Computing volumetric correspondences between 3D shapes is a prominent tool for medical and industrial applications. In this work, we pave the way for spectral volume mapping, extending for the first time the surface-based functional maps framework. We show that the eigenfunctions of the volumetric Laplace operator define a functional space that is suitable for high-quality signal transfer. We also experiment with various techniques that edit this functional space, porting them to volume domains. We validate our method on novel volumetric datasets and on tetrahedralizations of well established surface datasets, also showcasing practical applications involving both discrete and continuous signal mapping, for segmentation transfer, mesh connectivity transfer and solid texturing. Finally, we show that the volumetric spectrum greatly improves the accuracy for classical shape matching tasks among surfaces, consistently outperforming surface-only spectral methods.

翻译：计算三维形状之间的体积对应关系是医学与工业应用中的重要工具。本研究开创性地将基于表面的功能映射框架首次扩展到体积域，为光谱体积映射奠定基础。我们证明体积拉普拉斯算子的特征函数定义的功能空间适用于高质量信号传递。同时实验了多种编辑该功能空间的技术，并将其移植至体积域。通过在新型体积数据集及经典表面数据集的四面体化处理上进行验证，展示了涉及离散与连续信号映射的实际应用案例（包括分割传递、网格连通性传递及实体纹理映射）。最后证明体积光谱可显著提升经典表面形状匹配任务的精度，其性能始终优于纯表面光谱方法。