Multiscale Spectral Manifold Wavelet Regularizer for Unsupervised Deep Functional Maps

Abstract

In deep functional maps, the regularizer computing the functional map is especially crucial for ensuring the global consistency of the computed pointwise map. As the regularizers integrated into deep learning should be differentiable, it is not trivial to incorporate informative axiomatic structural constraints into the deep functional map, such as the orientation-preserving term. Although commonly used regularizers include the Laplacian-commutativity term and the resolvent Laplacian commutativity term, these are limited to single-scale analysis for capturing geometric information. To this end, we propose a novel and theoretically well-justified regularizer commuting the functional map with the multiscale spectral manifold wavelet operator. This regularizer enhances the isometric constraints of the functional map and is conducive to providing it with better structural properties with multiscale analysis. Furthermore, we design an unsupervised deep functional map with the regularizer in a fully differentiable way. The quantitative and qualitative comparisons with several existing techniques on the (near-)isometric and non-isometric datasets show our method's superior accuracy and generalization capabilities. Additionally, we illustrate that our regularizer can be easily inserted into other functional map methods and improve their accuracy.