球面拓扑曲面的不变形状分析

中文摘要 AI 辅助

闭合三维形状的球谐描述子依赖于曲面的参数化、姿态和尺度，而标准的旋转不变约简（功率谱和双谱）丢弃了谐波带的相对方向，无法区分形状与其镜像。我们构造了一个描述子，它精确地移除所有这三种依赖性，且不丢失其他信息：通过共形重心归一化的共形参数化，随后采用旋转群的多项式不变量。将每个谐波带视为二元形式，旋转商转化为经典不变量理论，并通过不变量的符号使镜像可见，从而记录手性。该描述子对截断展开是完备的，在轨道距离下稳定，并附带数值诊断。基准测试证实了这些保证，在双侧解剖结构上，该描述子能将镜像对与非对称对区分开来，而奇偶性盲描述子则无法做到。

英文摘要

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.

发表机构

• Oakland University（奥克兰大学）

机构由 AI 辅助整理，请以论文原文为准。