Integrating Efficient Optimal Transport and Functional Maps For Unsupervised Shape Correspondence Learning

TL;DR

This work addresses unsupervised non-rigid 3D shape correspondence by integrating deep functional maps with efficient optimal transport. It leverages the sliced Wasserstein distance to form fast, discriminative feature-alignment losses and couples them with a regularized FM objective, while an adaptive refinement stage employs entropic OT to yield sharper point-to-point mappings. The approach introduces two novel unsupervised OT-based losses (biSW and biEBSW) and a proper FM regularizer, achieving state-of-the-art results on near-isometric and non-isometric datasets and excelling at segmentation transfer. This yields a scalable, generalizable pipeline that improves correspondence accuracy and smoothness, with practical impact for downstream tasks in shape understanding and analysis.

Abstract

In the realm of computer vision and graphics, accurately establishing correspondences between geometric 3D shapes is pivotal for applications like object tracking, registration, texture transfer, and statistical shape analysis. Moving beyond traditional hand-crafted and data-driven feature learning methods, we incorporate spectral methods with deep learning, focusing on functional maps (FMs) and optimal transport (OT). Traditional OT-based approaches, often reliant on entropy regularization OT in learning-based framework, face computational challenges due to their quadratic cost. Our key contribution is to employ the sliced Wasserstein distance (SWD) for OT, which is a valid fast optimal transport metric in an unsupervised shape matching framework. This unsupervised framework integrates functional map regularizers with a novel OT-based loss derived from SWD, enhancing feature alignment between shapes treated as discrete probability measures. We also introduce an adaptive refinement process utilizing entropy regularized OT, further refining feature alignments for accurate point-to-point correspondences. Our method demonstrates superior performance in non-rigid shape matching, including near-isometric and non-isometric scenarios, and excels in downstream tasks like segmentation transfer. The empirical results on diverse datasets highlight our framework's effectiveness and generalization capabilities, setting new standards in non-rigid shape matching with efficient OT metrics and an adaptive refinement module.

Figures (11)

• Figure 1: Overview of unsupervised shape matching via efficient OT. Our framework takes as input a pair of shapes $\mathcal{X}$ and $\mathcal{Y}$ and outputs point-to-point correspondence. Firstly, the features extractor tasks the pair input and extracts vertex-wise features $\mathcal{F}_x$ and $\mathcal{F}_y$. Subsequently, the differentiable functional map solver is used to compute functional map given pre-computed eigenfunctions and the extracted features. In parallel, our framework estimates a soft feature similarity matrix, derived from the same extracted features. After that, an OT cost is computed given soft feature similarity and extracted feature $\mathcal{F}_x$ and $\mathcal{F}_y$. Finally, a proper loss is optimized together with regularized functional map loss and OT loss.
• Figure 2: Qualitative results of different methods evaluated on SHREC'19 datasets. Correspondence is visualized by texture transfer. The red arrow indicates poor mappings.
• Figure 3: Qualitative results of various methods on challenging non-isometric SMAL dataset. Our method demonstrates superior point mapping capabilities compared to previous works. More visualization is provided in Sup. \ref{['sec:additional-visualizations']}.
• Figure 4: Qualitative results of segmentation transfer. Our method exhibits a high-quality segmentation map via computed correspondence. More visualization is provided in Sup. \ref{['sec:additional-visualizations']}.
• Figure 5: Qualitative results of our method on FAUST dataset.