learn spherical parameterizations

This work addresses the challenge of achieving cross-shape consistency in spherical parameterizations for genus-zero 3D shape collections. The authors propose a continuous parameterization method based on neural generative modeling, which learns a continuous mapping from the unit sphere to each target shape and leverages its inverse to obtain consistent spherical parameterizations. To mitigate discretization artifacts, an intermediate shape is introduced to bridge the sphere and target geometry, while a generative tree in latent space propagates initial correspondences across the collection. By integrating neural generative modeling, continuous deformation representations, and latent structural analysis, the method significantly reduces geometric distortion and achieves markedly superior cross-shape parameterization consistency compared to existing approaches on the ShapeNet dataset.

To address the lack of native geometric processing capabilities in neural surface representations, this paper introduces spherical neural surface representation—a framework enabling seamless, mesh-free estimation of normals, first and second fundamental forms, gradients, divergence, and the Laplace–Beltrami operator on genus-0 neural surfaces. Our method leverages spherical parameterization with implicit neural representation, employs automatic differentiation to derive differential geometric operators, and establishes a numerical verification framework alongside neural spectral analysis tools. Key contributions include: (1) breaking the conventional “mesh-then-process” paradigm by establishing a systematic theoretical bridge between neural representations and classical differential geometry; (2) enabling geometric processing tasks—including neural heat flow and mean curvature flow—with robustness under isometric deformations; and (3) achieving numerical accuracy comparable to analytical solutions and mesh-based baselines, significantly outperforming existing neural surrogates.

Addressing the challenges of ensuring bijectivity and decoupling geometric from density distortion in volumetric parameterization of 3D simply-connected manifolds, this paper proposes a multi-objective co-optimization framework. For the first time in volumetric parameterization, it jointly models geometric distortion—via conformal/isometric energy—and density distortion—via Jacobian determinant distribution—integrating energy minimization, Jacobian constraints, and density control through tunable weights. The method employs nonlinear optimization coupled with adaptive mesh deformation for efficient computation. It enables user-controllable trade-offs among angle preservation, volume preservation, and bijectivity. Extensive evaluation on complex solid manifolds demonstrates strict bijectivity and significantly reduced composite distortion. Remeshing experiments further show that parameter domains generated by our method substantially improve downstream discretization accuracy and numerical stability.

This work addresses the problem of local parameterization of implicit surfaces—such as neural implicit fields and point clouds—at arbitrary query points. We propose a geodesic spline-based parameterization method that relies solely on the signed distance function (SDF) and its projection operator, without requiring meshes, surface normals, or other auxiliary geometric data. Our method employs a two-stage radial sampling and B-spline interpolation framework: first, neighborhood points are sampled along geodesic directions; second, a conformal, low-distortion explicit spline surface mapping is constructed. To our knowledge, this is the first unified local parameterization scheme supporting diverse geometric inputs—including neural implicit representations and unstructured point clouds. Experiments demonstrate significant improvements in robustness and generality for local texture mapping and interactive curve drawing on implicit surfaces. The approach establishes a new paradigm for real-time editing and visualization of implicit geometry.

Controllable geometric generation remains challenging in scenarios lacking large-scale 3D shape datasets. Method: This paper proposes a data-free neural implicit field generation framework that encodes user-specified design objectives—such as smoothness, genus (number of holes), and connectivity—as partial differential equation (PDE) constraints, geometric differential operator regularizers, and a multi-objective Lagrangian optimization objective, all directly embedded into neural field training. Contribution/Results: It establishes the first data-free paradigm for implicit shape generation; introduces explicit diversity constraints to mitigate mode collapse; and enables joint yet disentangled control over geometric and topological attributes. Experiments on multiple benchmarks and real-world engineering design tasks demonstrate precise, stable control over surface smoothness, connectivity, and genus, while consistently producing high-quality, diverse, and feasible shape ensembles.

This work proposes a novel method for constructing continuous, bijective, and orientation-preserving correspondences between genus-zero surfaces without requiring an initial bijection, mesh reparameterization, or barrier functions. The correspondence is implicitly represented as the zero level set of a complex-valued section defined over the four-dimensional product space of the two surfaces, and mapping distortion is minimized by optimizing a Ginzburg–Landau functional. The key innovation lies in formulating bijective surface correspondence as an implicit minimal surface for the first time, enabling automatic correction of non-bijective initializations—such as those obtained from functional maps—and naturally accommodating point and curve constraints. Compared to existing approaches, the proposed framework is more robust, concise, less sensitive to noise, and offers enhanced expressiveness and numerical stability.

This work addresses the challenge of preserving mapping continuity and bijectivity during complex remeshing processes, where conventional data transfer methods often induce geometric or attribute distortions. The authors propose a composite mapping framework based on local bijective atlases, enhanced by a Shared Scaffold structure that guarantees global bijectivity. The approach is generalized to support a variety of remeshing operations and, for the first time, enables the construction of bijective mappings on 3D tetrahedral remeshings by innovatively integrating Steinitz’s theorem with Maxwell–Cremona lifting theory. This framework facilitates precise tracking of geometric entities—including points, curves, and surfaces—across remeshing sequences, significantly improving fidelity in high-precision applications such as texture transfer and volumetric simulation.

This work addresses the challenge of parameterizing genus-0 closed surfaces, where existing methods struggle to simultaneously ensure bijectivity, geometric fidelity, and alignment with task-specific objectives such as landmark correspondence. The authors propose a spherical Beltrami differential (SBD) representation and introduce BOOST, a neural optimization framework built upon dual-hemisphere stereographic projection. For the first time, the SBD’s two-chart formulation is leveraged to characterize spherical homeomorphisms, combined with a spectral Beltrami network and explicit stitching consistency constraints to achieve globally consistent quasiconformal self-mappings. Evaluated on tasks like cortical surface registration, the method significantly improves sulcal alignment accuracy and depth map matching quality while effectively controlling distortion and rigorously preserving bijectivity.

This work addresses the high computational cost of geometric mapping under spatially varying fields at high resolutions by proposing a resolution-agnostic neural surrogate model. The method operates without reliance on fixed grids or ground-truth solution labels, leveraging coordinate-augmented multi-resolution field encoding to predict mapping positions at arbitrary point sets. A geometry-aware unsupervised loss is formulated by integrating variational energy, diffusion equilibration, and quasiconformal theory. Experimental results demonstrate that the approach achieves efficient, accurate, and resolution-flexible geometric parameterization in both quasiconformal mapping and density equilibration tasks, significantly enhancing computational efficiency and generalization capability.

研究通过五个方面分析了三维高斯点绘中神经网络参数化的趋势，并通过实验表明共享外观和不透明度能提高重建质量。