LebedevExpander

Lebedev grid-based spherical harmonics expander.

This class is a specialized version of the SphericalHarmonicsExpander that uses Lebedev grids for sampling points on the sphere.

param max_degree:

Maximum degree of spherical harmonics expansion.

type max_degree:

int

param n_quadrature_points:

Number of quadrature points to use for Lebedev sampling. Possible values are listed in the table below. If the value is larger than the maximum number of points available (5810) for the given degree, it is clipped to 5810.

type n_quadrature_points:

int

param expansion_type:

Type of expansion to perform. Can be either ‘cartesian’ or ‘radial’.

type expansion_type:

str

param use_minimal_point_set:

If True, use a minimal point set for the Lebedev quadrature. For every degree, a corresponding number of quadrature points is used, which is the minimum number of points. The table below lists the expansion degree and the corresponding minimal number of points:

Table 1 Lebedev quadrature point reference

Degree	Points
2	6
3	14
4	26
5	38
6	50
7	74
8	86
9	110
10	146
11	170
12	194
13	230
14	266
15	302
16	350
18	434
21	590
24	770
27	974
30	1202
33	1454
36	1730
39	2030
42	2354
45	2702
48	3074
51	3470
54	3890
57	4334
60	4802
63	5294
66	5810

If False, the number of quadrature points is set to the maximum number of points available for the given degree, which is 5810 for degree 66.

type use_minimal_point_set:

bool

param normalize_spectrum:

Normalize power spectrum sum to 1. If False, the power spectrum is not normalized.

type normalize_spectrum:

bool

napari_stress.approximation.LebedevExpander.coefficients_

Spherical harmonics coefficients of shape (3, max_degree + 1, max_degree + 1) for ‘cartesian’ expansion type or (1, max_degree + 1, max_degree + 1) for ‘radial’ expansion type.

Type:

np.ndarray

napari_stress.approximation.LebedevExpander.properties

Dictionary containing properties of the expansion.

• normals: np.ndarray

  Outward normals at Lebedev quadrature points.

• mean_curvature: np.ndarray

  Mean curvature \(H\) at Lebedev quadrature points, computed as

  \[H = \frac{1}{2}(k_1 + k_2)\]

  where \(k_1\) and \(k_2\) are the principal curvatures.

• H0_arithmetic: float

  Arithmetic average of mean curvature:

  \[H_0^{\mathrm{arith}} = \frac{1}{N} \sum_{i=1}^N H_i\]

  where \(N\) is the number of quadrature points.

• H0_surface_integral: float

  Surface-area-weighted average mean curvature:

  \[H_0^{\mathrm{surf}} = \frac{\int_S H \, dA}{\int_S dA}\]

  where \(S\) is the surface, \(H\) is the mean curvature, and \(dA\) is the surface element.

• H0_volume_integral: float

  Average mean curvature estimated via the volume integral of the manifold:

  \[H_0^{\mathrm{vol}} \approx \left( \frac{4\pi}{3V} \right)^{1/3}\]

  where \(V\) is the volume enclosed by the surface.

• S2_volume_integral: float

  Volume of the unit sphere, typically \(\frac{4}{3}\pi\).

• H0_radial_surface: float

  Surface-area-weighted average mean curvature on the radially expanded surface. Only calculated if expansion_type is ‘radial’.

• Gauss_Bonnet_error_radial: float

  Gauss-Bonnet error for the radial expansion, calculated as:

  \[\int_{\mathcal{M}} f(\theta, \phi) \, dA - 4\pi\]

  where \(K\) is the Gaussian curvature.

• Gauss_Bonnet_relative_error_radial: float

  Relative Gauss-Bonnet error for the radial expansion, calculated as:

  \[\frac{\int_{\mathcal{M}} f(\theta, \phi) \, dA - 4\pi}{4\pi}\]

• power_spectrum: np.ndarray

  Power spectrum of spherical harmonics coefficients. If ‘normalize_spectrum’ is set to True, the power spectrum is normalized to sum to 1.

  The spectrum \(P_l\) is calculated as:

  \[P_l = \sum_{m=-l}^{l} |a_{lm}|^2\]

  where \(a_{lm}\) are the spherical harmonics coefficients.

Type:

dict

napari_stress.approximation.LebedevExpander.fit(points: 'napari.types.PointsData')

Fit Lebedev quadrature points to input data.

napari_stress.approximation.LebedevExpander.expand() → 'napari.types.SurfaceData'

Expand spherical harmonics using Lebedev quadrature points. Calculates properties of the expansion as listed above.

napari_stress.approximation.LebedevExpander.fit_expand(points: 'napari.types.PointsData') → 'napari.types.SurfaceData'

Fit Lebedev quadrature points to input data and then expand them.