Integrating Efficient Optimal Transport and Functional Maps For
  Unsupervised Shape Correspondence Learning
Tung Le Khai Nguyen Shanlin Sun Nhat Ho Xiaohui Xie
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.