Cycle consistency has long been exploited as a powerful prior for jointly optimizing maps within a collection of shapes. In this paper, we investigate its utility in the approaches of Deep Functional Maps, which are considered state-of-the-art in non-rigid shape matching. We first justify that under certain conditions, the learned maps, when represented in the spectral domain, are already cycle consistent. Furthermore, we identify the discrepancy that spectrally consistent maps are not necessarily spatially, or point-wise, consistent. In light of this, we present a novel design of unsupervised Deep Functional Maps, which effectively enforces the harmony of learned maps under the spectral and the point-wise representation. By taking advantage of cycle consistency, our framework produces state-of-the-art results in mapping shapes even under significant distortions. Beyond that, by independently estimating maps in both spectral and spatial domains, our method naturally alleviates over-fitting in network training, yielding superior generalization performance and accuracy within an array of challenging tests for both near-isometric and non-isometric datasets. Codes are available at https://github.com/rqhuang88/Spatiallyand-Spectrally-Consistent-Deep-Functional-Maps.

翻译：循环一致性长期以来一直被用作联合优化形状集合内映射的强大先验。本文研究了其在深度函数映射方法中的效用，该方法被认为是非刚性形状匹配的前沿技术。我们首先证明，在特定条件下，当映射在谱域中表示时，它们已经具有循环一致性。此外，我们识别出谱一致性映射未必是空间（或逐点）一致性的差异。鉴于此，我们提出了一种无监督深度函数映射的新设计，该设计有效地强化了学习映射在谱表示和逐点表示下的和谐性。通过利用循环一致性，我们的框架即使在显著形变下也能产生最先进的形状映射结果。此外，通过独立估计谱域和空间域中的映射，我们的方法自然缓解了网络训练中的过拟合问题，在一系列具有挑战性的近等度量和非等度量数据集的测试中展现出卓越的泛化性能和准确性。代码可在 https://github.com/rqhuang88/Spatiallyand-Spectrally-Consistent-Deep-Functional-Maps 获取。