Although 3D shape matching and interpolation are highly interrelated, they are often studied separately and applied sequentially to relate different 3D shapes, thus resulting in sub-optimal performance. In this work we present a unified framework to predict both point-wise correspondences and shape interpolation between 3D shapes. To this end, we combine the deep functional map framework with classical surface deformation models to map shapes in both spectral and spatial domains. On the one hand, by incorporating spatial maps, our method obtains more accurate and smooth point-wise correspondences compared to previous functional map methods for shape matching. On the other hand, by introducing spectral maps, our method gets rid of commonly used but computationally expensive geodesic distance constraints that are only valid for near-isometric shape deformations. Furthermore, we propose a novel test-time adaptation scheme to capture both pose-dominant and shape-dominant deformations. Using different challenging datasets, we demonstrate that our method outperforms previous state-of-the-art methods for both shape matching and interpolation, even compared to supervised approaches.

翻译：尽管三维形状匹配与插值高度相关，但二者常被分开研究并依次应用于不同三维形状的关联中，导致性能次优。本文提出一个统一框架，可同时预测三维形状间的逐点对应关系与形状插值。为此，我们将深度函数映射框架与经典表面变形模型相结合，在谱域与空间域中同时映射形状。一方面，通过引入空间映射，我们的方法相较于先前用于形状匹配的函数映射方法，可获得更精确、更平滑的逐点对应关系。另一方面，通过引入谱映射，我们的方法摆脱了常见但计算昂贵、且仅适用于近似等距形状变形的测地距离约束。此外，我们提出一种新颖的测试时自适应方案，可同时捕捉姿势主导与形状主导的变形。在多个具有挑战性的数据集上，我们证明该方法在形状匹配与插值任务中均优于先前最先进的方法，甚至包括有监督方法。