** 2 * mu_003 ** 2
+ 10 * mu_021 * mu_012 ** 2 * mu_003
+ 2 * mu_021 * mu_003 ** 3
+ 4 * mu_012 ** 2 * mu_003 ** 2
+ 1 * mu_003 ** 4
+ 1 * mu_210 ** 4
+ 2 * mu_210 ** 3 * mu_012
+ 2 * mu_210 ** 2 * mu_201 ** 2
+ 2 * mu_210 ** 2 * mu_201 * mu_021
+ 8 * mu_210 ** 2 * mu_120 ** 2
+ 8 * mu_210 ** 2 * mu_120 * mu_102
+ 2 * mu_210 ** 2 * mu_111 ** 2
+ 2 * mu_210 ** 2 * mu_102 ** 2
+ 2 * mu_210 ** 2 * mu_021 ** 2
+ 2 * mu_210 ** 2 * mu_012 ** 2
+ 2 * mu_210 * mu_201 ** 2 * mu_012
+ 12 * mu_210 * mu_201 * mu_120 * mu_111
+ 12 * mu_210 * mu_201 * mu_111 * mu_102
+ 6 * mu_210 * mu_201 * mu_021 * mu_012
+ 8 * mu_210 * mu_120 ** 2 * mu_012
+ 12 * mu_210 * mu_120 * mu_111 * mu_021
+ 6 * mu_210 * mu_120 * mu_102 * mu_012
+ 4 * mu_210 * mu_111 ** 2 * mu_012
+ 12 * mu_210 * mu_111 * mu_102 * mu_021
+ 2 * mu_210 * mu_102 ** 2 * mu_012
+ 8 * mu_210 * mu_021 ** 2 * mu_012
+ 2 * mu_210 * mu_012 ** 3
+ 1 * mu_201 ** 4
+ 2 * mu_201 ** 3 * mu_021
+ 2 * mu_201 ** 2 * mu_120 ** 2
+ 8 * mu_201 ** 2 * mu_120 * mu_102
+ 2 * mu_201 ** 2 * mu_111 ** 2
+ 8 * mu_201 ** 2 * mu_102 ** 2
+ 2 * mu_201 ** 2 * mu_021 ** 2
+ 2 * mu_201 ** 2 * mu_012 ** 2
+ 2 * mu_201 * mu_120 ** 2 * mu_021
+ 12 * mu_201 * mu_120 * mu_111 * mu_012
+ 6 * mu_201 * mu_120 * mu_102 * mu_021
+ 4 * mu_201 * mu_111 ** 2 * mu_021
+ 12 * mu_201 * mu_111 * mu_102 * mu_012
+ 8 * mu_201 * mu_102 ** 2 * mu_021
+ 2 * mu_201 * mu_021 ** 3
+ 8 * mu_201 * mu_021 * mu_012 ** 2
+ 1 * mu_120 ** 4
+ 2 * mu_120 ** 3 * mu_102
+ 2 * mu_120 ** 2 * mu_111 ** 2
+ 2 * mu_120 ** 2 * mu_102 ** 2
+ 2 * mu_120 ** 2 * mu_021 ** 2
+ 2 * mu_120 ** 2 * mu_012 ** 2
+ 4 * mu_120 * mu_111 ** 2 * mu_102
+ 12 * mu_120 * mu_111 * mu_021 * mu_012
+ 2 * mu_120 * mu_102 ** 3
+ 2 * mu_120 * mu_102 * mu_021 ** 2
+ 2 * mu_120 * mu_102 * mu_012 ** 2
+ 2 * mu_111 ** 2 * mu_102 ** 2
+ 2 * mu_111 ** 2 * mu_021 ** 2
+ 2 * mu_111 ** 2 * mu_012 ** 2
+ 12 * mu_111 * mu_102 * mu_021 * mu_012
+ 1 * mu_102 ** 4
+ 2 * mu_102 ** 2 * mu_021 ** 2
+ 2 * mu_102 ** 2 * mu_012 ** 2
+ 1 * mu_021 ** 4
+ 8 * mu_021 ** 2 * mu_012 ** 2
+ 1 * mu_012 ** 4
)
[docs]@nb.njit
def phi_9(
mu_030,
mu_021,
mu_120,
mu_101,
mu_003,
mu_200,
mu_110,
mu_201,
mu_111,
mu_102,
mu_210,
mu_020,
mu_012,
mu_002,
mu_011,
mu_300,
):
return (
1 * mu_200 * mu_300 ** 2
+ 2 * mu_110 * mu_300 * mu_210
+ 2 * mu_110 * mu_120 * mu_030
+ 2 * mu_101 * mu_300 * mu_201
+ 2 * mu_101 * mu_102 * mu_003
+ 1 * mu_020 * mu_030 ** 2
+ 2 * mu_011 * mu_030 * mu_021
+ 2 * mu_011 * mu_012 * mu_003
+ 1 * mu_002 * mu_003 ** 2
+ 2 * mu_200 * mu_210 ** 2
+ 2 * mu_200 * mu_201 ** 2
+ 1 * mu_200 * mu_120 ** 2
+ 2 * mu_200 * mu_111 ** 2
+ 1 * mu_200 * mu_102 ** 2
+ 4 * mu_110 * mu_210 * mu_120
+ 4 * mu_110 * mu_201 * mu_111
