mlsanityData to 7 Oct 2026

Seeing Implicit Neural Representations as Fourier Series

arXiv 2109.00249 ↗First posted 1 Sep 2021cs.CVcited by 50

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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.

Key facts. Cited by 50 papers in the map (data through 7 Oct 2026). Builds on 21 of its 30 references that are in the map. First posted on arXiv 1 Sep 2021.

Builds on

21 of 30 references.

References · 21

  1. 2019DeepSDF: Learning Continuous Signed Distance Functions for Shape Representationcited by 2,682
  2. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,597
  3. 2020Implicit Neural Representations with Periodic Activation Functionscited by 2,214
  4. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,170
  5. 2018Occupancy Networks: Learning 3D Reconstruction in Function Spacecited by 2,037
  6. 2018Learning Implicit Fields for Generative Shape Modelingcited by 1,060
  7. 2019Scene Representation Networks: Continuous 3D-Structure-Aware Neural Scene Representationscited by 864
  8. 2019PIFu: Pixel-Aligned Implicit Function for High-Resolution Clothed Human Digitizationcited by 818
  9. 2020Convolutional Occupancy Networkscited by 760
  10. 2020Implicit Geometric Regularization for Learning Shapescited by 707
  11. 2019Differentiable Volumetric Rendering: Learning Implicit 3D Representations without 3D Supervisioncited by 699
  12. 2021BARF: Bundle-Adjusting Neural Radiance Fieldscited by 641
9 more
  1. 2019SAL: Sign Agnostic Learning of Shapes from Raw Datacited by 389
  2. 2020Local Implicit Grid Representations for 3D Scenescited by 372
  3. 2020Deep Local Shapes: Learning Local SDF Priors for Detailed 3D Reconstructioncited by 321
  4. 2019Learning Shape Templates with Structured Implicit Functionscited by 242
  5. 2019Texture Fields: Learning Texture Representations in Function Spacecited by 210
  6. 2019DIST: Rendering Deep Implicit Signed Distance Function with Differentiable Sphere Tracingcited by 179
  7. 2019NASA: Neural Articulated Shape Approximationcited by 149
  8. 2019Learning to Infer Implicit Surfaces without 3D Supervisioncited by 103
  9. 2019Learning a Neural 3D Texture Space from 2D Exemplarscited by 53

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 2; 2022: 13; 2023: 6; 2024: 12; 2025: 10; 2026: 7.

50 papers.

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  1. 2021Neural Fields in Visual Computing and Beyondcited by 539
  2. 2024On the expressiveness and spectral bias of KANscited by 36
  3. 2022Generalised Implicit Neural Representationscited by 31
  4. 2022Neural Fourier Filter Bankcited by 29
  5. 2022Exploring Differential Geometry in Neural Implicitscited by 26
  6. 2022HyperTime: Implicit Neural Representation for Time Seriescited by 24
  7. 2022Diffeomorphic Image Registration with Neural Velocity Fieldcited by 20
  8. 2022Intrinsic Neural Fields: Learning Functions on Manifoldscited by 20
  9. 2021LVAC: Learned Volumetric Attribute Compression for Point Clouds using Coordinate Based Networkscited by 19
  10. 2024Tuning the Frequencies: Robust Training for Sinusoidal Neural Networkscited by 18
  11. 2024Quantum Implicit Neural Representationscited by 16
  12. 2022Learning Local Implicit Fourier Representation for Image Warpingcited by 14
38 more
  1. 2022TINC: Tree-structured Implicit Neural Compressioncited by 14
  2. 2022SCI: A Spectrum Concentrated Implicit Neural Compression for Biomedical Datacited by 13
  3. 2023Hypernetworks build Implicit Neural Representations of Soundscited by 11
  4. 2023Siamese SIREN: Audio Compression with Implicit Neural Representationscited by 11
  5. 2023Frequency-Modulated Point Cloud Rendering with Easy Editingcited by 10
  6. 2024FINER++: Building a Family of Variable-periodic Functions for Activating Implicit Neural Representationcited by 10
  7. 2024Neural topology optimization: the good, the bad, and the uglycited by 9
  8. 2024Learning High-Frequency Functions Made Easy with Sinusoidal Positional Encodingcited by 8
  9. 2022A unified model for continuous conditional video predictioncited by 7
  10. 2024JDEC: JPEG Decoding via Enhanced Continuous Cosine Coefficientscited by 7
  11. 2024Spatiotemporal Implicit Neural Representation as a Generalized Traffic Data Learnercited by 7
  12. 2024Attention Beats Linear for Fast Implicit Neural Representation Generationcited by 6
  13. 2023Harnessing Low-Frequency Neural Fields for Few-Shot View Synthesiscited by 5
  14. 2025Data-Efficient Deep Operator Network for Unsteady Flow: A Multi-Fidelity Approach with Physics-Guided Subsamplingcited by 5
  15. 20223D Neural Sculpting (3DNS): Editing Neural Signed Distance Functionscited by 4
  16. 2022Understanding Sinusoidal Neural Networkscited by 4
  17. 2024Understanding the dynamics of the frequency bias in neural networkscited by 4
  18. 2025Vector Representations of Vessel Treescited by 3
  19. 2023Recovering Continuous Scene Dynamics from A Single Blurry Image with Eventscited by 2
  20. 2024Predicting the Encoding Error of SIRENscited by 2
  21. 2025Neural Functions for Learning Periodic Signalcited by 2
  22. 2025On Spectral Properties of Gradient-based Explanation Methodscited by 2
  23. 2026Calibrated Harmonic Overlaid Implicit Neural Representations for Multi-Dimensional Datacited by 2
  24. 2024Towards a Sampling Theory for Implicit Neural Representationscited by 1
  25. 2025Generating visual explanations from deep networks using implicit neural representationscited by 1
  26. 2025QFGN: A Quantum Approach to High-Fidelity Implicit Neural Representationscited by 1
  27. 2025Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditionscited by 1
  28. 2025Enhancing Robustness of Implicit Neural Representations Against Weight Perturbationscited by 1
  29. 2026Content-Aware Frequency Encoding for Implicit Neural Representations with Fourier-Chebyshev Featurescited by 1
  30. 2022WOLONet: Wave Outlooker for Efficient and High Fidelity Speech Synthesiscited by 0
  31. 2023Implicit neural representation for change detectioncited by 0
  32. 2025Pressure Field Reconstruction with SIREN: A Mesh-Free Approach for Image Velocimetry in Complex Noisy Environmentscited by 0
  33. 2025Fourier Learning Machines: Nonharmonic Fourier-Based Neural Networks for Scientific Machine Learningcited by 0
  34. 2026Looking Into the Water by Unsupervised Learning of the Surface Shapecited by 0
  35. 2026What Cohort INRs Encode and Where to Freeze Themcited by 0
  36. 2026AIR: Amortized Image Reconstruction Framework for Self-Supervised Feed-Forward 2D Gaussian Splattingcited by 0
  37. 2026Recurrent Sinusoidal INRs for Efficient High-Fidelity Representationcited by 0
  38. 2026Bernstein-Vazirani Networks: Quantum Machine Learning by Interferencecited by 0