Accurate and compact representation of signed distance functions (SDFs) of implicit surfaces is crucial for efficient storage, computation, and downstream processing of 3D geometry. In this work, we propose a general learning method for approximating precomputed SDF fields of implicit surfaces by a relatively small number of ellipsoidal radial basis functions (ERBFs). The SDF values could be computed from various sources, including point clouds, triangle meshes, analytical expressions, pretrained neural networks, etc. Given SDF values on spatial grid points, our method approximates the SDF using as few ERBFs as possible, achieving a compact representation while preserving the geometric shape of the corresponding implicit surface. To balance sparsity and approximation precision, we introduce a dynamic multi-objective optimization strategy, which adaptively incorporates regularization to enforce sparsity and jointly optimizes the weights, centers, shapes, and orientations of the ERBFs. For computational efficiency, a nearest-neighbor-based data structure restricts computations to points near each kernel center, and CUDA-based parallelism further accelerates the optimization. Furthermore, a hierarchical refinement strategy based on SDF spatial grid points progressively incorporates coarse-to-fine samples for parameter initialization and optimization, improving convergence and training efficiency. Extensive experiments on multiple benchmark datasets demonstrate that our method can represent SDF fields with significantly fewer parameters than existing sparse implicit representation approaches, achieving better accuracy, robustness, and computational efficiency. The corresponding executable program is publicly available at https://github.com/lianbobo/SE-RBFNet.git

翻译：隐式曲面的符号距离函数（SDF）的精确且紧凑表示，对于三维几何的高效存储、计算及下游处理至关重要。本文提出一种通用的学习方法，利用相对少量的椭球径向基函数（ERBF）来逼近预计算的隐式曲面SDF场。SDF值可从多种来源计算获得，包括点云、三角网格、解析表达式、预训练神经网络等。给定空间网格点上的SDF值，本方法使用尽可能少的ERBF来逼近SDF，在保持对应隐式曲面几何形状的同时实现紧凑表示。为平衡稀疏性与逼近精度，我们引入一种动态多目标优化策略，该策略自适应地融入正则化以增强稀疏性，并联合优化ERBF的权重、中心、形状与方向。为提高计算效率，一种基于最近邻的数据结构将计算限制在每个核中心附近的点上，而基于CUDA的并行化进一步加速了优化过程。此外，一种基于SDF空间网格点的分层细化策略逐步融入从粗到细的样本进行参数初始化与优化，从而提升收敛性与训练效率。在多个基准数据集上的大量实验表明，相较于现有稀疏隐式表示方法，本方法能够以显著更少的参数表示SDF场，并获得更好的精度、鲁棒性与计算效率。相应可执行程序已公开于 https://github.com/lianbobo/SE-RBFNet.git。