+ 4 * mu_110 * mu_111 * mu_021
+ 2 * mu_110 * mu_102 * mu_012
+ 4 * mu_101 * mu_210 * mu_111
+ 4 * mu_101 * mu_201 * mu_102
+ 2 * mu_101 * mu_120 * mu_021
+ 4 * mu_101 * mu_111 * mu_012
+ 1 * mu_020 * mu_210 ** 2
+ 2 * mu_020 * mu_120 ** 2
+ 2 * mu_020 * mu_111 ** 2
+ 2 * mu_020 * mu_021 ** 2
+ 1 * mu_020 * mu_012 ** 2
+ 2 * mu_011 * mu_210 * mu_201
+ 4 * mu_011 * mu_120 * mu_111
+ 4 * mu_011 * mu_111 * mu_102
+ 4 * mu_011 * mu_021 * mu_012
+ 1 * mu_002 * mu_201 ** 2
+ 2 * mu_002 * mu_111 ** 2
+ 2 * mu_002 * mu_102 ** 2
+ 1 * mu_002 * mu_021 ** 2
+ 2 * mu_002 * mu_012 ** 2
)
[docs]@nb.njit
def phi_10(
mu_030,
mu_021,
mu_120,
mu_101,
mu_003,
mu_200,
mu_110,
mu_201,
mu_111,
mu_102,
mu_210,
mu_020,
mu_012,
mu_002,
mu_011,
mu_300,
):
return (
1 * mu_200 * mu_300 ** 2
+ 1 * mu_200 * mu_300 * mu_120
+ 1 * mu_200 * mu_300 * mu_102
+ 1 * mu_200 * mu_210 * mu_030
+ 1 * mu_200 * mu_201 * mu_003
+ 2 * mu_110 * mu_300 * mu_210
+ 2 * mu_110 * mu_120 * mu_030
+ 2 * mu_110 * mu_111 * mu_003
+ 2 * mu_101 * mu_300 * mu_201
+ 2 * mu_101 * mu_111 * mu_030
+ 2 * mu_101 * mu_102 * mu_003
+ 1 * mu_020 * mu_300 * mu_120
+ 1 * mu_020 * mu_210 * mu_030
+ 1 * mu_020 * mu_030 ** 2
+ 1 * mu_020 * mu_030 * mu_012
+ 1 * mu_020 * mu_021 * mu_003
+ 2 * mu_011 * mu_300 * mu_111
+ 2 * mu_011 * mu_030 * mu_021
+ 2 * mu_011 * mu_012 * mu_003
+ 1 * mu_002 * mu_300 * mu_102
+ 1 * mu_002 * mu_201 * mu_003
+ 1 * mu_002 * mu_030 * mu_012
+ 1 * mu_002 * mu_021 * mu_003
+ 1 * mu_002 * mu_003 ** 2
+ 1 * mu_200 * mu_210 ** 2
+ 1 * mu_200 * mu_210 * mu_012
+ 1 * mu_200 * mu_201 ** 2
+ 1 * mu_200 * mu_201 * mu_021
+ 4 * mu_110 * mu_210 * mu_120
+ 2 * mu_110 * mu_210 * mu_102
+ 2 * mu_110 * mu_201 * mu_111
+ 2 * mu_110 * mu_120 * mu_012
+ 2 * mu_110 * mu_111 * mu_021
+ 2 * mu_101 * mu_210 * mu_111
+ 2 * mu_101 * mu_201 * mu_120
+ 4 * mu_101 * mu_201 * mu_102
+ 2 * mu_101 * mu_111 * mu_012
+ 2 * mu_101 * mu_102 * mu_021
+ 1 * mu_020 * mu_201 * mu_021
+ 1 * mu_020 * mu_120 ** 2
+ 1 * mu_020 * mu_120 * mu_102
+ 1 * mu_020 * mu_021 ** 2
+ 2 * mu_011 * mu_210 * mu_021
+ 2 * mu_011 * mu_201 * mu_012
+ 2 * mu_011 * mu_120 * mu_111
+ 2 * mu_011 * mu_111 * mu_102
+ 4 * mu_011 * mu_021 * mu_012
+ 1 * mu_002 * mu_210 * mu_012
+ 1 * mu_002 * mu_120 * mu_102
+ 1 * mu_002 * mu_102 ** 2
+ 1 * mu_002 * mu_012 ** 2
)
[docs]@nb.njit
def phi_11(
mu_030,
mu_021,
mu_120,
mu_101,
mu_003,
mu_200,
mu_110,
mu_201,
mu_102,
mu_210,
mu_012,
mu_020,
mu_002,
mu_011,
mu_300,
):
return (
1 * mu_200 * mu_300 ** 2
+ 2 * mu_200 * mu_300 * mu_120
+ 2 * mu_200 * mu_300 * mu_102
+ 2 * mu_110 * mu_300 * mu_210
+ 2 * mu_110 * mu_300 * mu_030
+ 2 * mu_110 * mu_300 * mu_012
+ 2 * mu_110 * mu_120 * mu_030
