Symmetry-Structured Neural Completion of Islamic Geometric Patterns from Sparse Control Geometry

Technical Summary: Symmetry-Structured Neural Completion of Islamic Geometric Patterns

Problem Formulation
The paper addresses the challenge of completing Islamic geometric patterns from sparse control geometry. Unlike standard image synthesis, where symmetry is often learned as a statistical regularity, Islamic ornamentation is defined by exact rotational symmetry and strict construction rules. A small deviation in symmetry renders a pattern invalid within the tradition. The task is framed as a vector-geometric completion problem: given a sparse set of control points (anchors, boundary markers) and a target rotational symmetry order $N$, the system must predict a full vector graph. The output must consist of edges selected from a geometrically meaningful candidate lattice and bounded curve refinements, all adhering to exact $N$-fold rotational symmetry.

Methodology
The proposed framework integrates a Graph Neural Network (GNN) with explicit geometric constraints derived from the domain's construction rules.

1. Candidate Lattice and Orbit Structure: The system does not generate arbitrary shapes. Instead, it constructs a deterministic candidate lattice based on the input control geometry and symmetry order $N$. Vertices are placed on concentric rings, and candidate edges (ring edges, radial spokes, star-polygon chords, etc.) are enumerated. Crucially, these edges are partitioned into rotational orbits under the cyclic group $C_N$. An edge $e_i$ belongs to an orbit $O(e_i) = \{g \cdot e_i : g \in C_N\}$.
2. Neural Predictor: A message-passing GNN operates over this candidate lattice. It takes featurized vertices and edges as input and outputs a score $s_i$ and a bounded refinement parameter $\alpha_i$ for each candidate edge. The refinement parameter displaces the midpoint of an edge along its normal within a fixed bound, ensuring the curve remains within prescribed limits.
3. Decoding Regimes: The paper evaluates four decoding strategies to isolate the impact of symmetry structure:
   • Orbit-tied: The network averages scores and refinement parameters over each orbit during both training and inference. This enforces $s_i = s_j$ and $\alpha_i = \alpha_j$ for all edges in the same orbit.
   • Selection Projection: The network is trained without constraints; at inference, scores are aggregated to the orbit level before selection.
   • Full Projection: Both scores and refinements are averaged over orbits at inference.
   • Unstructured (Free): The network selects edges independently without any orbit information, serving as a negative baseline.

Key Contributions
The paper makes four primary contributions:

1. Symmetry-Structured Framework: It introduces a neural completion framework where constructional knowledge defines the candidate lattice, orbit structure, and validity-preserving decoder. The network predicts within this constrained space rather than learning rules from data.
2. Constructive Guarantee: The orbit-tied variant provides a mathematical guarantee (Proposition 1) that for any input and any orbit-level selection rule, the output will exhibit exact $N$-fold symmetry, preserve anchor points, and keep refinements within bounds. This validity is intrinsic to the architecture, not dependent on the accuracy of the learned weights.
3. Controlled Comparison: The study compares the four decoding regimes. It demonstrates that while orbit-tied training offers no fidelity advantage over inference-time projection, both symmetry-structured methods preserve validity with no measurable loss in edge-selection fidelity compared to unstructured decoding.
4. Robustness and Evaluation Critique: The paper shows that augmentation (training on corrupted inputs) improves fidelity under missing geometry but fails to restore exact validity. Conversely, symmetry structure guarantees validity regardless of input corruption. Furthermore, it critiques standard overlap metrics (like F1), showing that a procedural template can achieve high F1 scores by over-selecting edges (low precision, high recall), proving that overlap alone is insufficient for evaluating geometric completion.

Results
Experiments were conducted on procedurally generated graphs inspired by Islamic design, using 600 training graphs across symmetry orders 6, 8, 10, and 12.

• Fidelity: On clean inputs, the orbit-tied model, projection methods, and unstructured models achieved comparable edge-selection F1 scores (approx. 0.52). The orbit-tied model incurred no measurable fidelity penalty for enforcing exact validity.
• Validity: The unstructured model violated orbit consistency (selecting only part of an orbit) in an average of 3.3 orbits per pattern, even on clean data. In contrast, the orbit-tied and projection methods recorded zero violations.
• Robustness: When 30% of the control geometry was removed, the unstructured model's F1 dropped significantly (to 0.415) and violations increased to 37.5 per pattern. Symmetry-structured methods maintained high fidelity (approx. 0.50) and zero violations.
• Augmentation: Training the unstructured model on corrupted inputs recovered its fidelity (restoring F1 to 0.517) but did not eliminate symmetry violations (remaining at 9.0 per pattern). The combination of augmentation and symmetry structure yielded the most robust configuration.

Significance and Claims
The paper claims that explicit geometric knowledge can constrain neural predictors to produce outputs that are not merely plausible but certifiably valid. The central finding is that exact geometric validity can be enforced without sacrificing statistical fidelity. The study distinguishes between two failure modes: fidelity loss under corruption (addressed by data augmentation) and structural invalidity (addressed by symmetry structure).

The authors position this work as a contribution to knowledge-driven artificial intelligence, demonstrating how embedding domain-specific structural rules (rotational orbits) into the inference process ensures that the output adheres to strict geometric definitions. They emphasize that while the current work is limited to rotational symmetry and procedurally generated patterns, the framework provides a scalable, knowledge-constrained approach for generating vector ornaments where validity is a structural requirement rather than a stylistic preference. The work does not claim superiority over traditional procedural systems for faithful reproduction but offers a learned approach capable of interpolating within a distribution and handling incomplete input geometry while maintaining mathematical rigor.