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Linear Spherical Sliced Optimal Transport: A Fast Metric for Comparing Spherical Data

Xinran Liu, Yikun Bai, Rocío Díaz Martín, Kaiwen Shi, Ashkan Shahbazi, Bennett A. Landman, Catie Chang, Soheil Kolouri

TL;DR

The Linear Spherical Sliced Optimal Transport (LSSOT) framework is introduced, which utilizes slicing to embed spherical distributions into L^2 spaces while preserving their intrinsic geometry, offering a computationally efficient metric for spherical probability measures.

Abstract

Efficient comparison of spherical probability distributions becomes important in fields such as computer vision, geosciences, and medicine. Sliced optimal transport distances, such as spherical and stereographic spherical sliced Wasserstein distances, have recently been developed to address this need. These methods reduce the computational burden of optimal transport by slicing hyperspheres into one-dimensional projections, i.e., lines or circles. Concurrently, linear optimal transport has been proposed to embed distributions into \( L^2 \) spaces, where the \( L^2 \) distance approximates the optimal transport distance, thereby simplifying comparisons across multiple distributions. In this work, we introduce the Linear Spherical Sliced Optimal Transport (LSSOT) framework, which utilizes slicing to embed spherical distributions into \( L^2 \) spaces while preserving their intrinsic geometry, offering a computationally efficient metric for spherical probability measures. We establish the metricity of LSSOT and demonstrate its superior computational efficiency in applications such as cortical surface registration, 3D point cloud interpolation via gradient flow, and shape embedding. Our results demonstrate the significant computational benefits and high accuracy of LSSOT in these applications.

Linear Spherical Sliced Optimal Transport: A Fast Metric for Comparing Spherical Data

TL;DR

The Linear Spherical Sliced Optimal Transport (LSSOT) framework is introduced, which utilizes slicing to embed spherical distributions into L^2 spaces while preserving their intrinsic geometry, offering a computationally efficient metric for spherical probability measures.

Abstract

Efficient comparison of spherical probability distributions becomes important in fields such as computer vision, geosciences, and medicine. Sliced optimal transport distances, such as spherical and stereographic spherical sliced Wasserstein distances, have recently been developed to address this need. These methods reduce the computational burden of optimal transport by slicing hyperspheres into one-dimensional projections, i.e., lines or circles. Concurrently, linear optimal transport has been proposed to embed distributions into spaces, where the distance approximates the optimal transport distance, thereby simplifying comparisons across multiple distributions. In this work, we introduce the Linear Spherical Sliced Optimal Transport (LSSOT) framework, which utilizes slicing to embed spherical distributions into spaces while preserving their intrinsic geometry, offering a computationally efficient metric for spherical probability measures. We establish the metricity of LSSOT and demonstrate its superior computational efficiency in applications such as cortical surface registration, 3D point cloud interpolation via gradient flow, and shape embedding. Our results demonstrate the significant computational benefits and high accuracy of LSSOT in these applications.

Paper Structure

This paper contains 45 sections, 15 theorems, 137 equations, 26 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Circular Optimal Transport and Linear Circular Optimal Transport
  4. Spherical Slicing
  5. Method
  6. Linear Spherical Sliced Optimal Transport (LSSOT)
  7. LSSOT Implementation
  8. Time Complexity and Runtime Analysis
  9. Experiments
  10. Rotation and Scaling in Spherical Geometry
  11. Cortical Surface Registration
  12. Experimental Setup
  13. Results
  14. Point Cloud Interpolation
  15. Mapping Point Clouds to Spherical Latent Space via LSSOT Autoencoder
  16. ...and 30 more sections

Key Result

Theorem 3.3

$LSSOT_2(\cdot,\cdot)$ defines a pseudo-metric in $\mathcal{P}(\mathbb{S}^{d-1})$, and a metric when restricting to probability measures with continuous density functions. We will refer to it as $LSSOT$ distance.

Figures (26)

  • Figure 1: Semicircle Transform projects non-polar points onto a great circle, and the north/south poles to everywhere on the circle.
  • Figure 2: Pairwise distances runtime (log scale) comparison w.r.t the number of distributions. $N$ denotes sample sizes in each distribution. The number of slices is 500 for all slice-based methods.
  • Figure 3: Generated VMF distributions and manifold learning results by LSSOT. Left: a group of 20 VMFs generated by rotating a source VMF, and the corresponding 3-dimensional visualizations by LSSOT and MDS. Right: 6 VMFs generated by scaling a source VMF with $\kappa=30$, and the same 3-dimensional visualizations.
  • Figure 4: Qualitative registration results (middle columns) from a moving surface (left column) to the fixed surface (right column). Both sulcal depth and parcellations are visualized with global and close-up views. This moving surface is from Subject A00065749 in the NKI dataset. Visualizations for more subjects can be found in the Appendix \ref{['app: visual-cortical-reg']}.
  • Figure 5: Top panel: the interpolation process between a pair of point clouds using gradient flow in the latent space $\mathbb{S}^2$. Bottom panel: gradient flow interpolations from a range hood to a bottle (left), and from a stool to a chair (right) using three metrics LSSOT, SSW and spherical OT. See Appendix \ref{['subsec: pc_interp']} for more interpolated pairs.
  • ...and 21 more figures

Theorems & Definitions (41)

  • Definition 3.1
  • Remark 3.2: LSSOT extends LCOT
  • Theorem 3.3: Metric property of LSSOT
  • Remark A.1
  • Lemma A.2
  • proof
  • Lemma A.3
  • proof
  • Proposition A.4
  • proof
  • ...and 31 more