+ 2 * mu_110 * mu_102 * mu_030
+ 2 * mu_101 * mu_300 * mu_201
+ 2 * mu_101 * mu_300 * mu_021
+ 2 * mu_101 * mu_300 * mu_003
+ 2 * mu_101 * mu_120 * mu_003
+ 2 * mu_101 * mu_102 * mu_003
+ 2 * mu_020 * mu_210 * mu_030
+ 1 * mu_020 * mu_030 ** 2
+ 2 * mu_020 * mu_030 * mu_012
+ 2 * mu_011 * mu_210 * mu_003
+ 2 * mu_011 * mu_201 * mu_030
+ 2 * mu_011 * mu_030 * mu_021
+ 2 * mu_011 * mu_030 * mu_003
+ 2 * mu_011 * mu_012 * mu_003
+ 2 * mu_002 * mu_201 * mu_003
+ 2 * mu_002 * mu_021 * mu_003
+ 1 * mu_002 * mu_003 ** 2
+ 1 * mu_200 * mu_120 ** 2
+ 2 * mu_200 * mu_120 * mu_102
+ 1 * mu_200 * mu_102 ** 2
+ 2 * mu_110 * mu_210 * mu_120
+ 2 * mu_110 * mu_210 * mu_102
+ 2 * mu_110 * mu_120 * mu_012
+ 2 * mu_110 * mu_102 * mu_012
+ 2 * mu_101 * mu_201 * mu_120
+ 2 * mu_101 * mu_201 * mu_102
+ 2 * mu_101 * mu_120 * mu_021
+ 2 * mu_101 * mu_102 * mu_021
+ 1 * mu_020 * mu_210 ** 2
+ 2 * mu_020 * mu_210 * mu_012
+ 1 * mu_020 * mu_012 ** 2
+ 2 * mu_011 * mu_210 * mu_201
+ 2 * mu_011 * mu_210 * mu_021
+ 2 * mu_011 * mu_201 * mu_012
+ 2 * mu_011 * mu_021 * mu_012
+ 1 * mu_002 * mu_201 ** 2
+ 2 * mu_002 * mu_201 * mu_021
+ 1 * mu_002 * mu_021 ** 2
)
[docs]@nb.njit
def phi_12(
mu_030,
mu_021,
mu_120,
mu_101,
mu_003,
mu_200,
mu_110,
mu_201,
mu_111,
mu_102,
mu_210,
mu_020,
mu_012,
mu_002,
mu_011,
mu_300,
):
return (
1 * mu_200 ** 2 * mu_300 ** 2
+ 4 * mu_200 * mu_110 * mu_300 * mu_210
+ 4 * mu_200 * mu_101 * mu_300 * mu_201
+ 2 * mu_200 * mu_020 * mu_300 * mu_120
+ 2 * mu_200 * mu_020 * mu_210 * mu_030
+ 4 * mu_200 * mu_011 * mu_300 * mu_111
+ 2 * mu_200 * mu_002 * mu_300 * mu_102
+ 2 * mu_200 * mu_002 * mu_201 * mu_003
+ 4 * mu_110 * mu_020 * mu_120 * mu_030
+ 4 * mu_110 * mu_002 * mu_111 * mu_003
+ 4 * mu_101 * mu_020 * mu_111 * mu_030
+ 4 * mu_101 * mu_002 * mu_102 * mu_003
+ 1 * mu_020 ** 2 * mu_030 ** 2
+ 4 * mu_020 * mu_011 * mu_030 * mu_021
+ 2 * mu_020 * mu_002 * mu_030 * mu_012
+ 2 * mu_020 * mu_002 * mu_021 * mu_003
+ 4 * mu_011 * mu_002 * mu_012 * mu_003
+ 1 * mu_002 ** 2 * mu_003 ** 2
+ 1 * mu_200 ** 2 * mu_210 ** 2
+ 1 * mu_200 ** 2 * mu_201 ** 2
+ 4 * mu_200 * mu_110 * mu_210 * mu_120
+ 4 * mu_200 * mu_110 * mu_201 * mu_111
+ 4 * mu_200 * mu_101 * mu_210 * mu_111
+ 4 * mu_200 * mu_101 * mu_201 * mu_102
+ 2 * mu_200 * mu_020 * mu_201 * mu_021
+ 4 * mu_200 * mu_011 * mu_210 * mu_021
+ 4 * mu_200 * mu_011 * mu_201 * mu_012
+ 2 * mu_200 * mu_002 * mu_210 * mu_012
+ 4 * mu_110 ** 2 * mu_210 ** 2
+ 4 * mu_110 ** 2 * mu_120 ** 2
+ 8 * mu_110 * mu_101 * mu_210 * mu_201
+ 8 * mu_110 * mu_101 * mu_120 * mu_111
+ 8 * mu_110 * mu_101 * mu_111 * mu_102
+ 4 * mu_110 * mu_020 * mu_210 * mu_120
+ 4 * mu_110 * mu_020 * mu_111 * mu_021
+ 8 * mu_110 * mu_011 * mu_210 * mu_111
