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[](https://colab.research.google.com/github/astro-informatics/s2fft/blob/main/notebooks/spherical_harmonic_transform.ipynb)
<imgalign="left"height="85"width="98"alt="s2fft logo - schematic representation of a tiled sphere"src="https://raw.githubusercontent.com/astro-informatics/s2fft/main//docs/assets/sax_logo.png">
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# Differentiable and accelerated spherical transforms
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angular resolution $L$. The diagram below illustrates the recursions
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(for further details see [Price & McEwen 2024]((https://arxiv.org/abs/2311.14670))).

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With this recursion to hand, the spherical harmonic coefficients of an
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isolatitudinally sampled map may be computed as a two step process. First,
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a 1D Fourier transform over longitude, for each latitudinal ring. Second,
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diagram below illustrates the separable spherical harmonic transform
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(for further details see [Price & McEwen 2024]((https://arxiv.org/abs/2311.14670))).

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## Sampling :earth_africa:
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precision but exhibit some error. However, the HEALPix sampling provides
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pixels of equal areas, which has many practical advantages.
<palign="center"><imgalt="Visualization of spherical sampling schemes"src="https://raw.githubusercontent.com/astro-informatics/s2fft/main/docs/assets/figures/spherical_sampling.png"width="700"></p>
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> [!NOTE]
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> For algorithmic reasons JIT compilation of HEALPix transforms can become slow at high bandlimits, due to XLA unfolding of loops which currently cannot be avoided. After compiling HEALPix transforms should execute with the efficiency outlined in the associated paper, therefore this additional time overhead need only be incurred once. We are aware of this issue and are working to fix it. A fix for CPU execution has now been implemented (see example [notebook](https://astro-informatics.github.io/s2fft/tutorials/spherical_harmonic/JAX_HEALPix_backend.html)). Fix for GPU execution is coming soon.
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