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import math
from typing import Optional, Tuple
import torch
import torch.nn as nn
from torch_harmonics import InverseRealSHT, RealSHT
from torch_harmonics.quadrature import QuadratureS2
from torch_harmonics.truncation import truncate_sht
[docs]
class SpectralConvS2(nn.Module):
r"""
Spectral convolution layer on :math:`S^2` implemented via real SHT
(Driscoll--Healy formulation, see https://api.semanticscholar.org/CorpusID:122817218).
Given a multi-channel input signal :math:`u^{c_i}(\theta, \lambda)` on the
sphere, the layer computes the output channels :math:`v^{c_o}(\theta, \lambda)`
in three steps:
1. **Forward SHT** (cf. :class:`~torch_harmonics.RealSHT`) -- transform
each input channel to spectral space:
.. math::
\hat{u}_l^{m,\,c_i} = \text{SHT}\!\left[\, u^{c_i}(\theta, \lambda) \,\right]
2. **Spectral contraction** -- mix channels with learnable weights
:math:`K_l^{c_o,\,c_i}` that are diagonal in :math:`(l, m)` (i.e.\ the
same weight is applied to every order :math:`m` at a given degree
:math:`l`):
.. math::
\hat{v}_l^{m,\,c_o}
= \sum_{c_i} K_l^{c_o,\,c_i}\; \hat{u}_l^{m,\,c_i}
3. **Inverse SHT** (cf. :class:`~torch_harmonics.InverseRealSHT`) --
transform back to the spatial domain:
.. math::
v^{c_o}(\theta, \lambda)
= \text{ISHT}\!\left[\, \hat{v}_l^{m,\,c_o} \,\right]
Because the spectral weights depend only on degree :math:`l` and not on order
:math:`m`, this corresponds to an **isotropic** (azimuthally symmetric)
convolution kernel on the sphere. When ``num_groups > 1``, the channel
contraction is performed independently within each group (grouped
convolution).
**Spectral bias.**
When ``bias=True``, a learnable spectral bias :math:`b_l^{m,\,c_i}` is
added to the SHT coefficients before the channel contraction. The bias is
modulated by the spatial integral (zeroth moment) of each input channel:
.. math::
I^{c_i} = \int_0^{2\pi}\!\int_0^{\pi}
u^{c_i}(\theta,\lambda)\,\sin\theta\;d\theta\;d\lambda
.. math::
\hat{u}_l^{m,\,c_i} \;\leftarrow\;
\hat{u}_l^{m,\,c_i} + I^{c_i}\, b_l^{m,\,c_i}
This allows the layer to learn a spectral response that depends on the
global mean of each input channel, effectively coupling the zero-frequency
content into all spectral modes.
Parameters
----------
in_shape : Tuple[int]
Spatial input grid shape ``(nlat, nlon)``.
out_shape : Tuple[int]
Spatial output grid shape ``(nlat, nlon)``.
in_channels : int
Number of input channels.
out_channels : int
Number of output channels.
num_groups : int, optional
Number of channel groups for grouped spectral weights, by default 1.
grid_in : str, optional
Grid used for the forward SHT (``"equiangular"``, ``"legendre-gauss"``,
``"lobatto"``, ``"equiangular-trapezoidal"``), by default ``"equiangular"``.
grid_out : str, optional
Grid used for the inverse SHT, same options as ``grid_in``.
bias : bool, optional
If ``True``, adds a learnable spectral bias computed from the spatial
integral, by default ``False``.
Examples
--------
>>> import torch
>>> import torch_harmonics as th
>>> conv = th.SpectralConvS2(
... in_shape=(128, 256), out_shape=(128, 256),
... in_channels=16, out_channels=32,
... ).cuda()
>>> x = torch.randn(4, 16, 128, 256, device="cuda")
>>> y = conv(x)
>>> y.shape
torch.Size([4, 32, 128, 256])
Raises
------
AssertionError
If ``in_channels`` or ``out_channels`` is not divisible by
``num_groups``.
Notes
-----
The SHT truncation ``lmax``/``mmax`` is the minimum of the input and output
truncations.
"""
def __init__(
self,
in_shape: Tuple[int],
out_shape: Tuple[int],
in_channels: int,
out_channels: int,
num_groups: Optional[int] = 1,
grid_in: Optional[str] = "equiangular",
grid_out: Optional[str] = "equiangular",
bias: Optional[bool] = False,
):
super().__init__()
if in_channels % num_groups != 0:
raise ValueError(f"in_channels ({in_channels}) must be divisible by num_groups ({num_groups})")
if out_channels % num_groups != 0:
raise ValueError(f"out_channels ({out_channels}) must be divisible by num_groups ({num_groups})")
# copy inputs
self.in_channels = in_channels
self.out_channels = out_channels
self.num_groups = num_groups
# compute truncation
lmax_in, mmax_in = truncate_sht(in_shape[0], in_shape[1], grid=grid_in)
lmax_out, mmax_out = truncate_sht(out_shape[0], out_shape[1], grid=grid_out)
# compute lmax and lmin
lmax = min(lmax_in, lmax_out)
mmax = min(mmax_in, mmax_out)
self.lmax = min(lmax, mmax)
self.mmax = self.lmax
# set up sht layers
self.sht = RealSHT(*in_shape, grid=grid_in, lmax=self.lmax, mmax=self.mmax)
self.isht = InverseRealSHT(*out_shape, grid=grid_out, lmax=self.lmax, mmax=self.mmax)
# weight shape
weight_shape = [num_groups, in_channels // num_groups, out_channels // num_groups, self.lmax]
# Compute scaling factor for correct initialization
scale = math.sqrt(1.0 / (in_channels // num_groups)) * torch.ones(self.lmax, dtype=torch.complex64)
# seemingly the first weight is not really complex, so we need to account for that
scale[0] *= math.sqrt(2.0)
self.weight = nn.Parameter(scale * torch.randn(*weight_shape, dtype=torch.complex64))
if bias:
self.spectral_bias = nn.Parameter(torch.zeros(1, self.in_channels, self.lmax, self.mmax, dtype=torch.complex64))
self.quadrature = QuadratureS2(img_shape=in_shape, grid=grid_in, normalize=False)
@torch.compile
def _contract_lwise(self, ac: torch.Tensor, bc: torch.Tensor) -> torch.Tensor:
resc = torch.einsum("bgixy,giox->bgoxy", ac, bc)
return resc
[docs]
def forward(self, x):
"""
Apply the spectral convolution.
Parameters
----------
x : torch.Tensor
Input signal of shape ``(batch, in_channels, nlat_in, nlon_in)``.
Returns
-------
torch.Tensor
Convolved signal of shape ``(batch, out_channels, nlat_out, nlon_out)``.
"""
dtype = x.dtype
with torch.amp.autocast(device_type=x.device.type, enabled=False):
x = x.to(torch.float32)
# compute integral in case if bias is used
if hasattr(self, "spectral_bias"):
integral = self.quadrature(x)
# perform SHT
x = self.sht(x).contiguous()
# store the shapes
B, C, H, W = x.shape
# deal with bias
if hasattr(self, "spectral_bias"):
x = x + integral.reshape(B, C, 1, 1) * self.spectral_bias
# perform contraction
x = x.reshape(B, self.num_groups, C // self.num_groups, H, W)
xp = self._contract_lwise(x, self.weight)
x = xp.reshape(B, self.out_channels, H, W).contiguous()
with torch.amp.autocast(device_type=x.device.type, enabled=False):
x = self.isht(x)
# convert datatype
x = x.to(dtype=dtype)
return x
