[](: Research Prototype](https://img.shields.io/badge/status-research%20prototypeL-UoHull/NeuSOGA)

NeuSOGA is a neuro-symbolic framework for transforming observations into explicit symbolic mathematical representations through topology-guided geometric abstraction.

NeuSOGA employs the Implicit Area Spline formulation (Li and Tian, ACM Transactions on Graphics, 2009) as its symbolic representation layer, enabling the generation of explicit analytical models from topology-guided geometric abstractions.

The framework implements the abstraction hierarchy:

Observation (O)
        ↓
Topology (T)
        ↓
Geometry (G)
        ↓
Symbol (S)

where observations are progressively transformed into topological abstractions, geometric abstractions, and ultimately symbolic mathematical representations.

Overview of the NeuSOGA framework.

Modern AI systems often encode knowledge within latent neural representations. While highly effective for perception and prediction, these representations are typically difficult to inspect, manipulate, or analyse mathematically.

NeuSOGA explores an alternative paradigm:

Observation
      ↓
Topology
      ↓
Geometry
      ↓
Symbol

Rather than learning geometry solely through statistical optimization, NeuSOGA extracts topology-aware structure, performs adaptive geometric abstraction, and synthesizes explicit symbolic mathematical representations using analytical geometry.

The central research question is:

How can symbolic mathematical representations emerge from observations through a neuro-symbolic abstraction process?

NeuSOGA follows a four-stage abstraction hierarchy.

Geometric observations are acquired from point clouds or projected views.

Current implementation supports:

• ModelNet40 point-cloud objects
• Arbitrary-view projections
• Normalized geometric observations

The observation layer provides the raw spatial data from which symbolic representations are ultimately constructed.

Intrinsic structure is extracted through topology-guided analysis.

NeuSOGA computes:

• Euclidean Distance Transforms (EDT)
• Topology-aware structural cores
• Distance-field maxima
• Internal void structures
• Global spatial organization

The resulting topology nodes provide compact abstractions of object structure and subsequently guide perception.

Topology-aware structure is transformed into sparse geometric abstractions.

The geometry stage combines:

• Topology-guided Segment Anything (SAM) prompting
• Contour extraction
• Adaptive multi-scale scale-space abstraction
• Hybrid coarse-to-fine boundary refinement
• Sparse control-polygon generation

Dense observations are compressed into compact and interpretable geometric representations while preserving important structural detail.

The final stage of NeuSOGA transforms sparse geometric abstractions into explicit symbolic mathematical representations using the Implicit Area Spline formulation introduced by Li and Tian in ACM Transactions on Graphics (TOG), 2009.

Unlike conventional parametric representations such as Bézier curves, B-splines, and NURBS, which primarily describe boundary trajectories, the Implicit Area Spline formulation represents spatial regions directly through analytical implicit fields. As a result, the generated representation is not merely a geometric approximation, but an explicit mathematical object suitable for analysis, interpretation, and symbolic manipulation.

Given a sparse control polygon produced by the geometric abstraction stage, NeuSOGA constructs a continuous analytical field whose boundary is represented by

F(x,y) = 0

This symbolic representation possesses several important properties:

• Arbitrary-order smoothness through controllable (C^n) continuity.
• Closed-form analytical evaluation without numerical reconstruction.
• Additive composition, enabling complex structures to be represented through algebraic combinations of simpler components.
• Explicit symbolic accessibility, allowing direct inspection and mathematical analysis.
• Topology-preserving representation, including support for internal voids and complex geometric structure.

Within NeuSOGA, the Implicit Area Spline serves as the symbolic endpoint of the

Observation (O)
        ↓
Topology (T)
        ↓
Geometry (G)
        ↓
Symbol (S)

abstraction hierarchy.

The resulting symbolic model therefore encodes geometric knowledge in an explicit analytical form rather than within latent neural parameters. This distinction is central to NeuSOGA's objective of transforming observations into symbolic mathematical representations and providing explainability through explicit representation.

NeuSOGA/
│
├── assets/
│   └── figures/
│
├── demo/
│   └── neusoga_demo.py
│
├── notebooks/
│
├── paper/
│
├── README.md
├── requirements-demo.txt
├── LICENSE
└── CITATION.cff

This repository contains the reference implementation accompanying the NeuSOGA paper.

NeuSOGA was developed and validated primarily in Google Colab. The easiest way to reproduce the results is through Colab or a compatible Jupyter environment.

