🤖 AI Summary
This study addresses the limitations of 3D shape descriptors that are sensitive to parameterization, pose, and scale while failing to distinguish mirror chirality. To overcome these issues, we propose a shape description method that is both complete and stable. The approach eliminates geometric dependencies through conformal mapping and conformal barycentric normalization. Furthermore, by leveraging classical invariant theory to identify harmonic bands as binary forms, it constructs polynomial invariants of the rotation group to precisely decouple transformation dependencies while preserving chirality information. Benchmark evaluations demonstrate that the proposed descriptor exhibits excellent stability and completeness, successfully achieving accurate separation between mirror-symmetric and asymmetric pairs within bilateral anatomical structures.
📝 Abstract
Spherical harmonic descriptors of closed 3D shapes depend on the parameterization, the pose and the scale of the surface, and the standard rotation-invariant reductions, the power spectrum and the bispectrum, discard the relative orientation of the harmonic bands and cannot distinguish a shape from its mirror image. We construct a descriptor that removes all three dependencies exactly and loses nothing else: a conformal parameterization normalized by its conformal barycenter, followed by polynomial invariants of the rotation group. Identifying each harmonic band with a binary form turns the rotation quotient into classical invariant theory and makes reflections visible as the sign of an invariant, so chirality is recorded. The descriptor is complete for the truncated expansion, stable in the orbit distance, and comes with numerical diagnostics. Benchmarks confirm the guarantees, and on bilateral anatomical structures the descriptor separates mirror-image pairs from asymmetric pairs, which parity-blind descriptors cannot.