Deep MSFOP: Multiple Spectral filter Operators Preservation in Deep
  Functional Maps for Unsupervised Shape Matching
Feifan Luo Qingsong Li Ling Hu Xinru Liu Haojun Xu Haibo Wang Ting Li Shengjun Liu
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.