Invariant Shape Analysis of Surfaces with Spherical Topology

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.