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arxiv: 2604.02846 · v2 · submitted 2026-04-03 · 💻 cs.CV · eess.IV

Adaptive Local Frequency Filtering for Fourier-Encoded Implicit Neural Representations

Ligen Shi , Jun Qiu , Yuhang Zheng , Zengyu Pang , Chang Liu This is my paper

Pith reviewed 2026-05-13 19:49 UTC · model grok-4.3

classification 💻 cs.CV eess.IV
keywords Fourier-encoded INRadaptive local filteringspatially varying frequencyneural tangent kernelimplicit neural representationssignal reconstructionnon-stationary signals
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The pith

Spatially varying modulation lets Fourier-encoded INRs adapt frequency response to local signal spectra.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes making frequency encoding in implicit neural representations adaptive rather than fixed across the whole domain. A single learned scalar field α(x) scales the sinusoidal features at each point, producing a smooth shift between low-pass, band-pass, and high-pass behavior depending on local signal content. This change is analyzed through the neural tangent kernel to show how it alters the effective spectrum seen by the optimizer. Experiments on image fitting, 3D shape representation, and sparse reconstruction report higher accuracy and quicker convergence than constant-frequency baselines. The learned α(x) also supplies a direct map of where the model chooses higher or lower frequencies.

Core claim

The central claim is that modulating the Fourier feature vector by a spatially varying scalar α(x) enables the INR to apply position-dependent frequency filtering, transitioning smoothly among low-pass, band-pass, and high-pass regimes and thereby fitting signals whose frequency content changes across space more accurately and with faster optimization than fixed mappings allow.

What carries the argument

The learnable scalar field α(x) that multiplies the encoded Fourier components to control local pass-band behavior.

If this is right

  • Reconstruction quality rises on 2D image fitting, 3D shape fitting, and sparse data tasks.
  • Optimization reaches target error in fewer steps than fixed-frequency encodings.
  • The learned α(x) field directly visualizes the model's preferred frequencies at each location.
  • NTK analysis shows the modulation reshapes the kernel spectrum to favor high-frequency components where they are needed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same local modulation idea could be attached to other positional encodings such as random Fourier features or positional encodings in transformers.
  • Making α also depend on time would let the method handle video or dynamic signals without retraining a new network for each frame.
  • Because α(x) is cheap to store and visualize, it offers a practical diagnostic for diagnosing under-fitting in high-frequency regions of any INR.

Load-bearing premise

A single spatially varying scalar α(x) produces stable, artifact-free transitions between frequency regimes during gradient descent.

What would settle it

If training runs using the adaptive α(x) exhibit visible ringing or slower convergence precisely at locations where the signal frequency content changes abruptly, the claim of smooth stable modulation would be falsified.

Figures

Figures reproduced from arXiv: 2604.02846 by Chang Liu, Jun Qiu, Ligen Shi, Yuhang Zheng, Zengyu Pang.

