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Neuro-Symbolic Geometric Abstraction (NeuSOGA): From Observations to Symbolic Mathematical Representations
Qingde Li, Qingqi Hong, Zihan Li, Jie Tian
TL;DR
NeuSOGA addresses the challenge of converting perceptual observations into explicit symbolic representations rather than leaving geometric knowledge in latent parameters. It combines topology-guided perception, multi-scale geometric abstraction, and Implicit Area Spline synthesis, producing compact analytical models across modalities and viewing directions while leaving higher-level symbolic reasoning for future work.
Problem
AI systems often encode perceptual and geometric knowledge in latent neural parameters, limiting direct analytical inspection and manipulation.
Method
NeuSOGA progressively transforms observations into topological abstractions, geometric abstractions, and symbolic mathematical representations using EDTs, foundation-model perception, multi-scale abstraction, and Implicit Area Splines.
Results
NeuSOGA produces compact symbolic representations that preserve essential geometric and topological structure across sensing modalities and viewing directions.
Takeaways & Limitations
The framework provides an interpretable and explainable pathway from observations to explicit mathematical objects whose structure can be inspected and manipulated.
Takeaways & Limitations
The current system does not yet perform higher-level reasoning directly on generated symbols, and evaluation should expand to larger and more diverse datasets.
Abstract
from arXiv · showhide
A fundamental challenge in artificial intelligence is the transformation of observations into explicit symbolic representations suitable for abstraction, interpretation, and reasoning. While modern AI systems achieve remarkable perceptual capabilities through large-scale statistical learning, the resulting knowledge is typically encoded within latent parameters that are difficult to inspect or manipulate analytically. Inspired by Neuro-Symbolic AI and theories of human abstraction, this paper investigates the formation of symbolic mathematical representations from geometric observations. We propose NeuSOGA (Neuro-Symbolic Geometric Abstraction), a framework that progressively transforms observations into topological abstractions, geometric abstractions, and ultimately symbolic mathematical representations. The architecture combines topology-guided structural discovery using Euclidean Distance Transforms, foundation-model perception using Segment Anything, adaptive multi-scale geometric abstraction, and symbolic synthesis through Implicit Area Splines. The resulting representation is an analytical implicit model supporting arbitrary-order smoothness, additive composition, and closed-form evaluation. Unlike neural latent encodings, the generated representation remains interpretable, editable, and mathematically explicit. Experiments on ModelNet40 point clouds, arbitrary-view projections, and segmented optical observations demonstrate that NeuSOGA transforms diverse observations into compact symbolic representations while preserving essential geometric and topological structure across sensing modalities and viewing directions. NeuSOGA provides an interpretable and explainable pathway from observation to symbol and establishes
I. INTRODUCTION
NeuSOGA addresses the gap between powerful but latent neural perception and explicit symbolic mathematical representation by structuring abstraction from observations through topology and geometry. It combines deterministic topology-guided processing, foundation-model perception, multi-scale simplification, and analytical spline synthesis.
- Modern deep-learning systems achieve strong perception but commonly encode knowledge in latent neural parameters that are difficult to inspect or manipulate analytically.
- NeuSOGA targets Scan-to-CAD abstraction by transforming unstructured geometric observations into compact, editable, and mathematically meaningful representations.
- The paper positions NeuSOGA against neural-symbolic and CAD reconstruction approaches that depend on statistical representations and training-data distributions.
- The framework follows an O → T → G → S hierarchy from observations to topological skeletal abstractions, sparse geometric control polygons, and analytical implicit spline fields.
- Implicit Area Splines provide an analytical representation with arbitrary-order smoothness, closed-form evaluation, editability, and additivity.
- Euclidean Distance Transforms provide topology-aware structural cores that guide foundation-model perception, while adaptive multi-scale abstraction compresses geometry into sparse representations.
D. Topological Shape Abstraction
NeuSOGA treats topological and multi-scale structure as intermediate abstractions between observation and symbolic representation. It uses invariant structural descriptions and scale-space analysis to separate persistent geometry from noise and measurement artefacts.
- D. Topological Shape Abstraction: Topology supplies invariant structural information by describing relationships that remain unchanged under continuous deformation.
- D. Topological Shape Abstraction: Medial Axis Transform and Euclidean Distance Transforms provide explicit skeletal and intrinsic-geometric descriptions that support invariant extraction.
- F. Scale-Space Theory and Multi-Scale Geometric Reasoning: Scale-space analysis separates persistent structural features from noise and local measurement artefacts by comparing geometry across observation scales.
- C. Toward Machine Mathematical Abstraction: This hierarchy reflects a view of mathematical intelligence based on structured priors and invariant relationships rather than statistical learning alone.
- C. Toward Machine Mathematical Abstraction: The framework interprets abstraction as transforming observations into invariants and then into symbolic representations or mathematical concepts.
IV. THE IMPLICIT AREA SPLINE
The Implicit Area Spline represents planar regions as analytical scalar fields derived from oriented control polygons and edge-based algebraic components. This representation supports smooth, differentiable, editable, and symbolically manipulable geometry.
