@@ -51,7 +59,7 @@ torch-harmonics uses PyTorch primitives to implement these operations, making it
torch-harmonics has been used to implement a variety of differentiable PDE solvers which generated the animations below. Moreover, it has enabled the development of Spherical Fourier Neural Operators (SFNOs) [1].
@@ -94,46 +103,46 @@ docker run --gpus all -it --rm --ipc=host --ulimit memlock=-1 --ulimit stack=671
### Spherical harmonics
The [spherical harmonics](https://en.wikipedia.org/wiki/Spherical_harmonics) are special functions defined on the two-dimensional sphere $S^2$ (embedded in three dimensions). They form an orthonormal basis of the space of square-integrable functions defined on the sphere ($L^2(S^2)$) and are comparable to the harmonic functions defined on a circle/torus. The spherical harmonics are defined as
The [spherical harmonics](https://en.wikipedia.org/wiki/Spherical_harmonics) are special functions defined on the two-dimensional sphere $S^2$ (embedded in three dimensions). They form an orthonormal basis of the space of square-integrable functions defined on the sphere $L^2(S^2)$ and are comparable to the harmonic functions defined on a circle/torus. The spherical harmonics are defined as
where $\theta$ and $\lambda$ are colatitude and longitude respectively, and $P_l^m$ the normalized, [associated Legendre polynomials](https://en.wikipedia.org/wiki/Associated_Legendre_polynomials).
realizes the projection of a signal $f(\theta, \lambda)$ on $S^2$ onto the spherical harmonics basis.
generalizes the Fourier transform on the sphere.
The truncated series expansion of a function $f$ defined on the surface of a sphere can be written as
realizes the projection of a signal $f(\theta, \lambda)$ on $S^2$ onto the spherical harmonics basis. The SHT generalizes the Fourier transform on the sphere. Conversely, a truncated series expansion of a function $f$ can be written in terms of spherical harmonics as
where $\theta$ is the colatitude, $\lambda$ the longitude, $\overline{P_l^m}$ the normalized, associated Legendre polynomials and $F_n^m$, the expansion coefficient associated to the mode $(m,n)$.
where $\hat{f}_l^m$, are the expansion coefficients associated to the mode $m$, $n$.
The implementation of the SHT follows the algorithm as presented in [2]. A direct spherical harmonic transform can be accomplished by a Fourier transform
The second integral, which computed the projection onto the Legendre polynomials is realized with quadrature. On the Gaussian grid, we use Gaussian quadrature in the $\cos \theta$ domain. The integral
$$
\tilde{f}_l^m = \frac{1}{2} \int_{-1}^1 \tilde{f}^m(\arccos x) \,\overline{P_l^m} (x) \;\mathrm{d} x
