# Seeing Implicit Neural Representations as Fourier Series

arXiv: https://arxiv.org/abs/2109.00249 · HTML: https://mlsanity.com/paper/2109.00249/

## Key facts

- Cited by 50 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 30 references that are in the map.
- First posted on arXiv 1 Sep 2021.

## Builds on (most cited first, 21 of 21)

- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [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)
- [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)
- [PIFu: Pixel-Aligned Implicit Function for High-Resolution Clothed Human Digitization](https://mlsanity.com/paper/1905.05172/) (2019)
- [Convolutional Occupancy Networks](https://mlsanity.com/paper/2003.04618/) (2020)
- [Implicit Geometric Regularization for Learning Shapes](https://mlsanity.com/paper/2002.10099/) (2020)
- [Differentiable Volumetric Rendering: Learning Implicit 3D Representations without 3D Supervision](https://mlsanity.com/paper/1912.07372/) (2019)
- [BARF: Bundle-Adjusting Neural Radiance Fields](https://mlsanity.com/paper/2104.06405/) (2021)
- [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)
- [Learning Shape Templates with Structured Implicit Functions](https://mlsanity.com/paper/1904.06447/) (2019)
- [Texture Fields: Learning Texture Representations in Function Space](https://mlsanity.com/paper/1905.07259/) (2019)
- [DIST: Rendering Deep Implicit Signed Distance Function with Differentiable Sphere Tracing](https://mlsanity.com/paper/1911.13225/) (2019)
- [NASA: Neural Articulated Shape Approximation](https://mlsanity.com/paper/1912.03207/) (2019)
- [Learning to Infer Implicit Surfaces without 3D Supervision](https://mlsanity.com/paper/1911.00767/) (2019)
- [Learning a Neural 3D Texture Space from 2D Exemplars](https://mlsanity.com/paper/1912.04158/) (2019)

## Built on by (most cited first, 30 of 50)

- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [On the expressiveness and spectral bias of KANs](https://mlsanity.com/paper/2410.01803/) (2024)
- [Generalised Implicit Neural Representations](https://mlsanity.com/paper/2205.15674/) (2022)
- [Neural Fourier Filter Bank](https://mlsanity.com/paper/2212.01735/) (2022)
- [Exploring Differential Geometry in Neural Implicits](https://mlsanity.com/paper/2201.09263/) (2022)
- [HyperTime: Implicit Neural Representation for Time Series](https://mlsanity.com/paper/2208.05836/) (2022)
- [Diffeomorphic Image Registration with Neural Velocity Field](https://mlsanity.com/paper/2202.12498/) (2022)
- [Intrinsic Neural Fields: Learning Functions on Manifolds](https://mlsanity.com/paper/2203.07967/) (2022)
- [LVAC: Learned Volumetric Attribute Compression for Point Clouds using Coordinate Based Networks](https://mlsanity.com/paper/2111.08988/) (2021)
- [Tuning the Frequencies: Robust Training for Sinusoidal Neural Networks](https://mlsanity.com/paper/2407.21121/) (2024)
- [Quantum Implicit Neural Representations](https://mlsanity.com/paper/2406.03873/) (2024)
- [Learning Local Implicit Fourier Representation for Image Warping](https://mlsanity.com/paper/2207.01831/) (2022)
- [TINC: Tree-structured Implicit Neural Compression](https://mlsanity.com/paper/2211.06689/) (2022)
- [SCI: A Spectrum Concentrated Implicit Neural Compression for Biomedical Data](https://mlsanity.com/paper/2209.15180/) (2022)
- [Siamese SIREN: Audio Compression with Implicit Neural Representations](https://mlsanity.com/paper/2306.12957/) (2023)
- [Hypernetworks build Implicit Neural Representations of Sounds](https://mlsanity.com/paper/2302.04959/) (2023)
- [FINER++: Building a Family of Variable-periodic Functions for Activating Implicit Neural Representation](https://mlsanity.com/paper/2407.19434/) (2024)
- [Frequency-Modulated Point Cloud Rendering with Easy Editing](https://mlsanity.com/paper/2303.07596/) (2023)
- [Neural topology optimization: the good, the bad, and the ugly](https://mlsanity.com/paper/2407.13954/) (2024)
- [Learning High-Frequency Functions Made Easy with Sinusoidal Positional Encoding](https://mlsanity.com/paper/2407.09370/) (2024)
- [A unified model for continuous conditional video prediction](https://mlsanity.com/paper/2210.05810/) (2022)
- [JDEC: JPEG Decoding via Enhanced Continuous Cosine Coefficients](https://mlsanity.com/paper/2404.05558/) (2024)
- [Spatiotemporal Implicit Neural Representation as a Generalized Traffic Data Learner](https://mlsanity.com/paper/2405.03185/) (2024)
- [Attention Beats Linear for Fast Implicit Neural Representation Generation](https://mlsanity.com/paper/2407.15355/) (2024)
- [Harnessing Low-Frequency Neural Fields for Few-Shot View Synthesis](https://mlsanity.com/paper/2303.08370/) (2023)
- [Data-Efficient Deep Operator Network for Unsteady Flow: A Multi-Fidelity Approach with Physics-Guided Subsampling](https://mlsanity.com/paper/2503.17941/) (2025)
- [3D Neural Sculpting (3DNS): Editing Neural Signed Distance Functions](https://mlsanity.com/paper/2209.13971/) (2022)
- [Understanding Sinusoidal Neural Networks](https://mlsanity.com/paper/2212.01833/) (2022)
- [Understanding the dynamics of the frequency bias in neural networks](https://mlsanity.com/paper/2405.14957/) (2024)
- [Vector Representations of Vessel Trees](https://mlsanity.com/paper/2506.11163/) (2025)

## Abstract (arXiv)

Implicit Neural Representations (INR) use multilayer perceptrons to represent high-frequency functions in low-dimensional problem domains. Recently these representations achieved state-of-the-art results on tasks related to complex 3D objects and scenes. A core problem is the representation of highly detailed signals, which is tackled using networks with periodic activation functions (SIRENs) or applying Fourier mappings to the input. This work analyzes the connection between the two methods and shows that a Fourier mapped perceptron is structurally like one hidden layer SIREN. Furthermore, we identify the relationship between the previously proposed Fourier mapping and the general d-dimensional Fourier series, leading to an integer lattice mapping. Moreover, we modify a progressive training strategy to work on arbitrary Fourier mappings and show that it improves the generalization of the interpolation task. Lastly, we compare the different mappings on the image regression and novel view synthesis tasks. We confirm the previous finding that the main contributor to the mapping performance is the size of the embedding and standard deviation of its elements.