Evaluation#

Evaluation of band-limited fields at arbitrary points of the sphere and rotation group.

Through the factorisation of Eq. (B25), a spin field band-limited at \(L\) is a finite two-dimensional Fourier series, \({}_sf(\theta, \varphi) = \sum_{m,k} H_{mk}\, e^{\mathrm{i}m\varphi} e^{-\mathrm{i}k\theta}\), which can be evaluated exactly at any point, including points that do not lie on a sampling grid. This is what the real-space checks of rotations and equivariance require.

s2conv.utils.evaluation.spin_fourier(flm, spin: int = 0) → Array[source]#

Fourier coefficients of a band-limited spin field on the torus.

Uses \({}_sY_{\ell m}(\theta, \varphi) = (-1)^s \sqrt{(2\ell+1)/(4\pi)}\, d^{\ell}_{m,-s}(\theta)\, e^{\mathrm{i}m\varphi}\) (Eq. A7) with the factorisation of Eq. (B25).

Parameters:

  • flm – Spin-\(s\) harmonic coefficients with shape [..., L, 2L - 1].

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

Returns:

Coefficients \(H_{mk}\) with shape [..., 2L - 1, 2L - 1], indexed [..., L - 1 + m, L - 1 + k].

Return type:

jnp.ndarray

s2conv.utils.evaluation.evaluate_fourier(h, theta, phi) → Array[source]#

Evaluate \(\sum_{m,k} H_{mk}\, e^{\mathrm{i}m\varphi} e^{-\mathrm{i}k\theta}\).

Parameters:

  • h – Coefficients with shape [..., 2L - 1, 2L - 1].

  • theta – Colatitudes (any shape broadcastable with phi).

  • phi – Longitudes.

Returns:

Values with shape [..., *points].

Return type:

jnp.ndarray

s2conv.utils.evaluation.evaluate(flm, spin: int, theta, phi) → Array[source]#

Evaluate a band-limited spin field at arbitrary points.

Parameters:

  • flm – Spin-\(s\) harmonic coefficients with shape [..., L, 2L - 1].

  • spin (int) – Spin.

  • theta – Colatitudes.

  • phi – Longitudes.

Returns:

Field values with shape [..., *points].

Return type:

jnp.ndarray

s2conv.utils.evaluation.evaluate_so3(flmn, alpha, beta, gamma) → Array[source]#

Evaluate a band-limited function on the rotation group at one rotation.

Computes \(F(\rho) = \sum_\ell \frac{2\ell+1}{8\pi^2} \sum_{m,n} F^{\ell}_{mn} D^{\ell *}_{mn}(\rho)\) (Eq. 7).

Parameters:

  • flmn – Wigner coefficients with shape [..., 2N - 1, L, 2L - 1].

  • alpha – First Euler angle.

  • beta – Second Euler angle.

  • gamma – Third Euler angle.

Returns:

The value(s) \(F(\alpha, \beta, \gamma)\), one per leading index.

Return type:

jnp.ndarray