Linear Spherical Sliced Optimal Transport: A Fast Metric for Comparing Spherical Data
This work addresses the need for fast and accurate metrics in fields such as computer vision, geosciences, and medicine, but it is incremental as it builds on existing sliced and linear optimal transport methods.
The paper tackles the problem of efficiently comparing spherical probability distributions by introducing the Linear Spherical Sliced Optimal Transport (LSSOT) framework, which embeds these distributions into L^2 spaces to achieve superior computational efficiency and high accuracy in applications like cortical surface registration and 3D point cloud interpolation.
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.