+ 8 * mu_110 * mu_011 * mu_120 * mu_021
+ 8 * mu_110 * mu_011 * mu_111 * mu_012
+ 4 * mu_110 * mu_002 * mu_210 * mu_102
+ 4 * mu_110 * mu_002 * mu_120 * mu_012
+ 4 * mu_101 ** 2 * mu_201 ** 2
+ 4 * mu_101 ** 2 * mu_102 ** 2
+ 4 * mu_101 * mu_020 * mu_201 * mu_120
+ 4 * mu_101 * mu_020 * mu_102 * mu_021
+ 8 * mu_101 * mu_011 * mu_201 * mu_111
+ 8 * mu_101 * mu_011 * mu_111 * mu_021
+ 8 * mu_101 * mu_011 * mu_102 * mu_012
+ 4 * mu_101 * mu_002 * mu_201 * mu_102
+ 4 * mu_101 * mu_002 * mu_111 * mu_012
+ 1 * mu_020 ** 2 * mu_120 ** 2
+ 1 * mu_020 ** 2 * mu_021 ** 2
+ 4 * mu_020 * mu_011 * mu_120 * mu_111
+ 4 * mu_020 * mu_011 * mu_021 * mu_012
+ 2 * mu_020 * mu_002 * mu_120 * mu_102
+ 4 * mu_011 ** 2 * mu_021 ** 2
+ 4 * mu_011 ** 2 * mu_012 ** 2
+ 4 * mu_011 * mu_002 * mu_111 * mu_102
+ 4 * mu_011 * mu_002 * mu_021 * mu_012
+ 1 * mu_002 ** 2 * mu_102 ** 2
+ 1 * mu_002 ** 2 * mu_012 ** 2
+ 4 * mu_110 ** 2 * mu_111 ** 2
+ 4 * mu_101 ** 2 * mu_111 ** 2
+ 4 * mu_011 ** 2 * mu_111 ** 2
)
[docs]@nb.njit
def phi_13(
mu_030,
mu_021,
mu_120,
mu_101,
mu_003,
mu_200,
mu_110,
mu_201,
mu_111,
mu_102,
mu_210,
mu_012,
mu_020,
mu_002,
mu_011,
mu_300,
):
return (
1 * mu_200 ** 2 * mu_300 ** 2
+ 2 * mu_200 * mu_110 * mu_300 * mu_210
+ 2 * mu_200 * mu_110 * mu_120 * mu_030
+ 2 * mu_200 * mu_101 * mu_300 * mu_201
+ 2 * mu_200 * mu_101 * mu_102 * mu_003
+ 1 * mu_110 ** 2 * mu_300 ** 2
+ 1 * mu_110 ** 2 * mu_030 ** 2
+ 2 * mu_110 * mu_101 * mu_030 * mu_021
+ 2 * mu_110 * mu_101 * mu_012 * mu_003
+ 2 * mu_110 * mu_020 * mu_300 * mu_210
+ 2 * mu_110 * mu_020 * mu_120 * mu_030
+ 2 * mu_110 * mu_011 * mu_300 * mu_201
+ 2 * mu_110 * mu_011 * mu_102 * mu_003
+ 1 * mu_101 ** 2 * mu_300 ** 2
+ 1 * mu_101 ** 2 * mu_003 ** 2
+ 2 * mu_101 * mu_011 * mu_300 * mu_210
+ 2 * mu_101 * mu_011 * mu_120 * mu_030
+ 2 * mu_101 * mu_002 * mu_300 * mu_201
+ 2 * mu_101 * mu_002 * mu_102 * mu_003
+ 1 * mu_020 ** 2 * mu_030 ** 2
+ 2 * mu_020 * mu_011 * mu_030 * mu_021
+ 2 * mu_020 * mu_011 * mu_012 * mu_003
+ 1 * mu_011 ** 2 * mu_030 ** 2
+ 1 * mu_011 ** 2 * mu_003 ** 2
+ 2 * mu_011 * mu_002 * mu_030 * mu_021
+ 2 * mu_011 * mu_002 * mu_012 * mu_003
+ 1 * mu_002 ** 2 * mu_003 ** 2
+ 2 * mu_200 ** 2 * mu_210 ** 2
+ 2 * mu_200 ** 2 * mu_201 ** 2
+ 1 * mu_200 ** 2 * mu_120 ** 2
+ 2 * mu_200 ** 2 * mu_111 ** 2
+ 1 * mu_200 ** 2 * mu_102 ** 2
+ 4 * mu_200 * mu_110 * mu_210 * mu_120
+ 4 * mu_200 * mu_110 * mu_201 * mu_111
+ 4 * mu_200 * mu_110 * mu_111 * mu_021
+ 2 * mu_200 * mu_110 * mu_102 * mu_012
+ 4 * mu_200 * mu_101 * mu_210 * mu_111
+ 4 * mu_200 * mu_101 * mu_201 * mu_102
+ 2 * mu_200 * mu_101 * mu_120 * mu_021
+ 4 * mu_200 * mu_101 * mu_111 * mu_012
+ 3 * mu_110 ** 2 * mu_210 ** 2
