We propose a novel constraint called Multiple Spectral filter Operators Preservation (MSFOR) to compute functional maps and based on it, develop an efficient deep functional map architecture called Deep MSFOP for shape matching. The core idea is that, instead of using the general descriptor preservation constraint, we require our maps to preserve multiple spectral filter operators. This allows us to incorporate more informative geometrical information, contained in different frequency bands of functions, into the functional map computing. This can be confirmed by that some previous techniques like wavelet preservation and LBO commutativity are actually our special cases. Moreover, we also develop a very efficient way to compute the maps with MSFOP constraint, which can be conveniently embedded into the deep learning, especially having learnable filter operators. Utilizing the above results, we finally design our Deep MSFOP pipeline, equipped with a suitable unsupervised loss jointly penalizing the functional map and the underlying pointwise map. Our deep functional map has notable advantages, including that the functional map is more geometrically informative and guaranteed to be proper, and the computing is numerically stable. Extensive experimental results on different datasets demonstrate that our approach outperforms the existing state-of-the-art methods, especially in challenging settings like non-isometric and inconsistent topology datasets.

翻译：摘要：我们提出了一种名为多频谱滤波器算子保持（MSFOR）的新型约束条件，用于计算函数映射，并基于此开发了一种高效的深度函数映射架构Deep MSFOP，应用于形状匹配。核心思想在于：不使用通用的描述子保持约束，而是要求映射保持多个频谱滤波器算子。这使我们能将不同频率波段中包含的更具信息量的几何信息融入函数映射计算中。这可以得以验证：诸如小波保持和拉普拉斯算子交换性等先前技术实际上是我们方法的特例。此外，我们还开发了一种极其高效的计算满足MSFOP约束的映射的方法，该方法可便捷地嵌入深度学习框架中，尤其是可学习的滤波器算子部分。利用上述成果，我们最终设计了Deep MSFOP流水线，并配备了合适的无监督损失函数，共同约束函数映射及其底层的逐点映射。我们的深度函数映射具有显著优势：函数映射具有更强的几何信息且保证正确性，同时计算过程数值稳定。在不同数据集上的大量实验表明，我们的方法超越了现有最先进技术，尤其在非同构和非一致拓扑数据集等具有挑战性的场景中表现突出。