Utility Functions#

HEALPix fast Fourier transform planning functions#
Function Name	Description
`healpix_ifft()`	Wrapper function for the Inverse Fast Fourier Transform with spectral folding in the polar regions to mitigate aliasing.
`healpix_ifft_numpy()`	Computes the Inverse Fast Fourier Transform with spectral folding in the polar regions to mitigate aliasing (NumPy).
`healpix_ifft_jax()`	Computes the Inverse Fast Fourier Transform with spectral folding in the polar regions to mitigate aliasing (JAX).
`healpix_ifft_torch()`	Computes the Inverse Fast Fourier Transform with spectral folding in the polar regions to mitigate aliasing (Torch).
`healpix_fft()`	Wrapper function for the Forward Fast Fourier Transform with spectral back-projection in the polar regions to manually enforce Fourier periodicity.
`healpix_fft_numpy()`	Computes the Forward Fast Fourier Transform with spectral back-projection in the polar regions (NumPy).
`healpix_fft_jax()`	Computes the Forward Fast Fourier Transform with spectral back-projection in the polar regions (JAX).
`healpix_fft_torch()`	Computes the Forward Fast Fourier Transform with spectral back-projection in the polar regions (Torch).
`spectral_folding()`	Folds higher frequency Fourier coefficients back onto lower frequency coefficients (NumPy).
`spectral_folding_jax()`	Folds higher frequency Fourier coefficients back onto lower frequency coefficients (JAX).
`spectral_folding_torch()`	Folds higher frequency Fourier coefficients back onto lower frequency coefficients (Torch).
`spectral_periodic_extension()`	Extends lower frequency Fourier coefficients onto higher frequency coefficients (NumPy).
`spectral_periodic_extension_jax()`	Extends lower frequency Fourier coefficients onto higher frequency coefficients (JAX).
`spectral_periodic_extension_torch()`	Extends lower frequency Fourier coefficients onto higher frequency coefficients (Torch).

Quadrature functions.#
Function Name	Description
`quad_weights_transform()`	Compute quadrature weights for \(\theta\) and \(\phi\) integration to use in transform for various sampling schemes.
`quad_weights()`	Compute quadrature weights for \(\theta\) and \(\phi\) integration for various sampling schemes.
`quad_weights_hp()`	Compute HEALPix quadrature weights for \(\theta\) and \(\phi\) integration.
`quad_weights_dh()`	Compute DH quadrature weights for \(\theta\) and \(\phi\) integration.
`quad_weights_mw()`	Compute MW quadrature weights for \(\theta\) and \(\phi\) integration.
`quad_weights_mwss()`	Compute MWSS quadrature weights for \(\theta\) and \(\phi\) integration.
`quad_weight_dh_theta_only()`	Compute DH quadrature weight for \(\theta\) integration (only), for given \(\theta\).
`quad_weights_mw_theta_only()`	Compute MW quadrature weights for \(\theta\) integration (only).
`quad_weights_mwss_theta_only()`	Compute MWSS quadrature weights for \(\theta\) integration (only).
`mw_weights()`	Compute MW weights given as a function of index m.

Note

JAX and Torch versions of these functions share an almost identical function trace and are simply accessed by the sub-modules quadrature_jax() and quadrature_torch() respectively.

Periodic resampling functions#
Function Name	Description
`periodic_extension()`	Perform period extension of MW/MWSS signal on the sphere in harmonic domain, extending \(\theta\) domain from \([0,\pi]\) to \([0,2\pi]\).
`periodic_extension_spatial_mwss()`	Perform period extension of MWSS signal on the sphere in spatial domain, extending \(\theta\) domain from \([0,\pi]\) to \([0,2\pi]\).
`upsample_by_two_mwss()`	Upsample MWSS sampled signal on the sphere defined on domain \([0,\pi]\) by a factor of two.
`upsample_by_two_mwss_ext()`	Upsample an extended MWSS sampled signal on the sphere defined on domain \([0,2\pi]\) by a factor of two.
`downsample_by_two_mwss()`	Downsample an MWSS sampled signal on the sphere.
`unextend()`	Unextend MW/MWSS sampled signal from \(\theta\) domain \([0,2\pi]\) to \([0,\pi]\).
`mw_to_mwss_phi()`	Convert \(\phi\) component of signal on the sphere from MW sampling to MWSS sampling.
`mw_to_mwss_theta()`	Convert \(\theta\) component of signal on the sphere from MW sampling to MWSS sampling.
`mw_to_mwss()`	Convert signal on the sphere from MW sampling to MWSS sampling.

Signal generating functions#
Function Name	Description
`generate_flm()`	Generate a 2D set of random harmonic coefficients.
`generate_flmn()`	Generate a 3D set of random Wigner coefficients.

Note

JAX and Torch versions of these functions share an almost identical function trace and are simply accessed by the sub-modules resampling_jax() and resampling_torch() respectively.

Rotation functions#
Function Name	Description
`rotate_flms()`	Euler rotates spherical harmonic coefficients by given angle in zyz convention.
`generate_rotate_dls()`	Generates an array of all reduced Wigner d-function coefficients for angle beta.