Fast Large Deformation Matching with the Energy Distance Kernel

TL;DR

This paper introduces a fast, scalable method for point cloud registration using the Energy-Distance kernel and bi-Lipschitz homeomorphisms, achieving efficient large-scale optimization without hyperparameter tuning.

Contribution

It presents a novel framework combining the Energy-Distance kernel with regularization models for efficient large deformation matching of point clouds.

Findings

Achieves O(n log n) complexity for registration tasks.

Demonstrates robustness and scalability on synthetic and real data.

Provides two regularization models for stable deformations.

Abstract

We propose an efficient framework for point cloud and measure registration using bi-Lipschitz homeomorphisms, achieving O(n log n) complexity, where n is the number of points. By leveraging the Energy-Distance (ED) kernel, which can be approximated by its sliced one-dimensional projections, each computable in O(n log n), our method avoids hyperparameter tuning and enables efficient large-scale optimization. The main issue to be solved is the lack of regularity of the ED kernel. To this goal, we introduce two models that regularize the deformations and retain a low computational footprint. The first model relies on TV regularization, while the second model avoids the non-smooth TV regularization at the cost of restricting its use to the space of measures, or cloud of points. Last, we demonstrate the numerical robustness and scalability of our models on synthetic and real data.