Run the following commands in the first notebook cell:

!pip install h5py opencv-python matplotlib scipy scikit-image numba
!pip install git+[https://github.com/facebookresearch/segment-anything.git](https://github.com/facebookresearch/segment-anything.git)
!wget -q [https://dl.fbaipublicfiles.com/segment_anything/sam_vit_b_01ec64.pth](https://dl.fbaipublicfiles.com/segment_anything/sam_vit_b_01ec64.pth)

The downloaded checkpoint:

sam_vit_b_01ec64.pth

will be used automatically by the NeuSOGA pipeline.

Runtime → Change runtime type → GPU

GPU acceleration is recommended but not required.

git clone https://github.com/QL-UoHull/NeuSOGA.git

cd NeuSOGA

python -m venv .venv

source .venv/bin/activate

python -m venv .venv

.venv\Scripts\activate

pip install rembg
pip install opencv-python matplotlib
pip install git+https://github.com/facebookresearch/segment-anything.git

wget https://dl.fbaipublicfiles.com/segment_anything/sam_vit_b_01ec64.pth

or manually download from:

https://github.com/facebookresearch/segment-anything

NeuSOGA automatically downloads and extracts the ModelNet40 dataset when executed for the first time.

The dataset is not bundled with this repository and no manual download is required.

NeuSOGA employs topology-guided perception using Meta's Segment Anything Model.

Required checkpoint:

sam_vit_b_01ec64.pth

Download:

wget -q https://dl.fbaipublicfiles.com/segment_anything/sam_vit_b_01ec64.pth

The framework automatically detects:

CPU
CUDA GPU

CPU execution is fully supported.

GPU acceleration is recommended for large-scale robustness experiments but is not required.

After executing the installation cell:

!python neusoga_demo.py

or execute the notebook cells sequentially.

Run:

python neusoga_demo.py

The script automatically:

1. Downloads ModelNet40 (if required).
2. Loads SAM.
3. Processes representative objects from all 40 ModelNet40 categories.
4. Generates arbitrary-view projections along:

[1, 1, 1]

5. Executes the complete:

O → T → G → S

abstraction hierarchy.

Results are written to:

robustness_results/

For each object, NeuSOGA generates an eight-stage visualization illustrating:

1. Observation (O)
2. Euclidean Distance Transform
3. Topology Nodes (T)
4. Topology-Guided Segmentation
5. Scale-Space Contour
6. Control Polygon (G)
7. Area Spline Field
8. Symbolic Boundary F(x,y)=0 (S)

These visualizations provide a transparent view of how symbolic mathematical representations emerge from geometric observations.

Recommended workflow:

1. Open the notebook in Colab.
2. Run the dependency installation cell shown above.
3. Ensure the SAM checkpoint has been downloaded.
4. Enable GPU runtime (optional but recommended).
5. Execute all notebook cells sequentially.

Launch Jupyter:

jupyter notebook

or

jupyter lab

Then:

1. Open the desired NeuSOGA notebook.
2. Install the required dependencies.
3. Download the SAM checkpoint.
4. Execute notebook cells in order.

notebooks/NeuSOGA_Demo.ipynb

End-to-end demonstration of the NeuSOGA pipeline.

notebooks/NeuSOGA_Robustness.ipynb

Robustness evaluation across object categories and viewpoints.

The current implementation evaluates NeuSOGA across:

• 40 object categories from ModelNet40
• Arbitrary-view projections
• Topology-aware segmentation
• Adaptive geometric abstraction
• Symbolic mathematical representation formation

The resulting visualizations demonstrate that the abstraction hierarchy remains stable across diverse geometric structures and viewing conditions.

A central objective of NeuSOGA is explainability through explicit representation.

Unlike conventional latent neural representations, every stage of the abstraction hierarchy remains observable:

Observation
      ↓
Topology
      ↓
Geometry
      ↓
Symbol

The generated symbolic models can be inspected, edited, analysed, and evaluated directly through their mathematical representation.

Neuro-Symbolic Geometric Abstraction (NeuSOGA): From Observations to Symbolic Mathematical Representations

arXiv:

https://arxiv.org/abs/2609.01408

@article{li2026neusoga,
  title={Neuro-Symbolic Geometric Abstraction (NeuSOGA):
         From Observations to Symbolic Mathematical Representations},
  author={Li, Qingde; et al},
  journal={arXiv preprint arXiv:2609.01408},
  year={2026},
  url={https://arxiv.org/abs/2609.01408}
}

@article{li20092d,
  title={2D Piecewise Algebraic Splines for Implicit Modeling},
  author={Li, Qiang and Tian, Jie},
  journal={ACM Transactions on Graphics (TOG)},
  volume={28},
  number={2},
  pages={1--19},
  year={2009}
}

This repository is distributed under the MIT License.

See LICENSE for details.