Differentiable and accelerated spherical transforms

S2FFT is a JAX package for computing Fourier transforms on the sphere and rotation group. It leverages autodiff to provide differentiable transforms, which are also deployable on modern hardware accelerators (e.g. GPUs and TPUs).

More specifically, S2FFT provides support for spin spherical harmonic and Wigner transforms (for both real and complex signals), with support for adjoint transformations where needed, and comes with different optimisations (precompute or not) that one may select depending on available resources and desired angular resolution \(L\).

Algorithms ⚡

S2FFT leverages new algorithmic structures that can he highly parallelised and distributed, and so map very well onto the architecture of hardware accelerators (i.e. GPUs and TPUs). In particular, these algorithms are based on new Wigner-d recursions that are stable to high angular resolution \(L\). The diagram below illustrates the recursions (for further details see Price & McEwen 2023).

Sampling 🌍

The structure of the algorithms implemented in S2FFT can support any isolattitude sampling scheme. A number of sampling schemes are currently supported.

The equiangular sampling schemes of McEwen & Wiaux (2012) and Driscoll & Healy (1995) are supported, which exhibit associated sampling theorems and so harmonic transforms can be computed to machine precision. Note that the McEwen & Wiaux sampling theorem reduces the Nyquist rate on the sphere by a factor of two compared to the Driscoll & Healy approach, halving the number of spherical samples required.

The popular HEALPix sampling scheme (Gorski et al. 2005) is also supported. The HEALPix sampling does not exhibit a sampling theorem and so the corresponding harmonic transforms do not achieve machine precision but exhibit some error. However, the HEALPix sampling provides pixels of equal areas, which has many practical advantages.

Contributors ✨

Thanks goes to these wonderful people (emoji key):

_{Matt Price}
💻 👀 🤔

_{Jason McEwen}
💻 👀 🤔

_{Matt Graham}
💻 👀

_sfmig
💻 👀

_{Devaraj Gopinathan}
💻

_{Francois Lanusse}
💻 🐛

We encourage contributions from any interested developers. A simple first addition could be adding support for more spherical sampling patterns!

Attribution 📚

Should this code be used in any way, we kindly request that the following article is referenced. A BibTeX entry for this reference may look like:

@article{price:s2fft,
   AUTHOR      = "Matthew A. Price and Jason D. McEwen",
   TITLE       = "TBA",
   YEAR        = "2023",
   EPRINT      = "arXiv:0000.00000"
}

You might also like to consider citing our related papers on which this code builds:

@article{mcewen:fssht,
    AUTHOR      = "Jason D. McEwen and Yves Wiaux",
    TITLE       = "A novel sampling theorem on the sphere",
    JOURNAL     = "IEEE Trans. Sig. Proc.",
    YEAR        = "2011",
    VOLUME      = "59",
    NUMBER      = "12",
    PAGES       = "5876--5887",
    EPRINT      = "arXiv:1110.6298",
    DOI         = "10.1109/TSP.2011.2166394"
}

@article{mcewen:so3,
    AUTHOR      = "Jason D. McEwen and Martin B{\"u}ttner and Boris ~Leistedt and Hiranya V. Peiris and Yves Wiaux",
    TITLE       = "A novel sampling theorem on the rotation group",
    JOURNAL     = "IEEE Sig. Proc. Let.",
    YEAR        = "2015",
    VOLUME      = "22",
    NUMBER      = "12",
    PAGES       = "2425--2429",
    EPRINT      = "arXiv:1508.03101",
    DOI         = "10.1109/LSP.2015.2490676"
}

License 📝

We provide this code under an MIT open-source licence with the hope that it will be of use to a wider community.

S2FFT is free software made available under the MIT License. For details see the LICENSE file.