+ 2 * mu_110 ** 2 * mu_201 ** 2
+ 3 * mu_110 ** 2 * mu_120 ** 2
+ 1 * mu_110 ** 2 * mu_102 ** 2
+ 2 * mu_110 ** 2 * mu_021 ** 2
+ 1 * mu_110 ** 2 * mu_012 ** 2
+ 2 * mu_110 * mu_101 * mu_210 * mu_201
+ 4 * mu_110 * mu_101 * mu_120 * mu_111
+ 4 * mu_110 * mu_101 * mu_111 * mu_102
+ 4 * mu_110 * mu_101 * mu_021 * mu_012
+ 4 * mu_110 * mu_020 * mu_210 * mu_120
+ 4 * mu_110 * mu_020 * mu_201 * mu_111
+ 4 * mu_110 * mu_020 * mu_111 * mu_021
+ 2 * mu_110 * mu_020 * mu_102 * mu_012
+ 4 * mu_110 * mu_011 * mu_210 * mu_111
+ 4 * mu_110 * mu_011 * mu_201 * mu_102
+ 2 * mu_110 * mu_011 * mu_120 * mu_021
+ 4 * mu_110 * mu_011 * mu_111 * mu_012
+ 2 * mu_101 ** 2 * mu_210 ** 2
+ 3 * mu_101 ** 2 * mu_201 ** 2
+ 1 * mu_101 ** 2 * mu_120 ** 2
+ 3 * mu_101 ** 2 * mu_102 ** 2
+ 1 * mu_101 ** 2 * mu_021 ** 2
+ 2 * mu_101 ** 2 * mu_012 ** 2
+ 4 * mu_101 * mu_011 * mu_210 * mu_120
+ 4 * mu_101 * mu_011 * mu_201 * mu_111
+ 4 * mu_101 * mu_011 * mu_111 * mu_021
+ 2 * mu_101 * mu_011 * mu_102 * mu_012
+ 4 * mu_101 * mu_002 * mu_210 * mu_111
+ 4 * mu_101 * mu_002 * mu_201 * mu_102
+ 2 * mu_101 * mu_002 * mu_120 * mu_021
+ 4 * mu_101 * mu_002 * mu_111 * mu_012
+ 1 * mu_020 ** 2 * mu_210 ** 2
+ 2 * mu_020 ** 2 * mu_120 ** 2
+ 2 * mu_020 ** 2 * mu_111 ** 2
+ 2 * mu_020 ** 2 * mu_021 ** 2
+ 1 * mu_020 ** 2 * mu_012 ** 2
+ 2 * mu_020 * mu_011 * mu_210 * mu_201
+ 4 * mu_020 * mu_011 * mu_120 * mu_111
+ 4 * mu_020 * mu_011 * mu_111 * mu_102
+ 4 * mu_020 * mu_011 * mu_021 * mu_012
+ 1 * mu_011 ** 2 * mu_210 ** 2
+ 1 * mu_011 ** 2 * mu_201 ** 2
+ 2 * mu_011 ** 2 * mu_120 ** 2
+ 2 * mu_011 ** 2 * mu_102 ** 2
+ 3 * mu_011 ** 2 * mu_021 ** 2
+ 3 * mu_011 ** 2 * mu_012 ** 2
+ 2 * mu_011 * mu_002 * mu_210 * mu_201
+ 4 * mu_011 * mu_002 * mu_120 * mu_111
+ 4 * mu_011 * mu_002 * mu_111 * mu_102
+ 4 * mu_011 * mu_002 * mu_021 * mu_012
+ 1 * mu_002 ** 2 * mu_201 ** 2
+ 2 * mu_002 ** 2 * mu_111 ** 2
+ 2 * mu_002 ** 2 * mu_102 ** 2
+ 1 * mu_002 ** 2 * mu_021 ** 2
+ 2 * mu_002 ** 2 * mu_012 ** 2
+ 4 * mu_110 ** 2 * mu_111 ** 2
+ 4 * mu_101 ** 2 * mu_111 ** 2
+ 4 * mu_011 ** 2 * mu_111 ** 2
)
[docs]@nb.njit
def CI(mu_000,
mu_200,
mu_020,
mu_002,
mu_110,
mu_101,
mu_011,
mu_111,
mu_210,
mu_201,
mu_120,
mu_021,
mu_012,
mu_102,
mu_003,
mu_030,
mu_300,
mu_013,
mu_103,
mu_130,
mu_310,
mu_301,
mu_031,
mu_112,
mu_121,
mu_211,
mu_022,
mu_202,
mu_220,
mu_400,
mu_040,
mu_004):
alpha_1 = mu_002 - mu_020
alpha_2 = mu_020 - mu_200
alpha_3 = mu_200 - mu_002
beta_1 = mu_021 - mu_201
beta_2 = mu_102 - mu_120
beta_3 = mu_210 - mu_012
beta_4 = mu_003 - mu_201 - 2 * mu_021
beta_5 = mu_030 - mu_012 - 2 * mu_210
beta_6 = mu_300 - mu_120 - 2 * mu_102
beta_7 = mu_021 - mu_003 + 2 * mu_201
beta_8 = mu_102 - mu_300 + 2 * mu_120
