Wigner FFT + IFFT is not the identity for 'DH' sampling and low bandwidths

[PhilippMisofCH]

First of all, thanks again for this nice library. While working with it, I've noticed some unexpected results in my project, which eventually boil down to the fact that applying the inverse Wigner FFT on the Wigner FFT does not yield the original input, not even at low bandwidth (L=8) and dh sampling (This is actually one of the only cases that I've tested thoroughly). In contrast, an equivalent setup for the $S^2$ transforms seems to work as expected.

So I wondered if this is indeed a problem, or it is to be expected.

I've also attached a small example where I test this out:

L = 8
sampling = 'dh'
a = s2fft.sampling.s2_samples.phis_equiang(L, sampling=sampling)
b = s2fft.sampling.s2_samples.thetas(L, sampling=sampling)
g = s2fft.sampling.s2_samples.phis_equiang(L, sampling=sampling)
f_grid = jnp.meshgrid(g, b, a, sparse=True, indexing='ij')
f_s2 = jnp.squeeze(jnp.sin(f_grid[1]) * jnp.cos(f_grid[2]))
flm = s2fft.transforms.spherical.forward_jax(f_s2, L, sampling=sampling, reality=True)
f_s2_back = s2fft.transforms.spherical.inverse_jax(flm, L, sampling=sampling)
assert jnp.allclose(f_s2, f_s2_back, atol=1e-4), ("Spherical forward + inverse transform is"
                                                  " not the identity")

f_so3 = jnp.sin(f_grid[0]) * jnp.cos(f_grid[1]) * jnp.sin(f_grid[2])
flmn = s2fft.transforms.wigner.forward_jax(f_so3, L, L, sampling=sampling, reality=True)
f_back = s2fft.transforms.wigner.inverse_jax(flmn, L, L, sampling=sampling)
assert jnp.allclose(f_so3, f_back, atol=1e-2), ("Wigner forward + inverse transform is not "
                                                "the identity")

The first assertion passes, but the second one fails.

I've also looked around at other reported issues and found e.g.

• 209, but this seems to only appear for higher values of N than I'm testing here

• https://github.com/astro-informatics/s2fft/pull/170#issuecomment-1831649757 but this is specific to the HEALPix as far as I understand.

It would be great if you could tell me if I misunderstood something or if this is indeed a bug / numerical issue. Thanks in advance!

[PhilippMisofCH]

Adding on this: At first sight it looks like one of the notebooks is demonstrating explicitly that this works, but here, one starts in the Fourier space, applies the inverse transformation and then goes back to the Fourier space, where all values agree.

Similarly, doing the following also passes the assertion:

rng = np.random.default_rng(83459)
flmn = s2fft.utils.signal_generator.generate_flmn(rng, L, L, reality=True)
f_so3 = s2fft.transforms.wigner.inverse_jax(flmn, L, L, sampling=sampling)
flmn = s2fft.transforms.wigner.forward_jax(f_so3, L, L, sampling=sampling, reality=True)
f_so3_back = s2fft.transforms.wigner.inverse_jax(flmn, L, L, sampling=sampling)
assert jnp.allclose(f_so3, f_so3_back, atol=1e-8), ("Wigner forward + inverse transform is not "
                                                "the identity")

But for an arbitrary (smooth) signal generated on $SO(3)$ this does not work as shown before.