RFMNet: Robust Deep Functional Maps for unsupervised non-rigid shape correspondence

Abstract

In traditional deep functional maps for non-rigid shape correspondence, estimating a functional map including high-frequency information requires enough linearly independent features via the least square method, which is prone to be violated in practice, especially at an early stage of training, or costly post-processing, e.g. ZoomOut. In this paper, we propose a novel method called RFMNet (Robust Deep Functional Map Networks), which jointly considers training stability and more geometric shape features than previous works. We directly first produce a pointwise map by resorting to optimal transport and then convert it to an initial functional map. Such a mechanism mitigates the requirements for the descriptor and avoids the training instabilities resulting from the least square solver. Benefitting from the novel strategy, we successfully integrate a state-of-the-art geometric regularization for further optimizing the functional map, which substantially filters the initial functional map. We show our novel computing functional map module brings more stable training even under encoding the functional map with high-frequency information and faster convergence speed. Considering the pointwise and functional maps, an unsupervised loss is presented for penalizing the correspondence distortion of Delta functions between shapes. To catch discretization-resistant and orientation-aware shape features with our network, we utilize DiffusionNet as a feature extractor. Experimental results demonstrate our apparent superiority in correspondence quality and generalization across various shape discretizations and different datasets compared to the state-of-the-art learning methods.

Keywords

• Shape correspondence
• Functional maps
• Unsupervised learning
• Optimal transport