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，但现有研究通常将它们分开处理，并依序应用于不同三维形状的关联任务中，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曲面形变模型相结合，在谱域与空间域双域空间中对形状进行映射。一方面，通过引入空间映射，相较于传统的形状匹配函数映射方法，我们的方法能获得更精确且平滑的逐点对应关系。另一方面，通过引入谱域映射，我们摆脱了仅适用于近等距形变、且计算成本高昂的测地距离约束。此外，我们提出一种新型测试时自适应方案，可同时捕获姿态主导形变与形状主导形变。在多个具有挑战性的数据集上的实验表明，即使与监督学习方法相比，我们的方法在形状匹配与插值两项任务上均超越了现有最优方法。