- A. Closed-Form Symbolic Representation: Area Splines convert topological abstractions into analytical expressions that can be manipulated independently of the original sensory data.
- A. Closed-Form Symbolic Representation: A planar shape is defined by a counter-clockwise control polygon, whose zero level-set yields the represented spatial region.
- A. Closed-Form Symbolic Representation: Each directed polygon edge contributes a signed algebraic distance component based on Green’s integral theorem and localized Cn-smooth blending weights.
- A. Closed-Form Symbolic Representation: The resulting field supports arbitrary-order smoothness and globally Cn-continuous piecewise-polynomial evaluation.
- A. Closed-Form Symbolic Representation: Analytical gradients and higher-order derivatives are available in closed form without finite differences.
- A. Closed-Form Symbolic Representation: Directly perturbing control-point coordinates enables symbolic geometric editing while preserving analytical evaluability and differentiability.
B. Additivity and Symbolic Composition
Area Splines encode regions as additive implicit fields, allowing composite shapes to be formed analytically while preserving symbolic structure beyond conventional boundary-based splines.
- Shared internal boundaries cancel through equal and opposite signed terms, so composite fields retain only external-boundary contributions.The cancellation follows from the additive area formulation.
- Additive field operations support symbolic composition without requiring explicit reconstruction of boundary topology.This provides an analytical alternative to numerically intensive CSG operations such as intersection, trimming, and topology maintenance.
- Area Splines serve as the final symbolic layer, projecting topological and geometric abstractions into explicit mathematical objects.The formulation connects sensory observations to an analytical domain suitable for mathematical abstraction.
- Area Splines represent spatial regions directly as implicit fields rather than parameterized boundary trajectories.This makes the region itself, rather than only its boundary, the represented object.
- The representation makes containment, spatial relationships, and topological structure intrinsic properties of the field.These properties support compositional symbolic reasoning beyond conventional parametric spline formulations.
V. METHODOLOGY: A NEURO-SYMBOLIC PIPELINE FOR GEOMETRIC ABSTRACTION
NeuSOGA progressively transforms observations into topological, geometric, and symbolic representations, explicitly separating structural reasoning from perception and latent neural encoding.
- The pipeline maps observational data through topological and geometric abstractions into symbolic mathematical representations.Its four stages are topological abstraction, perceptual isolation, geometric abstraction, and symbolic synthesis.
- NeuSOGA derives structural representations explicitly from topological and geometric principles rather than encoding them in neural network parameters.This distinguishes the framework from conventional learning-based Scan-to-CAD systems.
- An observed point cloud is first projected onto a principal viewing plane to produce a two-dimensional occupancy mask.
- Euclidean Distance Transforms identify local maxima corresponding to centres of maximal inscribed regions and topology-aware structural primitives.These primitives analytically summarize intrinsic spatial organization without task-specific training.
- The structural primitives automatically guide SAM/SAM2 segmentation, combining foundation-model perception with deterministic geometric reasoning.Topology guides perception rather than being discovered only through perception.
C. Stage 3: Adaptive Multi-Scale Geometric Abstraction
NeuSOGA uses adaptive multi-scale contour analysis to preserve global smoothness while selectively retaining local geometric detail before producing sparse control polygons.
- Each segmented contour is analyzed with Gaussian scale-space representations at coarse and fine resolutions.The coarse representation captures dominant global structure, while the fine representation retains local detail.
- Large deviations between coarse and fine contours identify regions where smoothing would remove significant geometric detail.An importance mask marks these regions using a deviation threshold.
- Gaussian smoothing converts the binary importance mask into a continuous blending function for spatially coherent fusion.
- Adaptive feature fusion preserves coarse geometry across most of the contour and reintroduces fine structure only where needed.The resulting contour combines global smoothness with local fidelity.
- Douglas-Peucker approximation compresses the hybrid contour into a sparse control polygon used as the symbolic geometric abstraction.
D. Stage 4: Symbolic Mathematical Synthesis
NeuSOGA synthesizes sparse geometric abstractions into explicit implicit Area Spline fields whose additive structure supports smooth, editable, and topology-preserving symbolic composition.
- Each control polygon is converted into an analytical implicit Area Spline using a Piecewise Algebraic Spline formulation.
- Complete objects are synthesized by additive accumulation of component fields.Strict additivity allows composition without explicit boundary merging or topological repair.
- The output has explicit mathematical structure, arbitrary-order smoothness, exact analytical evaluation, and direct editability through control points.
- The abstraction hierarchy transforms raw observations into structural cores, perceptual components, geometric primitives, and analytical symbolic representations.
- Strict additivity assumes decomposed semantic parts form a quasi-disjoint partition with oppositely oriented shared boundaries.For overlapping components, smooth implicit blending operators can control boundary behavior.
G. Symbolic Representation as the Outcome of Abstraction
NeuSOGA treats symbolic representation as the endpoint of a hierarchy that converts observations into topological and geometric abstractions before producing an explicit mathematical model.
