Combinative Matching for Geometric Shape Assembly

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.