beta_9 = mu_210 - mu_030 + 2 * mu_012
beta_10 = mu_021 + mu_201 - 3 * mu_003
beta_11 = mu_012 + mu_210 - 3 * mu_030
beta_12 = mu_102 + mu_120 - 3 * mu_300
beta_13 = mu_021 + mu_003 + 3 * mu_201
beta_14 = mu_102 + mu_300 + 3 * mu_120
beta_15 = mu_210 + mu_030 + 3 * mu_012
beta_16 = mu_012 + mu_030 + 3 * mu_210
beta_17 = mu_201 + mu_003 + 3 * mu_021
beta_18 = mu_120 + mu_300 + 3 * mu_102
gamma_1 = mu_022 - mu_400
gamma_2 = mu_202 - mu_040
gamma_3 = mu_220 - mu_004
gamma_4 = mu_112 + mu_130 + mu_310
gamma_5 = mu_121 + mu_103 + mu_301
gamma_6 = mu_211 + mu_013 + mu_031
gamma_7 = mu_022 - mu_220 + mu_004 - mu_400
gamma_8 = mu_202 - mu_022 + mu_400 - mu_040
gamma_9 = mu_220 - mu_202 + mu_040 - mu_004
r_gyr = np.sqrt((mu_200 + mu_020 + mu_002) / (3 * mu_000))
if r_gyr == 0:
return np.nan
s_3 = 1 / (mu_000 ** 3 * r_gyr ** 9)
s_4 = 1 / (mu_000 ** 4 * r_gyr ** 9)
return 4 * s_3 * (mu_110 * (mu_021 * (3 * gamma_2 - 2 * gamma_3 - gamma_1)
- mu_201 * (3 * gamma_1 - 2 * gamma_3 - gamma_2) + beta_12 * gamma_5
- beta_11 * gamma_6 + mu_003 * gamma_8)
+ mu_101 * (mu_210 * (3 * gamma_1 - 2 * gamma_2 - gamma_3)
- mu_012 * (
3 * gamma_3 - 2 * gamma_2 - gamma_1) + beta_10 * gamma_6 - beta_12 * gamma_4
+ mu_030 * gamma_7)
+ mu_011 * (mu_102 * (3 * gamma_3 - 2 * gamma_1 - gamma_2)
- mu_120 * (3 * gamma_2 - 2 * gamma_1 - gamma_3)
+ beta_11 * gamma_4 - beta_10 * gamma_5 + mu_300 * gamma_9)
+ mu_002 * (beta_18 * gamma_6 - beta_15 * gamma_5 - 2 * (mu_111 * gamma_8 + beta_1 * gamma_4))
+ mu_020 * (beta_17 * gamma_4 - beta_14 * gamma_6 - 2 * (mu_111 * gamma_7 + beta_3 * gamma_5))
+ mu_200 * (beta_16 * gamma_5 - beta_13 * gamma_4 - 2 * (mu_111 * gamma_9 + beta_2 * gamma_6))) \
- 16 * s_4 * (mu_011 * alpha_2 * alpha_3 * beta_2 + mu_101 * alpha_1 * alpha_2 * beta_3
+ mu_110 * alpha_1 * alpha_3 * beta_1 - mu_111 * alpha_1 * alpha_2 * alpha_3
- mu_011 * mu_011 * (mu_111 * alpha_1 - mu_011 * beta_2 - mu_101 * beta_5 - mu_110 * beta_7)
- mu_101 * mu_101 * (mu_111 * alpha_3 - mu_101 * beta_3 - mu_110 * beta_4 - mu_011 * beta_8)
- mu_110 * mu_110 * (mu_111 * alpha_2 - mu_110 * beta_1 - mu_011 * beta_6 - mu_101 * beta_9)
+ mu_011 * mu_101 * (mu_002 * beta_1 + mu_020 * beta_4 + mu_200 * beta_7)
+ mu_011 * mu_110 * (mu_020 * beta_3 + mu_200 * beta_5 + mu_002 * beta_9)
+ mu_101 * mu_110 * (mu_200 * beta_2 + mu_002 * beta_6 + mu_020 * beta_8))
[docs]class MomentType(Enum):
"""
Different rotation invariant moments (order 2 and order 3)
Choose from ['O_3', 'O_4', 'O_5', 'F', 'phi_2', 'phi_3', 'phi_4', 'phi_5', 'phi_6', 'phi_7', 'phi_8', 'phi_9', 'phi_10', 'phi_11', 'phi_12', 'phi_13']
O_3, O_4, and O_5 are second order moments from [1] and F is a third order moment from [2].
