from tqdm import tqdm
from typing import Tuple
import numpy as np
from numpy.typing import NDArray
from mumott.core.wigner_d_utilities import load_d_matrices, calculate_sph_coefficients_rotated_by_euler_angles
[docs]def find_approximate_symmetry_axis(coefficients: NDArray[float],
ell_max: int,
resolution: int = 10,
filter: str = None) -> Tuple[NDArray[float]]:
r""" Find the axis of highest apparent symmetry voxel-by-voxel for a voxel map of sperical harmonics
coefficients. As a default, the measure of degree of symmetry is a power of the function rotationally
averaged around the given axis.
Parameters
----------
coefficients
Voxel map of spherical harmonic coefficients.
ell_max
Largest order :math:`\ell` used in the fitting. If :attr:`coefficients` contains higher orders,
it will be truncated.
resolution
Number or angular steps along a half-circle, used in the search for the optimal axis.
filter
Weighing of different orders used to calculate the degree of symmetry.
Possible values are "ramp" and "square". By default no filter is applied (`None`).
Returns
---------
optimal_zonal_coeffs
Zonal harmonics coefficients in the frame-of-reference corresponding
to the axis of highest degree of symmetry.
optimal_theta
Voxel-by-voxel polar angles of the axis with highest degree of symmetry.
optimal_phi
Voxel-by-voxel azimuthal angles of the axis with highest degree of symmetry.
"""
# Use only the coefficients up until ell_max
truncated_coefficients = coefficients[..., :(ell_max+1)*(ell_max+2)//2]
volume_shape = coefficients.shape[:-1]
# Find the zonal coefficients
zonal_indexes = np.zeros((ell_max+1)*(ell_max+2)//2, dtype=bool)
inc = 0
f = np.ones(ell_max//2+1)
for ell in range(0, ell_max+1, 2):
zonal_indexes[inc + ell] = True
if filter == 'ramp':
f[ell//2] = ell
elif filter == 'square':
f[ell//2] = ell**2
inc += 2*ell + 1
# Load d matrices
d_matrices = load_d_matrices(ell_max)
optimal_theta = np.zeros(volume_shape)
optimal_phi = np.zeros(volume_shape)
maximum_degree_of_symmetry = np.zeros(volume_shape)
optimal_zonal_coeffs = np.zeros((*volume_shape, ell_max//2 + 1))
# Loop through grid of directions
theta_points = np.linspace(0, np.pi/2, resolution//2, endpoint=True)
phi_points = np.linspace(0, 2*np.pi, 2*resolution, endpoint=False)
for theta in tqdm(theta_points):
for phi in phi_points:
rotated_coefficients = calculate_sph_coefficients_rotated_by_euler_angles(
truncated_coefficients,
Psi=-phi,
Theta=-theta,
Phi=None,
d_matrices=d_matrices)
# Check if degree of symmetry (DOS) is higher than maximum so far
zonal_coeffs = rotated_coefficients[..., zonal_indexes]
degree_of_symmetry = np.sum(zonal_coeffs**2*f[np.newaxis, np.newaxis, np.newaxis, :], axis=-1)
indices = degree_of_symmetry > maximum_degree_of_symmetry
maximum_degree_of_symmetry[indices] = degree_of_symmetry[indices]
optimal_theta[indices] = theta
optimal_phi[indices] = phi
optimal_zonal_coeffs[indices, :] = zonal_coeffs[indices, :]
return optimal_zonal_coeffs, optimal_theta, optimal_phi
[docs]def degree_of_symmetry_map(coefficients: NDArray[float],
ell_max: int,
resolution: int = 10,
filter: str = None) -> Tuple[NDArray[float]]:
r""" Make a longitude-latitude map of the degree of symmetry. Can be used to make illustrating
plots and to decide which filter is more appropriate.
Parameters
----------
coefficients
Spherical harmonic coefficients of a single voxel.
ell_max
Largest order :math:`\ell` used in the fitting. If :attr:`coefficients` contains higher
orders, it will be truncated.
resolution
Number or angular steps along a half-circle used in the search for the optimal axis.
filter
Weighing of different orders used to calculate the degree of symmetry.
Possible values are "ramp" and "square". By default no filter is applied (`None`).
Returns
---------
dos
Map of the calculated degree of symmetry.
theta
Polar angle coordinates of the map.
phi
Azimuthal angle coordinates of the map.
"""
# Use only the coefficients up until ell_max
truncated_coefficients = coefficients[..., :(ell_max+1)*(ell_max+2)//2]
# Find the zonal coefficients
zonal_indexes = np.zeros((ell_max+1)*(ell_max+2)//2, dtype=bool)
inc = 0
f = np.ones(ell_max//2+1)
for ell in range(0, ell_max+1, 2):
zonal_indexes[inc + ell] = True
if filter == 'ramp':
f[ell//2] = ell
elif filter == 'square':
f[ell//2] = ell**2
inc += 2*ell + 1
# Load d matrices
d_matrices = load_d_matrices(ell_max)
# Loop through grid of directions
theta_points = np.linspace(0, np.pi, resolution, endpoint=True)
phi_points = np.linspace(0, 2*np.pi, 2*resolution, endpoint=True)
dos = np.zeros((resolution, 2*resolution))
for ii, theta in enumerate(theta_points):
for jj, phi in enumerate(phi_points):
rotated_coefficients = calculate_sph_coefficients_rotated_by_euler_angles(
truncated_coefficients,
Psi=-phi,
Theta=-theta,
Phi=None,
d_matrices=d_matrices)
# Check if degree of symmetry (DOS) is higher than maximum so far
zonal_coeffs = rotated_coefficients[..., zonal_indexes]
degree_of_symmetry = np.sum(zonal_coeffs**2*f, axis=-1)
dos[ii, jj] = degree_of_symmetry
theta, phi = np.meshgrid(theta_points, phi_points, indexing='ij')
dos = dos / np.sum(truncated_coefficients**2)
return dos, theta, phi
[docs]def symmetric_part_along_given_direction(coefficients: NDArray[float],
theta: NDArray[float],
phi: NDArray[float],
ell_max: int) -> NDArray[float]:
r""" Find the zonal harmonic coefficients along the given input directions. This can be
used if eigenvector analysis is used to generate the symmetry directions used for a
further zonal-harmonics refinement step.
Parameters
----------
coefficients
Voxel map of spherical harmonic coefficients.
theta
Voxel map of polar angles.
phi
Voxel map of azimuthal angles.
ell_max
Largest order :math:`\ell` used in the fitting. If :attr:`coefficients` contains higher orders,
it will be truncated.
Returns
---------
Zonal harmonics coefficients in the frame of reference corresponding
to the axis of highest degree of symmetry.
"""
# Use only the coefficients up until ell_max
truncated_coefficients = coefficients[..., :(ell_max+1)*(ell_max+2)//2]
# Find the zonal coefficients
zonal_indexes = np.zeros((ell_max+1)*(ell_max+2)//2, dtype=bool)
inc = 0
for ell in range(0, ell_max+1, 2):
zonal_indexes[inc + ell] = True
inc += 2*ell + 1
# Load d matrices
d_matrices = load_d_matrices(ell_max)
rotated_coefficients = calculate_sph_coefficients_rotated_by_euler_angles(
truncated_coefficients,
Psi=-phi,
Theta=-theta,
Phi=None,
d_matrices=d_matrices)
# Pick out symmetric part
zonal_coeffs = rotated_coefficients[..., zonal_indexes]
return zonal_coeffs