efunc: An Efficient Function Representation without Neural Networks

TL;DR

This work proposes a neural-network-free, parameter-efficient framework for continuous function modeling and applies it to 3D signed distance functions (SDFs). The core idea is to represent functions on regular grids via radial-basis interpolation with polynomial-valued values, augmented with optional offsets that shift grid points toward surface regions. An analytic gradient and a CUDA-accelerated forward/backward pipeline enable efficient training and inference, achieving competitive Chamfer distances with far fewer parameters than state-of-the-art neural or hierarchical approaches. The method supports scalar, vector, and function-valued targets, and permits straightforward shape manipulation, normal estimation, and function decomposition, with potential extensions to 2D/4D and generative modeling. Although slower than highly optimized baselines and tied to GPU hardware, the approach offers simplicity, transparency, and efficiency advantages for SDF fitting and geometry processing.

Abstract

Function fitting/approximation plays a fundamental role in computer graphics and other engineering applications. While recent advances have explored neural networks to address this task, these methods often rely on architectures with many parameters, limiting their practical applicability. In contrast, we pursue high-quality function approximation using parameter-efficient representations that eliminate the dependency on neural networks entirely. We first propose a novel framework for continuous function modeling. Most existing works can be formulated using this framework. We then introduce a compact function representation, which is based on polynomials interpolated using radial basis functions, bypassing both neural networks and complex/hierarchical data structures. We also develop memory-efficient CUDA-optimized algorithms that reduce computational time and memory consumption to less than 10% compared to conventional automatic differentiation frameworks. Finally, we validate our representation and optimization pipeline through extensive experiments on 3D signed distance functions (SDFs). The proposed representation achieves comparable or superior performance to state-of-the-art techniques (e.g., octree/hash-grid techniques) with significantly fewer parameters.

Figures (14)

• Figure 1: We analyze how we interpolate. The target can be scalars, vectors, or functions (top). The support can be local or global (bottom). In this work, we are using functions as the interpolating targets with global support.
• Figure 2: Quality of the representations. We start from a simple trilinear method applied on a 32-resolution grid (gray). We show the results obtained using the proposed representations (blue). The ground-truth mesh is shown on the left (purple) as a reference. The best results are obtained using the proposed $\mathcal{O}^{+\Delta}$ in \ref{['eq:func-offset']}. For the other figures, we can see that the results are getting better when using a higher order of $f$. However, the differences in visual results are not as significant as the metrics.
• Figure 3: We learn an offset $\Delta_i$ for each grid point $\textcolor{textred}{\mathbf{k}_i}$ (second column). The initialization (third column) is obtained using mean shift. They are around the surface region. After convergence, the offsets moved a little bit (fourth column). The final reconstruction is shown on the right (last column).
• Figure 4: Diagram for the forward and backward pass algorithm.
• Figure 5: We compare the input (left) to the main competitors, FFN tancik2020fourier, NGLOD takikawa2021neural, Instant NGP muller2022instant, Dif-Grid chen2023dictionary, and our representation.