These four moments are used in the original Geometricus manuscript [3].
phi_{2-13} are independent third order moments from [4].
CI is the chiral invariant moment from [5].
[1] Mamistvalov, Alexander G. "N-dimensional moment invariants and conceptual mathematical theory of recognition n-dimensional solids."
IEEE Transactions on pattern analysis and machine intelligence 20.8 (1998): 819-831.
[2] Flusser, Jan, Jirí Boldys, and Barbara Zitová. "Moment forms invariant to rotation and blur in arbitrary number of dimensions."
IEEE Transactions on Pattern Analysis and Machine Intelligence 25.2 (2003): 234-246.
[3] Durairaj, Janani, et al. "Geometricus represents protein structures as shape-mers derived from moment invariants." Bioinformatics 36.Supplement_2 (2020): i718-i725.
[4] Flusser, Jan, Tomas Suk, and Barbara Zitová. 2D and 3D image analysis by moments. John Wiley & Sons, 2016.
[5] Hattne, Johan, and Victor S. Lamzin. "A moment invariant for evaluating the chirality of three-dimensional objects." Journal of The Royal Society Interface 8.54 (2011): 144-151.
"""
O_3 = MomentInfo(O_3, [(2, 0, 0), (0, 2, 0), (0, 0, 2)])
O_4 = MomentInfo(
O_4, [(2, 0, 0), (0, 2, 0), (0, 0, 2), (1, 1, 0), (1, 0, 1), (0, 1, 1)]
)
O_5 = MomentInfo(
O_5, [(2, 0, 0), (0, 2, 0), (0, 0, 2), (1, 1, 0), (1, 0, 1), (0, 1, 1)]
)
F = MomentInfo(
F,
[
(2, 0, 1),
(0, 2, 1),
(2, 1, 0),
(3, 0, 0),
(1, 1, 1),
(0, 1, 2),
(0, 0, 3),
(0, 3, 0),
(1, 0, 2),
(1, 2, 0),
],
)
phi_2 = MomentInfo(
phi_2, [(0, 2, 0), (0, 1, 1), (1, 1, 0), (2, 0, 0), (0, 0, 2), (1, 0, 1)]
)
phi_3 = MomentInfo(
phi_3, [(0, 2, 0), (0, 1, 1), (1, 1, 0), (2, 0, 0), (0, 0, 2), (1, 0, 1)]
)
phi_4 = MomentInfo(
phi_4,
[
(0, 3, 0),
(0, 2, 1),
(1, 2, 0),
(0, 0, 3),
(1, 1, 1),
(2, 0, 1),
(1, 0, 2),
(2, 1, 0),
(0, 1, 2),
(3, 0, 0),
],
)
phi_5 = MomentInfo(
phi_5,
[
(0, 3, 0),
(0, 2, 1),
(1, 2, 0),
(0, 0, 3),
(2, 0, 1),
(1, 0, 2),
(2, 1, 0),
(0, 1, 2),
(3, 0, 0),
],
)
phi_6 = MomentInfo(
phi_6,
[
(0, 3, 0),
(0, 2, 1),
(1, 2, 0),
(0, 0, 3),
(1, 1, 1),
(2, 0, 1),
(1, 0, 2),
(2, 1, 0),
(0, 1, 2),
(3, 0, 0),
],
)
phi_7 = MomentInfo(
phi_7,
[
(0, 3, 0),
(0, 2, 1),
(1, 2, 0),
(0, 0, 3),
(1, 1, 1),
(2, 0, 1),
(1, 0, 2),
(2, 1, 0),
(0, 1, 2),
(3, 0, 0),
],
)
phi_8 = MomentInfo(
phi_8,
[
(0, 3, 0),
(0, 2, 1),
(1, 2, 0),
(0, 0, 3),
(1, 1, 1),
(2, 0, 1),
(1, 0, 2),
(2, 1, 0),
(0, 1, 2),
(3, 0, 0),
],
)
phi_9 = MomentInfo(
phi_9,
[
(0, 3, 0),
(0, 2, 1),
(1, 2, 0),
(1, 0, 1),
(0, 0, 3),
(2, 0, 0),
(1, 1, 0),
