quadrature

s2fft.utils.quadrature.mw_weights(m: int) → float

Compute MW weights given as a function of index m.

MW weights are defined by

\[w(m^\prime) = \int_0^\pi \text{d} \theta \sin \theta \exp(i m^\prime\theta),\]

which can be computed analytically.

Parameters:

m (int) – Harmonic weight index.

Returns:

MW weight.

Return type:

float

s2fft.utils.quadrature.quad_weight_dh_theta_only(theta: float, L: int) → float

Compute DH quadrature weight for \(\theta\) integration (only), for given \(\theta\).

Parameters:

• theta (float) – \(\theta\) angle for which to compute weight.

• L (int) – Harmonic band-limit.

Returns:

Weight computed for each \(\theta\).

Return type:

float

s2fft.utils.quadrature.quad_weights(L: int | None = None, sampling: str = 'mw', spin: int = 0, nside: int | None = None) → ndarray

Compute quadrature weights for \(\theta\) and \(\phi\) integration for various sampling schemes.

Parameters:

• L (int, optional) – Harmonic band-limit. Required if sampling not healpix. Defaults to None.

• sampling (str, optional) – Sampling scheme. Supported sampling schemes include {“mw”, “mwss”, “dh”, “gl”, “healpix”}. Defaults to “mw”.

• spin (int, optional) – Harmonic spin. Defaults to 0.

• nside (int, optional) – HEALPix Nside resolution parameter. Only required if sampling=”healpix”. Defaults to None.

Raises:

ValueError – Invalid sampling scheme.

Returns:

Quadrature weights for sampling scheme for each \(\theta\) (weights are identical as \(\phi\) varies for given \(\theta\)).

Return type:

np.ndarray

s2fft.utils.quadrature.quad_weights_dh(L: int) → ndarray

Compute DH quadrature weights for \(\theta\) and \(\phi\) integration.

Parameters:

L (int) – Harmonic band-limit.

Returns:

Weights computed for each \(\theta\) (weights are identical as \(\phi\) varies for given \(\theta\)).

Return type:

np.ndarray

s2fft.utils.quadrature.quad_weights_gl(L: int) → ndarray

Compute GL quadrature weights for \(\theta\) and \(\phi\) integration.

Parameters:

L (int) – Harmonic band-limit.

Returns:

Weights computed for each \(\theta\) (weights are identical as \(\phi\) varies for given \(\theta\)).

Return type:

np.ndarray

s2fft.utils.quadrature.quad_weights_hp(nside: int) → ndarray

Compute HEALPix quadrature weights for \(\theta\) and \(\phi\) integration.

Note

HEALPix weights are identical for all pixels. Nevertheless, an array of weights is returned (with identical values) for consistency of interface across other sampling schemes.

Parameters:

nside (int) – HEALPix Nside resolution parameter.

Returns:

Weights computed for each \(\theta\) (all weights in array are identical).

Return type:

np.ndarray

s2fft.utils.quadrature.quad_weights_mw(L: int, spin: int = 0) → ndarray

Compute MW quadrature weights for \(\theta\) and \(\phi\) integration.

Parameters:

• L (int) – Harmonic band-limit.

• spin (int, optional) – Harmonic spin. Defaults to 0.

Returns:

Weights computed for each \(\theta\) (weights are identical as \(\phi\) varies for given \(\theta\)).

Return type:

np.ndarray

s2fft.utils.quadrature.quad_weights_mw_theta_only(L: int, spin: int = 0) → ndarray

Compute MW quadrature weights for \(\theta\) integration (only).

Parameters:

• L (int) – Harmonic band-limit.

• spin (int, optional) – Harmonic spin. Defaults to 0.

Returns:

Weights computed for each \(\theta\).

Return type:

np.ndarray

s2fft.utils.quadrature.quad_weights_mwss(L: int, spin: int = 0) → ndarray

Compute MWSS quadrature weights for \(\theta\) and \(\phi\) integration.

Parameters:

• L (int) – Harmonic band-limit.

• spin (int, optional) – Harmonic spin. Defaults to 0.

Returns:

Weights computed for each \(\theta\) (weights are identical as \(\phi\) varies for given \(\theta\)).

Return type:

np.ndarray

s2fft.utils.quadrature.quad_weights_mwss_theta_only(L: int, spin: int = 0) → ndarray

Compute MWSS quadrature weights for \(\theta\) integration (only).

Parameters:

• L (int) – Harmonic band-limit.

• spin (int, optional) – Harmonic spin. Defaults to 0.

Returns:

Weights computed for each \(\theta\).

Return type:

np.ndarray

s2fft.utils.quadrature.quad_weights_transform(L: int, sampling: str = 'mwss', spin: int = 0, nside: int = 0) → ndarray

Compute quadrature weights for \(\theta\) and \(\phi\) integration to use in transform for various sampling schemes.

Quadrature weights to use in transform for MWSS correspond to quadrature weights are twice the base resolution, i.e. 2 * L.

Parameters:

• L (int) – Harmonic band-limit.

• sampling (str, optional) – Sampling scheme. Supported sampling schemes include {“mwss”, “dh”, “gl”, “healpix}. Defaults to “mwss”.

• spin (int, optional) – Harmonic spin. Defaults to 0.

• nside (int, optional) – HEALPix Nside resolution parameter. Only required if sampling=”healpix”. Defaults to None.

Raises:

ValueError – Invalid sampling scheme.

Returns:

Quadrature weights to use in transform for sampling scheme for each \(\theta\) (weights are identical as \(\phi\) varies for given \(\theta\)).

Return type:

np.ndarray