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Fast Large Deformation Matching with the Energy Distance Kernel

Siwan Boufadene, François-Xavier Vialard, Jean Feydy

TL;DR

This work tackles efficient registration of point clouds and measures under large deformations by embedding the Energy-Distance kernel into the LDDMM framework. The key insight is that ED-based losses and their convolutions with finite measures can be approximated by sums of 1D projections, enabling $O(n \log n)$ per projection computation and scalable optimization without hyperparameter tuning. To address the ED kernel's lack of regularity, the authors introduce two regularization schemes that enforce bi-Lipschitz deformations: (i) TV regularization on the dual momentum and (ii) an $L^2$ momentum regularization on the evolving measure, both preserving the existence of minimizers and enabling complete deformation groups. A sliced-ED variant further accelerates computations, yielding near-linear time complexity and robust, translation-invariant registrations on large datasets. Overall, the paper delivers a hyperparameter-free, scalable registration framework with strong theoretical foundations and practical performance on synthetic and real data.

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.

Fast Large Deformation Matching with the Energy Distance Kernel

TL;DR

This work tackles efficient registration of point clouds and measures under large deformations by embedding the Energy-Distance kernel into the LDDMM framework. The key insight is that ED-based losses and their convolutions with finite measures can be approximated by sums of 1D projections, enabling per projection computation and scalable optimization without hyperparameter tuning. To address the ED kernel's lack of regularity, the authors introduce two regularization schemes that enforce bi-Lipschitz deformations: (i) TV regularization on the dual momentum and (ii) an momentum regularization on the evolving measure, both preserving the existence of minimizers and enabling complete deformation groups. A sliced-ED variant further accelerates computations, yielding near-linear time complexity and robust, translation-invariant registrations on large datasets. Overall, the paper delivers a hyperparameter-free, scalable registration framework with strong theoretical foundations and practical performance on synthetic and real data.

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.
Paper Structure (28 sections, 39 theorems, 163 equations, 6 figures, 2 algorithms)

This paper contains 28 sections, 39 theorems, 163 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Introduction to Large Deformation by Diffeomorphisms
  3. Deformation group
  4. Kernel methods
  5. General matching problem
  6. Measure matching using a positive definite kernel.
  7. Reduction of the optimization set.
  8. Interpolation using the Energy-Distance kernel, induced metrics and regularity.
  9. Interpolation using a CPD kernel.
  10. Case of a positive definite kernel.
  11. Case of a $\mathbb{P}$ conditionally positive definite kernel.
  12. Case of the Energy-Distance kernel, induced norm on $\mathcal{H}_{K,\mathbb{P}}$ .
  13. Regularity and embeddings.
  14. A few possible choices for the Energy-Distance kernel.
  15. The Energy-Distance as a pseudo-polynomial spline.
  16. ...and 13 more sections

Key Result

Theorem 1.3

Let $v \in L_V^1$. Then, for all $x \in \mathbb{R}^d$, there exists a unique continuous solution $t \mapsto \psi_t^v(x)$ that verifies the Lagrangian flow equation Moreover, for all $t\geq 0$, the application $\psi_t^v$ is a $\mathcal{C}^1$ diffeomorphism of $\mathbb{R}^d$. Finally, the applications $(t,x) \mapsto \psi_t^v(x)$ are uniformly continuous, in a uniform way regarding $v$ on every boun

Figures (6)

  • Figure 1: Obtained deformation map while matching two particles to $0$ (left), and between the landmarks $\left(x_+ + \left( 0\varepsilon \right), x_- - \left( 0\varepsilon \right)\right)$ and $\left(x_- + \left( 0\varepsilon \right), x_+ - \left( 0\varepsilon \right)\right)$ (right).
  • Figure 2: In the left figure, we plot the error convergence to zero in $P^{-1/2}$ between the approximated vector and the ground-truth as $P$ grows. In the middle, the error as the dimension increases. In the right-most graph, we plot the computing time against the number of points in dimension 5 on a standard 8-core CPU.
  • Figure 3: Registration with Gaussian kernel at the top, Energy-Distance at the bottom. The final losses are comparable.
  • Figure 4: Same registration problem using the sliced computation of the kernel. From left to right $P = 2, 10, 50, 500\,.$
  • Figure 5: Visualization of translated registration results: Energy-Distance kernel (left) and a tuned Gaussian kernel results (right).
  • ...and 1 more figures

Theorems & Definitions (81)

  • Definition 1.1: Admissible Hilbert spaces
  • Example 1.2
  • Theorem 1.3: see younes2010shapes
  • Proposition 1.4: (see tro98)
  • Proposition 1.5: Existence of constant speed geodesics
  • Example 1.6
  • Definition 1.7: General matching problem
  • Theorem 1.8: Existence of a minimizer
  • Proposition 1.9
  • Example 1.10
  • ...and 71 more