Approximating Signed Distance Fields of Implicit Surfaces with Sparse Ellipsoidal Radial Basis Function Networks

TL;DR

SE-RBFNet introduces a sparse ellipsoidal radial basis function network to compress and accurately represent precomputed SDF samples of implicit surfaces. By employing anisotropic ERBF kernels, a dynamic multi-objective loss, hierarchical coarse-to-fine optimization, adaptive basis addition, and nearest-neighbor acceleration, it achieves high geometric fidelity with far fewer parameters than traditional sparse RBF approaches. The method demonstrates strong sparsity, efficiency, and generality across diverse datasets and even when applied to neural implicit SDFs, outperforming baselines like SparseRBF and approaching the performance of neural-QNN-style interpolants while using a fraction of the parameters. This yields a practical, storage-efficient, and fast representation suitable for downstream 3D processing and transmission tasks, with broad potential for integration with existing SDF generation pipelines.

Abstract

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

Figures (8)

• Figure 1: The workflow and results of the SE-RBFNet on SDFs of implicit surfaces. (a) takes the octree structure as an example, where the signed distances of all grid points can be obtained using arbitrary SDF-generation methods and used as training data; (b) shows the initial structure of SE-RBFNet and the structure after sparse optimization. SE-RBFNet takes the SDF values on the octree grids and the sampling points on the implicit surface as input and outputs the optimized parameters of ERBFs; (c) shows the explicit surfaces extracted from the original SDF values on octree grids (in green) and from the SE-RBFNet approximated SDF values on octree grids (in purple); (d) shows the process of extracting the initial ERBF centers using maximum inscribed spheres. The left part illustrates the interior (in green) and exterior (in black) octree grid points. The right part shows the computed maximum inscribed spheres (in light blue); (e) shows that the number of optimized ERBF bases is dramatically reduced while the surface shape is preserved.
• Figure 2: The neural network structure of SE-RBFNet.
• Figure 3: Comparison of implicit surface approximation results obtained by our method and SparseRBF. (a), (b), and (d) are from the Famous dataset; (c) is from the Thingi10k dataset; (e) and (f) correspond to two real-world scanned datasets provided by Erler et al. Erler_2020.
• Figure 4: Comparison of average number of basis functions and parameters on four benchmark datasets.
• Figure 5: Comparison of the implicit surfaces extracted from the approximated SDFs by our method, SparseRBF, and RBF-QNN. (a) and (b) are from the MeshSeg dataset, (c) is from the ABC dataset, while (d) is from the Thingi10k dataset.