Approximating Signed Distance Fields With Sparse Ellipsoidal Radial Basis Function Networks: A Dynamic Multi-Objective Optimization Strategy

Abstract

In this paper, a general learning method is proposed for approximating precomputed signed distance function (SDF) fields of implicit surfaces using a relatively small number of ellipsoidal radial basis functions (ERBFs). The SDF values may be obtained from various sources, including point clouds, triangle meshes, analytical expressions, pretrained neural networks, and others. Given SDF values at spatial grid points, the proposed method approximates the SDF with a significantly reduced number of ERBFs, yielding a compact representation while preserving the geometric shape of the corresponding implicit surface. To balance sparsity and approximation precision, a dynamic multi-objective optimization strategy inspired by Pareto multi-task learning is introduced. Instead of relying on fixed coefficients, this mechanism dynamically adjusts the trade-off during training by computing an optimal convex combination based on the gradients of the accuracy and sparsity loss terms. Simultaneous optimization is performed over the weights, centers, shapes, and orientations of the ERBFs. For enhanced computational efficiency, a nearest-neighbor-based data structure is employed to confine computations to points in the vicinity of each kernel center, and CUDA-based parallelism is utilized to further accelerate the optimization process. In addition, a hierarchical refinement strategy is applied based on the SDF spatial grid points, where coarse-to-fine samples are progressively incorporated for parameter initialization and optimization. This approach improves both convergence behavior and training efficiency. Extensive experiments on multiple benchmark datasets demonstrate that the proposed method represents SDF fields with substantially fewer parameters than existing sparse implicit representation methods, while attaining superior accuracy, robustness, and computational efficiency.