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

Qingde Li, Qingqi Hong, Zihan Li, Jie Tian

arXiv:2609.01408v2cs.AIcs.CVcs.GR

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.

Abstract

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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.Source: §I · p. 1
  • NeuSOGA targets Scan-to-CAD abstraction by transforming unstructured geometric observations into compact, editable, and mathematically meaningful representations.Source: §I · p. 2
  • The paper positions NeuSOGA against neural-symbolic and CAD reconstruction approaches that depend on statistical representations and training-data distributions.Source: §I · p. 2, B. Neuro-Symbolic AI and Machine… · p. 3, C. Learning-Based Scan-to-CAD… · pp. 3–4
  • The framework follows an O → T → G → S hierarchy from observations to topological skeletal abstractions, sparse geometric control polygons, and analytical implicit spline fields.Source: §I · p. 2
  • Implicit Area Splines provide an analytical representation with arbitrary-order smoothness, closed-form evaluation, editability, and additivity.Source: §I · pp. 2–3
  • Euclidean Distance Transforms provide topology-aware structural cores that guide foundation-model perception, while adaptive multi-scale abstraction compresses geometry into sparse representations.Source: §I · pp. 2–3

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

F. Scale-Space Theory and Multi-Scale Geometric Reasoning

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.Source: §III · pp. 4–5
  • The framework interprets abstraction as transforming observations into invariants and then into symbolic representations or mathematical concepts.Source: C. Toward Machine Mathematical… · p. 5

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

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.

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.Source: §V · p. 7
  • 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.Source: §V · p. 7
  • An observed point cloud is first projected onto a principal viewing plane to produce a two-dimensional occupancy mask.Source: A. Stage 1: Topological Abstraction… · p. 7
  • 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.Source: A. Stage 1: Topological Abstraction… · p. 7
  • 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.Source: B. Stage 2: Topology-Guided Perceptual… · p. 7

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.Source: C. Stage 3: Adaptive Multi-Scale… · p. 8
  • 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.Source: C. Stage 3: Adaptive Multi-Scale… · p. 8
  • Gaussian smoothing converts the binary importance mask into a continuous blending function for spatially coherent fusion.Source: C. Stage 3: Adaptive Multi-Scale… · p. 8
  • 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.Source: C. Stage 3: Adaptive Multi-Scale… · p. 8
  • Douglas-Peucker approximation compresses the hybrid contour into a sparse control polygon used as the symbolic geometric abstraction.Source: C. Stage 3: Adaptive Multi-Scale… · p. 8

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.

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.Source: G. Symbolic Representation as the… · p. 9, §VI · p. 9
  • Implicit Area Splines form the symbolic layer by representing geometry analytically as a scalar field with boundary F(x, y) = 0.Source: G. Symbolic Representation as the… · p. 9
  • The resulting symbolic object supports explicit mathematical structure, closed-form evaluation, arbitrary-order continuity, and additive composition.Source: G. Symbolic Representation as the… · p. 9
  • Experiments evaluate whether NeuSOGA transforms sensory observations into compact, accurate, and editable analytical representations without category-specific training distributions.Source: §VI · p. 9

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.Source: A. Experimental Setup · p. 9
  • Each object is normalized and projected onto the top XY, front XZ, and side YZ planes.Source: A. Experimental Setup · p. 9
  • For every projection, the complete neuro-symbolic pipeline is executed from observation through topological and geometric abstraction to symbolic representation.Source: A. Experimental Setup · p. 9
  • No category-specific training, CAD templates, geometric priors, or reconstruction supervision are used.Source: A. Experimental Setup · p. 9
  • Evaluation considers structural preservation, topological consistency, representation compactness, and symbolic expressiveness.Source: B. Evaluation Criteria · p. 9

Airplane Reconstruction

Chair Reconstruction

  • The chair results preserve seat, backrest, and support-leg structures while maintaining correspondence with the original topology after simplification.Source: D. Chair Reconstruction · p. 10

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

Robustness Across Modalities and 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.

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.Source: H. Explainability Through Symbolic… · p. 15, §VII · p. 16

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.Source: H. Explainability Through Symbolic… · p. 15, §VII · p. 16

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.Source: §I · p. 15, §VII · p. 16

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.Source: §I · p. 15

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.Source: J. Limitations and Future Directions · p. 15
  • 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.Source: J. Limitations and Future Directions · p. 16

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