Learning Bijective Surface Parameterization for Inferring Signed Distance Functions from Sparse Point Clouds with Grid Deformation

TL;DR

The paper tackles the problem of reconstructing continuous surfaces from sparse point clouds by introducing Bijective Surface Parameterization (BSP), which maps sparse data into a unit-sphere canonical domain to create dense local patches that form a global surface, and Grid Deformation Optimization (GDO), which deforms a tetrahedral grid to infer a precise signed distance field. BSP provides dense, coherent parametric supervision, while GDO refines the implicit field and yields watertight surfaces via Deep Marching Tetrahedra. End-to-end training couples BSP-derived supervision with deformable-grid optimization, achieving state-of-the-art performance on ShapeNet, DFAUST, SRB, 3DScene, and KITTI under challenging sparse-input conditions. The approach reduces reliance on surface priors and demonstrates robust reconstruction of complex geometries in both synthetic and real-world datasets, with potential for broad impact in 3D vision applications.

Abstract

Inferring signed distance functions (SDFs) from sparse point clouds remains a challenge in surface reconstruction. The key lies in the lack of detailed geometric information in sparse point clouds, which is essential for learning a continuous field. To resolve this issue, we present a novel approach that learns a dynamic deformation network to predict SDFs in an end-to-end manner. To parameterize a continuous surface from sparse points, we propose a bijective surface parameterization (BSP) that learns the global shape from local patches. Specifically, we construct a bijective mapping for sparse points from the parametric domain to 3D local patches, integrating patches into the global surface. Meanwhile, we introduce grid deformation optimization (GDO) into the surface approximation to optimize the deformation of grid points and further refine the parametric surfaces. Experimental results on synthetic and real scanned datasets demonstrate that our method significantly outperforms the current state-of-the-art methods. Project page: https://takeshie.github.io/Bijective-SDF

Figures (11)

• Figure 1: Overview of Our method. Given a sparse point cloud $Q$, we first learn a mapping function $\Phi$ to encode $Q$ into a unit sphere parametric domain. We consider each point as center point and sample local patches on the parametric surface. Next, we learn the inverse mapping $\Psi$ to predict the positions of these local patches in 3D space and integrate them to obtain $S$. We leverage $S$ as the supervision for the grid deformation network $g$ and predict the signed distance field through the GDO optimization strategy. We further extract dense point cloud $\bar{V}$ from the implicit field and optimize the parameterized surface $S$.
• Figure 2: Illustration of BSP. The white points indicate the sparse input $Q$. For each point $q \in Q$, we learn mapping function $\Phi$ to map $q$ to $\mathcal{U}(q)$, and sample a local patch $\mathcal{U}(p)$ on the parametric surface. Subsequently, we employ an inverse mapping $\Psi$ to assembles these patches into a global surface (red points).
• Figure 3: Visual comparison of GDO (a) and direct offset optimization (b), the red lines indicate the offset direction.
• Figure 4: Visual comparison on ShapeNet. The input contains 300 points.
• Figure 5: Visual comparison on D-FAUST. The input contains 300 points.