Non-Euclidean Sliced Optimal Transport Sampling

TL;DR

This work extends sliced optimal transport sampling to non-Euclidean domains by formulating geodesic-slice projections and Riemannian gradient updates on $\mathbb{S}^d$, $\mathbb{H}^d$, and $\mathbb{P}^d$ to produce blue-noise samples. It combines explicit Exp/Log mappings, group actions, and a stochastic Riemannian gradient framework (with geometric-median aggregation) to generate well-dispersed samples for meshes, directions, and rotations, including intrinsic mesh sampling via conformal uniformization. The approach enables intrinsic blue-noise sampling on surfaces of arbitrary genus through global spherical embedding or local hyperbolic patches, with applications to mesh stippling, direction sampling, and quaternion-based rotations. Limitations include sensitivity to global conformal distortions and the need to operate in genus-specific regimes, while future work points to variance reduction, broader geometric applications, and improved remeshing/line-sampling tasks.

Abstract

In machine learning and computer graphics, a fundamental task is the approximation of a probability density function through a well-dispersed collection of samples. Providing a formal metric for measuring the distance between probability measures on general spaces, Optimal Transport (OT) emerges as a pivotal theoretical framework within this context. However, the associated computational burden is prohibitive in most real-world scenarios. Leveraging the simple structure of OT in 1D, Sliced Optimal Transport (SOT) has appeared as an efficient alternative to generate samples in Euclidean spaces. This paper pushes the boundaries of SOT utilization in computational geometry problems by extending its application to sample densities residing on more diverse mathematical domains, including the spherical space Sd , the hyperbolic plane Hd , and the real projective plane Pd . Moreover, it ensures the quality of these samples by achieving a blue noise characteristic, regardless of the dimensionality involved. The robustness of our approach is highlighted through its application to various geometry processing tasks, such as the intrinsic blue noise sampling of meshes, as well as the sampling of directions and rotations. These applications collectively underscore the efficacy of our methodology.

Figures (13)

• Figure 1: Sliced optimal transport sampling and notations: from left to right, on the Euclidean domain (zero curvature metric space), on the spherical one (positive constant curvature metric space), and on the hyperbolic model (Lorentz's model with only a part of the hyperboloid, negative curvature metric space). We only illustrate the assignment and the associated advection for a single sample (yellow bars).
• Figure 2: Exp and Log maps: on $\mathbb{S}^2$, the orange point is the point obtained by iteratively going in the average of the Logs $x_{n+1} = \text{Exp}_{x_n}(\frac{\gamma}{n} \sum_i \text{Log}_{x_n}(y_i))$, which is equivalent to Fréchet means, whereas the red one is obtained by going in the geometric median of the directions $x_{n+1} = \text{Exp}_{x_n}(\gamma\,\,\text{GeoMed}(\{\text{Log}_{x_n}(y_i)\}_i))$.
• Figure 3: Blue noise on the sphere. On $\mathbb{S}^2$, we evaluate the blue noise property of our sampling (2048 samples). Our result as to be compared to a uniform sampling, a stratified sampling using a healpix spherical structure Pilleboue:2015:VAMCI, a Poisson disk sampling, a spherical Fibonacci sequence keinert2015spherical, and a Lloyd's relaxation approach (Centroidal Voronoi Tesselation, CVT) liu2009centroidal, and a geodesic farthest point greedy strategy peyre2006geodesic (FP) . The graph corresponds to the angular power spectra of the spherical harmonic transform of the point sets (except for spherical Fibonacci whose regular patterns make the spectral analysis less relevant) . As discussed in Pilleboue et al. Pilleboue:2015:VAMCI, our sampler exhibits correct blue noise property with low energy for low frequencies, a peak at the average distance between samples and a plateau with few oscillations for higher frequencies.
• Figure 4: Non-uniform measure sampling: given a non-uniform probability measure $\phi$ in $\mathbb{S}^2$$(a)$, we first construct a discrete measure $\nu\sim\phi$ with a large number of samples, 2048 samples here $(b)$. Figures $(c)$ and $(d)$ are the output of the NESOTS algorithm for 2048 samples ($L=32$, $K=300$), when averaging the directions during the advection $(c)$, or using the geometric median $(d)$. While both distributions approximate the density, the latter provides a more stable result without sample alignment artifacts.
• Figure 5: Overall pipeline of our intrinsic discrete manifold sampling: starting from an input shape, we conformally embed the discrete structure onto either $\mathbb{S}^2$ for 0-genus surfaces, or local patches to $\mathbb{H}^2$ for higher genus one. Then, the NESOTS (Alg. \ref{['alg:nesots']}) is used (globally or locally) to blue noise sample the embedded structure targeting a measure taking into account the metric distortion.