We present a framework for uncertainty-aware geometry processing on Gaussian Process Implicit Surfaces (GPIS), enabling computations directly on such probabilistic representations of shapes. In contrast to classical geometry processing pipelines that assume deterministic surface meshes or point clouds, our approach considers uncertainty in the input data and defines analogs of fundamental differential operators-gradient, divergence, and Laplacian- that account for the distribution of plausible geometries encoded by the GPIS. Leveraging the Kac-Rice formula, we embed computations from random surfaces into a volumetric Cartesian domain, enabling efficient evaluation of expected integrals and differential operators. The proposed approach bridges classical surface PDE-based geometry processing and volumetric representations, enabling a principled handling of noise and ambiguity for various downstream geometry processing tasks.

Abstract Implicit representations of shapes are broadly used in computer graphics since they offer many valuable properties in design, modeling, and animation. However, their implicit and volumetric nature makes applying 2D textures fundamentally challenging. We propose a method to compute point‐wise and parallelizable semi‐global parameterizations of implicit surfaces for texturing, rendering, and modeling purposes. Our method not only defines local patches of parameterization, but also enables the merging of multiple adjacent patches into large and spatially coherent ones that conform to the geometry. Implemented in shaders into a sphere‐tracing pipeline, our method allows users to edit the uv‐fields with real‐time visualization. We demonstrate how to add rendering details (texture, normal, displacement, etc.) using our parameterization, as well extending modeling tools with implicit shell maps. Furthermore, the textured objects remain implicit and can still be used in a modeling pipeline.

To solve the optimal transport problem between two uniform discrete measures of the same size, one seeks a bijective assignment that minimizes some matching cost. For this task, exact algorithms are intractable for large problems, while approximate ones may lose the bijectivity of the assignment. We address this issue and the more general cases of non-uniform discrete measures with different total masses, where partial transport may be desirable. The core of our algorithm is a variant of the Quicksort algorithm that provides an efficient strategy to randomly explore many relevant and easy-to-compute couplings, by matching BSP trees in loglinear time. The couplings we obtain are as sparse as possible, in the sense that they provide bijections, injective partial matchings or sparse couplings depending on the nature of the matched measures. To improve the transport cost, we propose efficient strategies to merge k sparse couplings into a higher quality one. For k = 64, we obtain transport plans with typically less than 1% of relative error in a matter of seconds between hundreds of thousands of points in 3D on the CPU. We demonstrate how these high-quality approximations can drastically speed-up usual pipelines involving optimal transport, such as shape interpolation, intrinsic manifold sampling, color transfer, topological data analysis, rigid partial registration of point clouds and image stippling.

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.