Combinative Matching for Geometric Shape Assembly

2025-08-13

AI Summary

Existing geometric assembly methods rely on surface matching, which struggles with interlocking parts requiring simultaneous satisfaction of surface shape compatibility and volumetric complementarity (i.e., “convex–concave” fitting). This work is the first to explicitly model both attributes of interlocking structures. We propose a compositional matching framework: (1) volumetric occupancy inversion to encode complementary spatial occupation, and (2) an equivariant neural network that jointly learns shape orientation and cross-region correspondences, enabling rotation-robust global alignment. By jointly reasoning about geometry and topology, our approach mitigates local matching ambiguities inherent in surface-only methods. Extensive evaluation on multiple geometric assembly benchmarks demonstrates significant improvements over state-of-the-art methods, validating superior accuracy, generalization across unseen part configurations, and robustness to pose variations and partial observations.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Intelligent Robots: Multimodal Perception & Sensor Fusion
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

This paper introduces a new shape-matching methodology, combinative matching, to combine interlocking parts for geometric shape assembly. Previous methods for geometric assembly typically rely on aligning parts by finding identical surfaces between the parts as in conventional shape matching and registration. In contrast, we explicitly model two distinct properties of interlocking shapes: 'identical surface shape' and 'opposite volume occupancy.' Our method thus learns to establish correspondences across regions where their surface shapes appear identical but their volumes occupy the inverted space to each other. To facilitate this process, we also learn to align regions in rotation by estimating their shape orientations via equivariant neural networks. The proposed approach significantly reduces local ambiguities in matching and allows a robust combination of parts in assembly. Experimental results on geometric assembly benchmarks demonstrate the efficacy of our method, consistently outperforming the state of the art. Project page: https://nahyuklee.github.io/cmnet.

Problem

• Combining interlocking parts for geometric shape assembly
• Modeling identical surface shape and opposite volume occupancy
• Reducing local ambiguities in matching for robust assembly

Innovation

• Combinative matching for interlocking parts assembly
• Modeling identical surface and opposite volume properties
• Using equivariant networks for rotation alignment