Non-isometric shape correspondence remains a fundamental challenge in computer vision. Traditional methods using Laplace-Beltrami operator (LBO) eigenmodes face limitations in characterizing high-frequency extrinsic shape changes like bending and creases. We propose a novel approach of combining the non-orthogonal extrinsic basis of eigenfunctions of the elastic thin-shell hessian with the intrinsic ones of the LBO, creating a hybrid spectral space in which we construct functional maps. To this end, we present a theoretical framework to effectively integrate non-orthogonal basis functions into descriptor- and learning-based functional map methods. Our approach can be incorporated easily into existing functional map pipelines across varying applications and is able to handle complex deformations beyond isometries. We show extensive evaluations across various supervised and unsupervised settings and demonstrate significant improvements. Notably, our approach achieves up to 15% better mean geodesic error for non-isometric correspondence settings and up to 45% improvement in scenarios with topological noise.

翻译：非等距形状对应仍然是计算机视觉中的一项基本挑战。使用拉普拉斯-贝尔特拉米算子本征模的传统方法在表征弯曲和折痕等高频外在形状变化时存在局限性。我们提出了一种新方法，将弹性薄壳海森矩阵本征函数的非正交外在基与拉普拉斯-贝尔特拉米算子的内在基相结合，构建了一个混合谱空间，并在其中构造函数映射。为此，我们提出了一个理论框架，有效地将非正交基函数整合到基于描述符和学习的函数映射方法中。我们的方法可以轻松地融入跨不同应用的现有函数映射流程，并能够处理超出等距的复杂变形。我们在各种监督和无监督设置下进行了广泛评估，并展示了显著的改进。值得注意的是，在非等距对应设置中，我们的方法实现了高达15%的平均测地误差改善，而在拓扑噪声场景中实现了高达45%的改进。