- The framework converts observations into topological invariants, geometric abstractions, and symbolic mathematical representations.
- Implicit Area Splines form the symbolic layer by representing geometry analytically as a scalar field with boundary F(x, y) = 0.
- The resulting symbolic object supports explicit mathematical structure, closed-form evaluation, arbitrary-order continuity, and additive composition.
- Experiments evaluate whether NeuSOGA transforms sensory observations into compact, accurate, and editable analytical representations without category-specific training distributions.
A. Experimental Setup
The experimental setup evaluates NeuSOGA on normalized ModelNet40 point clouds across three orthographic views, using qualitative criteria centered on structural and symbolic fidelity.
- Experiments use ModelNet40 point-cloud models to test robustness across substantially different structural configurations.
- Each object is normalized and projected onto the top XY, front XZ, and side YZ planes.
- For every projection, the complete neuro-symbolic pipeline is executed from observation through topological and geometric abstraction to symbolic representation.
- No category-specific training, CAD templates, geometric priors, or reconstruction supervision are used.
- Evaluation considers structural preservation, topological consistency, representation compactness, and symbolic expressiveness.
- Airplane Reconstruction: The airplane abstraction preserves high-curvature features and fuselage geometry using 97 to 132 control points across projections.
- Chair Reconstruction: The chair results preserve seat, backrest, and support-leg structures while maintaining correspondence with the original topology after simplification.
E. Lamp Reconstruction
Lamp experiments show NeuSOGA capturing smooth curved and symmetric geometry, while broader evaluations test whether the same abstraction process transfers across modalities and viewing directions.
- Lamp Reconstruction: The lamp provides a complementary test case through strong rotational symmetry and smooth curved surfaces.
- Lamp Reconstruction: NeuSOGA preserves the lamp’s hemispherical shade, cylindrical support, and circular base across three projections.
- Lamp Reconstruction: The lamp experiment demonstrates transformation of noisy point-cloud observations into smooth analytical geometric descriptions.
- Robustness Across Modalities and Viewing Directions: Robustness studies evaluate segmented COCO optical observations and non-canonical ModelNet40 projections along the (1, 1, 1) direction.
- Robustness Across Modalities and Viewing Directions: For binary observations, EDT-derived topology guides segmentation, adaptive abstraction produces sparse control polygons, and Area Splines synthesize the symbolic output.
- Robustness Across Modalities and Viewing Directions: Across modalities, the representation layer remains unchanged and produces explicit analytical outputs with smoothness, additive composition, and symbolic interpretability.
- Robustness Across Modalities and Viewing Directions: The abstraction operates on structural and topological information that remains largely invariant across viewing directions.
G. Why Symbolic Expressiveness in 2D Matters
NeuSOGA frames 2D symbolic contour representations as meaningful intermediate abstractions that preserve semantic structure and support later manipulation and reasoning.
- Visual theories describe structured 2D contours, boundaries, edges, and local geometry as intermediate representations in visual understanding.
- NeuSOGA transforms observations into explicit 2D symbolic representations rather than directly reconstructing complete 3D CAD models.
- The resulting implicit functions possess semantic structure, analytical properties, and compositional behavior, unlike latent neural representations.
- Compact Area Spline representations demonstrate formation of explicit geometric symbols that may support higher-level reasoning, concept formation, and symbolic manipulation.
- Area Splines encode spatial regions, topological relationships, and compositional structure within one analytical implicit field.
H. Explainability Through Symbolic Representation
NeuSOGA maintains explicit, inspectable representations from topology through geometry to symbolic mathematical fields, making the transformation interpretable and editable. Its Area Spline endpoint supports compositional symbolic manipulation, while higher-level reasoning and broader evaluation remain future work.
- Explicit Representation: NeuSOGA keeps topological primitives, geometric control polygons, and symbolic Area Spline formulations explicit throughout its abstraction hierarchy.These stages remain observable rather than being encoded solely in hidden neural parameters.
- Explainability: The resulting representations can be inspected, edited, differentiated, composed, and mathematically analysed without hidden neural parameters.This provides intrinsic explainability by construction rather than post-hoc explanation.
- Empirical Scope: Experiments across object categories, sensing modalities, and viewing directions generated compact symbolic representations while preserving essential geometric and topological structure.The reported evaluations include ModelNet40 point clouds, arbitrary-view projections, and segmented optical observations.
- Symbolic Representation: Area Splines represent spatial regions directly through analytical implicit fields and support additive symbolic composition for subsequent reasoning and manipulation.This distinguishes them from boundary-focused Bézier, B-spline, and NURBS formulations.
- Limitations: The current framework forms symbolic representations but does not yet perform higher-level reasoning directly on them.Automatic discovery of geometric properties, structural invariants, symbolic relations, and conceptual hierarchies remains open.
- Limitations: Future evaluation should test larger and more diverse datasets under changes in observation quality, sensing modality, occlusion, and geometric complexity.These conditions define the stated boundary for assessing representation stability.