FOSCAT

FOSCAT (Field cOmpression via Scattering CovariAnce Transform) is a Python library for wavelet/scattering-based statistical analysis, synthesis, denoising, and deep learning on 2D fields and on the sphere with HEALPix pixelisation.

The central idea: any spatial field can be summarised by a compact vector of scattering-covariance statistics Φ(d) — encoding multi-scale, cross-orientation correlations beyond what a power spectrum captures. FOSCAT can then synthesise a new field matching those statistics, use them as a loss for denoising or component separation, or feed them into neural-network architectures that operate directly on spherical geometry.

---

Install

pip install foscat

Full environment for running the example notebooks:

micromamba create -n foscat python=3.10
micromamba activate foscat
pip install foscat torch healpy xarray gcsfs zarr jupyterlab matplotlib scipy

Backend note: FOSCAT uses PyTorch as its only active backend (TensorFlow and NumPy backends are no longer maintained).

---

Minimal HEALPix synthesis

import numpy as np
import foscat.scat_cov as sc
from foscat.Synthesis import Loss, Synthesis

nside = 64
target_map = np.random.randn(12 * nside**2)

# build the scattering operator and compute target statistics
scat_op     = sc.funct(KERNELSZ=5, NORIENT=4, nstep_max=4, all_type='float64')
target_stat = scat_op.eval(target_map)

# define a differentiable loss
def synth_loss(x, scat_op, args):
    ref  = args[0]
    stat = scat_op.eval(x)
    return stat.reduce_mean_batch((stat - ref) ** 2)

loss   = Loss(synth_loss, scat_op, target_stat)
solver = Synthesis([loss], eta=0.03)

x0     = np.random.randn(12 * nside**2)
result = solver.run(x0, NUM_EPOCHS=300, EVAL_FREQUENCY=10)

---

What FOSCAT can do

Capability	Entry point	Example notebook
HEALPix synthesis	foscat.scat_cov.funct + Synthesis	Demo_Synthesis.ipynb
2D image synthesis	foscat.scat_cov2D.funct + Synthesis	Synthesis2D.ipynb
Denoising	two-loss synthesis (statistics + fidelity)	Denoising-2D.ipynb
Component separation	multi-loss synthesis	Remove_CMB.ipynb
Local spherical analysis	cell_ids in FoCUS / HealpixUNet	local_foscat.ipynb
Spherical U-Net (regression/segmentation)	HealpixUNet	CNN_local.ipynb, CNN_ecmwf.ipynb

Example notebooks are maintained in demo-foscat-pangeo-eosc.

---

Documentation

Full documentation (installation, mathematical background, API reference, user guide) is published at:

https://jmdelouis.github.io/FOSCAT/

To build the docs locally:

cd docs_sphinx
make install   # pip install requirements.txt
make html      # → _build/html/index.html
make livehtml  # live-reload on save (requires sphinx-autobuild)

---

Citation

If you use FOSCAT in a publication, please cite:

@software{foscat,
  author  = {Delouis, Jean-Marc and Foulquier, Theo},
  title   = {{FOSCAT}: Field cOmpression via Scattering CovariAnce Transform},
  year    = {2026},
  url     = {https://github.com/jmdelouis/FOSCAT},
  version = {2026.04.1},
}

The underlying statistical framework is described in:

• Allys et al. (2019), Rwst, a comprehensive statistical description of the non-Gaussian structures in the ISM, A&A 629, A115.
• Allys et al. (2020), New interpretable statistics for large-scale structure analysis and generation, Phys. Rev. D 102, 103506.

---

License

BSD 3-Clause. See LICENSE.