# Spline Positional Encoding for Learning 3D Implicit Signed Distance Fields

arXiv: https://arxiv.org/abs/2106.01553 · HTML: https://mlsanity.com/paper/2106.01553/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 29 references that are in the map.
- First posted on arXiv 3 Jun 2021.

## Builds on (most cited first, 18 of 18)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Learning Implicit Fields for Generative Shape Modeling](https://mlsanity.com/paper/1812.02822/) (2018)
- [Scene Representation Networks: Continuous 3D-Structure-Aware Neural Scene Representations](https://mlsanity.com/paper/1906.01618/) (2019)
- [Convolutional Occupancy Networks](https://mlsanity.com/paper/2003.04618/) (2020)
- [Implicit Geometric Regularization for Learning Shapes](https://mlsanity.com/paper/2002.10099/) (2020)
- [SAL: Sign Agnostic Learning of Shapes from Raw Data](https://mlsanity.com/paper/1911.10414/) (2019)
- [Local Implicit Grid Representations for 3D Scenes](https://mlsanity.com/paper/2003.08981/) (2020)
- [Deep Local Shapes: Learning Local SDF Priors for Detailed 3D Reconstruction](https://mlsanity.com/paper/2003.10983/) (2020)
- [Texture Fields: Learning Texture Representations in Function Space](https://mlsanity.com/paper/1905.07259/) (2019)
- [Training behavior of deep neural network in frequency domain](https://mlsanity.com/paper/1807.01251/) (2018)
- [SALD: Sign Agnostic Learning with Derivatives](https://mlsanity.com/paper/2006.05400/) (2020)
- [Reconstructing continuous distributions of 3D protein structure from cryo-EM images](https://mlsanity.com/paper/1909.05215/) (2019)

## Built on by (most cited first, 21 of 21)

- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [E-NeRV: Expedite Neural Video Representation with Disentangled Spatial-Temporal Context](https://mlsanity.com/paper/2207.08132/) (2022)
- [NeuRBF: A Neural Fields Representation with Adaptive Radial Basis Functions](https://mlsanity.com/paper/2309.15426/) (2023)
- [Surface Reconstruction from Point Clouds: A Survey and a Benchmark](https://mlsanity.com/paper/2205.02413/) (2022)
- [Intrinsic Neural Fields: Learning Functions on Manifolds](https://mlsanity.com/paper/2203.07967/) (2022)
- [PINs: Progressive Implicit Networks for Multi-Scale Neural Representations](https://mlsanity.com/paper/2202.04713/) (2022)
- [Learning Local Implicit Fourier Representation for Image Warping](https://mlsanity.com/paper/2207.01831/) (2022)
- [From Activation to Initialization: Scaling Insights for Optimizing Neural Fields](https://mlsanity.com/paper/2403.19205/) (2024)
- [PREF: Phasorial Embedding Fields for Compact Neural Representations](https://mlsanity.com/paper/2205.13524/) (2022)
- [Training-Free Hierarchical Scene Understanding for Gaussian Splatting with Superpoint Graphs](https://mlsanity.com/paper/2504.13153/) (2025)
- [BF-STVSR: B-Splines and Fourier-Best Friends for High Fidelity Spatial-Temporal Video Super-Resolution](https://mlsanity.com/paper/2501.11043/) (2025)
- [Curvature-Aware Training for Coordinate Networks](https://mlsanity.com/paper/2305.08552/) (2023)
- [RBF-PINN: Non-Fourier Positional Embedding in Physics-Informed Neural Networks](https://mlsanity.com/paper/2402.08367/) (2024)
- [CoNeS: Conditional neural fields with shift modulation for multi-sequence MRI translation](https://mlsanity.com/paper/2309.03320/) (2023)
- [How Learnable Grids Recover Fine Detail in Low Dimensions: A Neural Tangent Kernel Analysis of Multigrid Parametric Encodings](https://mlsanity.com/paper/2504.13412/) (2025)
- [A Simple And Effective Filtering Scheme For Improving Neural Fields](https://mlsanity.com/paper/2201.13013/) (2022)
- [Feature Mapping in Physics-Informed Neural Networks (PINNs)](https://mlsanity.com/paper/2402.06955/) (2024)
- [MetricGrids: Arbitrary Nonlinear Approximation with Elementary Metric Grids based Implicit Neural Representation](https://mlsanity.com/paper/2503.10000/) (2025)
- [On Optimal Sampling for Learning SDF Using MLPs Equipped with Positional Encoding](https://mlsanity.com/paper/2401.01391/) (2024)
- [Dual-Prior Guided Null-Space Learning with Mixture-of-Splines for Arbitrary Medical Slice Super-Resolution](https://mlsanity.com/paper/2606.26716/) (2026)
- [CARA: Collision-Aware Resolution Adaptation for Multiresolution Hash Encoding Based Image Fitting](https://mlsanity.com/paper/2609.18554/) (2026)

## Abstract (arXiv)

Multilayer perceptrons (MLPs) have been successfully used to represent 3D shapes implicitly and compactly, by mapping 3D coordinates to the corresponding signed distance values or occupancy values. In this paper, we propose a novel positional encoding scheme, called Spline Positional Encoding, to map the input coordinates to a high dimensional space before passing them to MLPs, for helping to recover 3D signed distance fields with fine-scale geometric details from unorganized 3D point clouds. We verified the superiority of our approach over other positional encoding schemes on tasks of 3D shape reconstruction from input point clouds and shape space learning. The efficacy of our approach extended to image reconstruction is also demonstrated and evaluated.