  This work investigates the use of robust optimal transport (OT) for shape
matching. Specifically, we show that recent OT solvers improve both
optimization-based and deep learning methods for point cloud registration,
boosting accuracy at an affordable computational cost. This manuscript starts
with a practical overview of modern OT theory. We then provide solutions to the
main difficulties in using this framework for shape matching. Finally, we
showcase the performance of transport-enhanced registration models on a wide
range of challenging tasks: rigid registration for partial shapes; scene flow
estimation on the Kitti dataset; and nonparametric registration of lung
vascular trees between inspiration and expiration. Our OT-based methods achieve
state-of-the-art results on Kitti and for the challenging lung registration
task, both in terms of accuracy and scalability. We also release PVT1010, a new
public dataset of 1,010 pairs of lung vascular trees with densely sampled
points. This dataset provides a challenging use case for point cloud
registration algorithms with highly complex shapes and deformations. Our work
demonstrates that robust OT enables fast pre-alignment and fine-tuning for a
wide range of registration models, thereby providing a new key method for the
computer vision toolbox. Our code and dataset are available online at:
https://github.com/uncbiag/robot.