(2, 0, 1),
(1, 1, 1),
(1, 0, 2),
(2, 1, 0),
(0, 2, 0),
(0, 1, 2),
(0, 0, 2),
(0, 1, 1),
(3, 0, 0),
],
)
phi_10 = MomentInfo(
phi_10,
[
(0, 3, 0),
(0, 2, 1),
(1, 2, 0),
(1, 0, 1),
(0, 0, 3),
(2, 0, 0),
(1, 1, 0),
(2, 0, 1),
(1, 1, 1),
(1, 0, 2),
(2, 1, 0),
(0, 2, 0),
(0, 1, 2),
(0, 0, 2),
(0, 1, 1),
(3, 0, 0),
],
)
phi_11 = MomentInfo(
phi_11,
[
(0, 3, 0),
(0, 2, 1),
(1, 2, 0),
(1, 0, 1),
(0, 0, 3),
(2, 0, 0),
(1, 1, 0),
(2, 0, 1),
(1, 0, 2),
(2, 1, 0),
(0, 1, 2),
(0, 2, 0),
(0, 0, 2),
(0, 1, 1),
(3, 0, 0),
],
)
phi_12 = MomentInfo(
phi_12,
[
(0, 3, 0),
(0, 2, 1),
(1, 2, 0),
(1, 0, 1),
(0, 0, 3),
(2, 0, 0),
(1, 1, 0),
(2, 0, 1),
(1, 1, 1),
(1, 0, 2),
(2, 1, 0),
(0, 2, 0),
(0, 1, 2),
(0, 0, 2),
(0, 1, 1),
(3, 0, 0),
],
)
phi_13 = MomentInfo(
phi_13,
[
(0, 3, 0),
(0, 2, 1),
(1, 2, 0),
(1, 0, 1),
(0, 0, 3),
(2, 0, 0),
(1, 1, 0),
(2, 0, 1),
(1, 1, 1),
(1, 0, 2),
(2, 1, 0),
(0, 1, 2),
(0, 2, 0),
(0, 0, 2),
(0, 1, 1),
(3, 0, 0),
],
)
CI = MomentInfo(
CI,
[
(0, 0, 0),
(2, 0, 0),
(0, 2, 0),
(0, 0, 2),
(1, 1, 0),
(1, 0, 1),
(0, 1, 1),
(1, 1, 1),
(2, 1, 0),
(2, 0, 1),
(1, 2, 0),
(0, 2, 1),
(0, 1, 2),
(1, 0, 2),
(0, 0, 3),
(0, 3, 0),
(3, 0, 0),
(0, 1, 3),
(1, 0, 3),
(1, 3, 0),
(3, 1, 0),
(3, 0, 1),
(0, 3, 1),
(1, 1, 2),
(1, 2, 1),
(2, 1, 1),
(0, 2, 2),
(2, 0, 2),
(2, 2, 0),
(4, 0, 0),
(0, 4, 0),
(0, 0, 4),
],
)
[docs] def get_moments_from_coordinates(self, mus: ty.List[float]):
return self.value.moment_function(*mus)
[docs]def get_moments_from_coordinates(
coordinates: np.ndarray,
moment_types: ty.List[MomentType] = (
MomentType.O_3,
MomentType.O_4,
MomentType.O_5,
MomentType.F,
),
) -> ty.List[float]:
"""
Gets rotation-invariant moments for a set of coordinates
Parameters
----------
coordinates
moment_types
Which moments to calculate
Choose from ['O_3', 'O_4', 'O_5', 'F', 'phi_2', 'phi_3', 'phi_4', 'phi_5', 'phi_6', 'phi_7', 'phi_8', 'phi_9', 'phi_10', 'phi_11', 'phi_12', 'phi_13', 'CI']
Returns
-------
list of moments
"""
if coordinates.shape[0] < 2:
return [np.nan] * len(moment_types)
all_moment_mu_types: ty.Set[ty.Tuple[int, int, int]] = set(
m for moment_type in moment_types for m in moment_type.value.mu_arguments
)
centroid = nb_mean_axis_0(coordinates)
mus = {
(x, y, z): mu(float(x), float(y), float(z), coordinates, centroid)
for (x, y, z) in all_moment_mu_types
}
moments = [
moment_type.get_moments_from_coordinates(
[mus[m] for m in moment_type.value.mu_arguments]
)
for moment_type in moment_types
]
return moments
