DGNO: Discontinuous Galerkin Neural Operator for Pathology Defocus Deblurring

Shaoqing Duan, Haofei Song, Xintian Mao, Qingli Li, Yan Wang

East China Normal University

Official implementation of "Discontinuous Galerkin Neural Operator for Pathology Defocus Deblurring" (ICML 2026).

Defocus deblurring in pathological microscopy remains challenging due to the spatially varying and locally discontinuous nature of optical blur induced by a position-dependent integral imaging process. Existing deep learning methods, constrained by shift-invariance assumptions and limited interpretability, are not well suited to such heterogeneous blur patterns. Neural operators provide a principled alternative by modeling defocus formation directly as an integral operator. However, most existing neural operator architectures rely on globally parameterized kernels that assume smoothness and stationarity, limiting their ability to model heterogeneous and locally discontinuous blur. To address this, we propose the Discontinuous Galerkin Neural Operator (DGNO), which parameterizes the integral kernel using a discontinuous Galerkin formulation with element-local volume operators and interface numerical fluxes. DGNO provides a principled combination of locality, heterogeneity modeling, and global coherence while preserving the underlying physics of optical image formation.

Model variants

DGNO comes in two operator variants (see paper), both produced by a single training architecture DGNO (basicsr/models/archs/dgno_arch.py); the yml field network_g.variant selects which discontinuous Galerkin operator is instantiated:

Variant	network_g.variant	Description
DGNO-Face	face	Face-wise discontinuous Galerkin operator (DG_Win_SRNO_FaceWise)
DGNO-Cell	cell	Cell (element)-wise discontinuous Galerkin operator (DG_Win_SRNO)

Inference uses a single unified evaluation architecture DGNO_eval (basicsr/models/archs/dgno_eval_arch.py), into which both Face and Cell checkpoints load.

Installation

conda create -n dgno python=3.9 -y
conda activate dgno

pip install -r requirements.txt
pip install mamba-ssm          # compiled CUDA extension; needs a matching torch/CUDA toolchain
python setup.py develop --no_cuda_ext

A CUDA GPU is required (mamba-ssm's selective-scan kernels run on GPU).

Datasets

The paper evaluates DGNO on three microscopy defocus-deblurring datasets:

• BBBC006 (Broad Bioimage Benchmark Collection) — fluorescence microscopy in two channels: w1 (Hoechst-stained nuclei) and w2 (Phalloidin-stained actin). Single-channel grayscale, 696×520. Images at the optimal focal plane (z-stack = 16) are the sharp ground truth; defocused planes (z-stack = [2, 6, 10]) are the blurry inputs. 6,144 pairs, 4:1 train/test split. The preprocessed copy used in this work is mirrored on Hugging Face as BBBC006.tar.gz at Duane245/DGNO; extracting it yields the BBBC006/{train,test}/{GT,blur}/*.tif layout below directly.
• 3DHistech (Geng et al., Cervical cytopathology image refocusing via multi-scale attention features and domain normalization, Medical Image Analysis 2022) — cytopathology dataset captured with a 3DHistech digital scanner across multiple focal planes; the in-focus image is the sharp ground truth. RGB, 94,973 patches of 256×256, split 66,976 / 9,088 / 18,909 for train / val / test. Released by the original authors at ShenghuaCheng/Cytopathology-image-refocusing; download from Baidu Netdisk (access code: refo).

Place (or symlink) the data under datasets/:

datasets/
  BBBC006/
    train/{GT,blur}/*.tif
    test/{GT,blur}/*.tif
  3DHistech/
    train/{sharp,blur}/*.png
    val/{sharp,blur}/*.png
    test/{sharp,blur}/*.png

For BBBC006, download the preprocessed tarball (~3.2 GB, both w1 and w2 channels) from Hugging Face and extract it under datasets/:

pip install -U huggingface_hub
huggingface-cli download Duane245/DGNO BBBC006.tar.gz --local-dir datasets
tar -xzf datasets/BBBC006.tar.gz -C datasets && rm datasets/BBBC006.tar.gz

This yields datasets/BBBC006/{train,test}/{GT,blur}/*.tif directly — no further preprocessing needed. A single download covers both channels; the w1 / w2 split is selected at run time from the yml datasets.*.BBBCw field.

Pretrained checkpoints

Six checkpoints (3 datasets × 2 variants) are hosted on Hugging Face: https://huggingface.co/Duane245/DGNO

Dataset	DGNO-Face	DGNO-Cell
BBBC006 w1	DGNO_BBBC006_w1_Face.pth	DGNO_BBBC006_w1_Cell.pth
BBBC006 w2	DGNO_BBBC006_w2_Face.pth	DGNO_BBBC006_w2_Cell.pth
3DHistech	DGNO_3DHistech_Face.pth	DGNO_3DHistech_Cell.pth

The checkpoints are stored under pretrained/ inside the HF repo, so download into the repo root (--local-dir .) and they land in pretrained/:

pip install -U huggingface_hub

# all six checkpoints (the --include filter skips the BBBC006.tar.gz dataset)
huggingface-cli download Duane245/DGNO --include "pretrained/*" --local-dir .

# or a single checkpoint — e.g. 3DHistech / Face
huggingface-cli download Duane245/DGNO pretrained/DGNO_3DHistech_Face.pth --local-dir .

Training

Distributed training on 4 GPUs (basicsr/train.py under torch.distributed.run):

bash scripts/train_gpu0-3.sh options/DGNO_BBBC006_w1_Face.yml   # GPUs 0-3
bash scripts/train_gpu4-7.sh options/DGNO_BBBC006_w1_Cell.yml   # GPUs 4-7

Available configs in options/: DGNO_BBBC006_w1_{Face,Cell}.yml, DGNO_BBBC006_w2_{Face,Cell}.yml, DGNO_3DHistech_{Face,Cell}.yml.

Testing

# BBBC006 (w1 / w2) — channel is read from the yml
python test/test_bbbc.py \
    --config  options/DGNO_BBBC006_w1_Face.yml \
    --weights pretrained/DGNO_BBBC006_w1_Face.pth \
    --input_dir ./datasets/BBBC006/test --device cuda:0

# 3DHistech (RGB)
python test/test_3dhistech.py \
    --config  options/DGNO_3DHistech_Face.yml \
    --weights pretrained/DGNO_3DHistech_Face.pth \
    --input_dir ./datasets/3DHistech/test --device cuda:0

Add --save_images to write restored images to results/. Each script reports PSNR / SSIM / LPIPS.

Repository structure

basicsr/            trimmed BasicSR framework (training engine, data, metrics, utils)
  models/archs/     dgno_arch.py (training; variant=face|cell)  dgno_eval_arch.py
options/            6 training/eval configs (3 datasets × Face/Cell)
scripts/            distributed training launchers
test/               test_bbbc.py  test_3dhistech.py
pretrained/         place downloaded checkpoints here
datasets/           place datasets here

Acknowledgements

This codebase is built on BasicSR and Restormer. We thank the authors for their work.

Citation

@inproceedings{duan2026dgno,
  title         = {Discontinuous Galerkin Neural Operator for Pathology Defocus Deblurring},
  author        = {Duan, Shaoqing and Song, Haofei and Mao, Xintian and Li, Qingli and Wang, Yan},
  booktitle     = {International Conference on Machine Learning (ICML)},
  year          = {2026},
  eprint        = {2605.23282},
  archivePrefix = {arXiv},
  primaryClass  = {eess.IV},
  url           = {https://arxiv.org/abs/2605.23282}
}

Preprint: arXiv:2605.23282.