Rotate a signal

This tutorial demonstrates how to use S2FFT to rotate a signal on the sphere.

If you are working on this notebook in Google Colab, you will need to have Google Colab install s2fft. You can do this by adding a cell to the top of the notebook with the following content:

!pip install s2fft &> /dev/null

and then running that cell.

A signal can be rotated in pixel space, however this can introduce artifacts. The best way to perform a rotation is through spherical harmonic space using the Wigner d-matrices, that is:

• forward spherical harmonic transform,

• rotation on the flm coefficients,

• inverse spherical harmonic transform.

Specifically, we will adopt the sampling scheme of McEwen & Wiaux (2012). For our purposes here we’ll just generate a random bandlimited signal.

import jax
import numpy as np

import s2fft

jax.config.update("jax_enable_x64", True)

L = 128
sampling = "mw"
rng = np.random.default_rng(12346161)
flm = s2fft.utils.signal_generator.generate_flm(rng, L)
f = s2fft.inverse(flm, L)

JAX is not using 64-bit precision. This will dramatically affect numerical precision at even moderate L.

Execute the rotation steps

First, we will run the JAX function to compute the spherical harmonic transform of our signal

flm = s2fft.forward_jax(f, L, reality=True)

Now apply the rotation (here \(\pi/2\) in each of alpha, beta, gamma) on the harmonic coefficients flm.

flm_rotated = s2fft.rotate_flms(flm, L, (np.pi / 2, np.pi / 2, np.pi / 2))

Finally, we will run the JAX function to compute the inverse spherical harmonic transform to get back to pixel space.

f_rotated = s2fft.inverse_jax(flm_rotated, L, reality=True)