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问题高度相关，但两者通常被分别研究并顺序应用于关联不同三维形状，导致性能次优。本文提出统一框架，可同时预测三维形状间的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，实现谱域与空域的双域映射。一方面，通过引入空域映射，相较于此前基于函数映射的形状匹配方法，本方法可获得更精确且平滑的点对应关系；另一方面，通过引入谱域映射，本方法摆脱了仅适用于近等距形变场景的测地距离约束（此类约束计算成本高昂）。此外，我们提出新颖的测试时自适应方案，可同时捕捉姿态主导形变与形状主导形变。在多个挑战性数据集上的实验表明，本方法在形状匹配与插值任务中均超越现有最先进方法（包括监督学习方法）。