Geographic Location Encoding with Spherical Harmonics and Sinusoidal Representation Networks

Paper TL;DR:

We recommend to use Siren for any problem involving geographic coordinates and Spherical Harmonic embeddings for problems that involve data at a global scale or in polar regions, which are not represented well by existing approaches.

if you want to cite our work, use:

@misc{rußwurm2023locationencoding,
      title={Geographic Location Encoding with Spherical Harmonics and Sinusoidal Representation Networks}, 
      author={Marc Rußwurm and Konstantin Klemmer and Esther Rolf and Robin Zbinden and Devis Tuia},
      year={2023},
      eprint={2310.06743},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

Getting Started

Check this Colab Notebook to get started.

It

1. clones the current repository
2. installs the required packages
3. trains Siren(SH) on the landocean dataset
4. predicts the following map:

Using Siren(SH)

To use Spherical Harmonic embedding function with SirenNet, i.e., Siren(SH(lon,lat)), in your code, instantiate the LocationEncoder class in locationencoder/locationencoder.py with Siren and Spherical Harmonics and a hyperparameter dict hparams as follows:

from locationencoder import LocationEncoder

hparams = dict(
    legendre_polys=10,
    dim_hidden=64,
    num_layers=2,
    optimizer=dict(lr=1e-4, wd=1e-3),
    num_classes=1
)

# Pytorch Lightning Model
model = LocationEncoder("sphericalharmonics", "siren", hparams)

# input longitude latitude in degrees
import torch
lonlat = torch.tensor([[51.9,5.6]], dtype=torch.float32)

# forward pass
output = model(lonlat)

Using Spherical Harmonics

We provide the analytically calculated Spherical Harmonic Basis Functions in locationencoder/pe/spherical_harmonics_ylm.py up to L=100 (! 40k lines, 7.7MB source code file). This file was generated with SymPy using locationencoder/pe/spherical_harmonics_generate_ylms.py.

To instantiate only the Spherical Harmonic positional encoder (no trainable weights) run

from locationencoder.pe import SphericalHarmonics
sh = SphericalHarmonics(legendre_polys=20)

# use lonlat coordinates, as above
embedded_lonlat = sh(lonlat) 

Reproduce Results

Run the following lines to reproduce results from the paper. The hyperparameter are configured in hparams.yaml.

Tables 1a (--dataset checkerboard) and 1b (--dataset landoceandataset) with

python experiments/exp_quantitative.py \ 
    --dataset landoceandataset \
    --nn linear fcnet siren \ 
    --pe sphericalharmonics theory grid spherec spherecplus spherem spheremplus direct cartesian3d wrap 

For Fig 4b) run

python experiments/exp_longitudinal_accuracy.py

For Fig 4a) run

python experiments/exp_resolution.py

Code Acknowledgements

This repository uses and relies on

• SymPy for the analytic calculation of spherical harmonic basis functions
• Pytorch Lightning for smooth model fitting
• the SirenNet implementation (Sitzmann et al., 2020) by Phil Wang/lucidrains