Figure 1
Figure 1. Figure 1: Overview of the proposed AL-Filter. A learnable grid stores the adaptive parameter 𝛼(𝐱), which modulates the local frequency response of Fourier features before MLP-based reconstruction. • We propose adaptive local frequency filtering for Fourier-encoded implicit neural representations, en￾abling spatially varying frequency modulation through a learnable parameter 𝛼(𝐱). • We analyze the proposed method fro… view at source ↗
Figure 2
Figure 2. Figure 2: Frequency responses of the proposed adaptive filter on the encoded-channel axis under different values of the parameter 𝛼(𝐱). The filter transitions smoothly among low-pass, band-pass, and high-pass behaviors. Here, 𝛼(𝐱) controls the center of the effective pass region, while 𝐵 denotes the bandwidth measured in channel units. where 𝑗 ∈ {0, ⋯ , 𝐿−1} indexes the dyadic frequency scales introduced by the Four… view at source ↗
Figure 3
Figure 3. Figure 3: , we consider representative natural images containing both complex textures and relatively smooth background regions. 30 32 34 36 38 40 42 44 46 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10-4 Ground Truth Ours-ReLU 𝛼 𝐱 Map Absolute Difference 30 35 40 45 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10-4 28 30 32 34 36 38 40 42 44 46 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10-4 [PITH_FULL_IMAGE:figures/full_fig_… view at source ↗
Figure 4
Figure 4. Figure 4: Empirical NTK analysis. (a) Comparison of normal￾ized eigenspectra. (b) Relative retention ratio of normalized eigenvalues. The empirical spectra are consistent with the NTK￾inspired analysis in Section 3.5. As shown in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Convergence curves of INR models on 2D image fitting. Left: PSNR during the first 100 iterations. Right: PSNR over 5,000 iterations. We first examine the training curves to evaluate the effect of the proposed filter on optimization behavior. As shown in [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Qualitative comparison of the proposed method and baseline approaches on 2D image fitting. the compared baselines during the early stage of training and also reach higher final PSNR values. This behavior is consistent with the intended role of the adaptive filter in improving local frequency selection during optimization [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Qualitative comparison of signed distance field representations for the Armadillo model. in the legs, but also introduces visible roughness in the smoother region. PE-MLP does not reconstruct either region as accurately. In contrast, the proposed method preserves finer geometric detail in the leg region while maintaining smoothness in the pectoral area, which is consistent with its goal of adapting local f… view at source ↗
Figure 8
Figure 8. Figure 8: Sparse data reconstruction under 5% and 35% pixel observations. Columns from left to right: ground truth, masked input, masked error, reconstructed result (Ours), reconstruction error, and learned 𝛼(𝐱) map. Rows 1–2 correspond to 5% observations, and Rows 3–4 correspond to 35% observations. 6. Conclusion This paper presented an adaptive local frequency filter￾ing method for Fourier-encoded implicit neural … view at source ↗
read the original abstract

Fourier-encoded implicit neural representations (INRs) have shown strong capability in modeling continuous signals from discrete samples. However, conventional Fourier feature mappings use a fixed set of frequencies over the entire spatial domain, making them poorly suited to signals with spatially varying local spectra and often leading to slow convergence of high-frequency details. To address this issue, we propose an adaptive local frequency filtering method for Fourier-encoded INRs. The proposed method introduces a spatially varying parameter $\alpha(\mathbf{x})$ to modulate encoded Fourier components, enabling a smooth transition among low-pass, band-pass, and high-pass behaviors at different spatial locations. We further analyze the effect of the proposed filter from the neural tangent kernel (NTK) perspective and provide an NTK-inspired interpretation of how it reshapes the effective kernel spectrum. Experiments on 2D image fitting, 3D shape representation, and sparse data reconstruction demonstrate that the proposed method consistently improves reconstruction quality and leads to faster optimization compared with fixed-frequency baselines. In addition, the learned $\alpha(\mathbf{x})$ provides an intuitive visualization of spatially varying frequency preferences, which helps explain the behavior of the model on non-stationary signals. These results indicate that adaptive local frequency modulation is a practical enhancement for Fourier-encoded INRs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes an adaptive local frequency filtering method for Fourier-encoded implicit neural representations (INRs). It introduces a spatially varying parameter α(x) to modulate encoded Fourier components, enabling location-specific transitions among low-pass, band-pass, and high-pass behaviors. An NTK-based analysis interprets how this reshapes the effective kernel spectrum. Experiments on 2D image fitting, 3D shape representation, and sparse data reconstruction report consistent improvements in quality and optimization speed over fixed-frequency baselines, with learned α(x) providing visualizations of frequency preferences.

Significance. If the stability of the learned α(x) holds and the reported gains are not offset by artifacts, the method offers a practical enhancement for handling non-stationary signals in INRs, with added interpretability via α(x) maps. The NTK perspective strengthens the theoretical grounding beyond purely empirical claims.

major comments (2)
  1. [§3] §3 (method description): The claim of 'smooth transition' among filtering behaviors relies on α(x) being learned jointly with the INR, but no smoothness regularizer, Lipschitz constraint, or post-processing is specified to bound spatial gradients of α(x). This leaves open the possibility of discontinuous effective Fourier kernels, which would invalidate the NTK spectrum-reshaping interpretation in §4.
  2. [§5] §5 (experiments, sparse reconstruction): The central claim of improved quality and faster convergence on sparse data is load-bearing, yet no quantitative diagnostics (e.g., gradient norms of α(x), frequency of high-frequency ringing, or ablation with enforced smoothness on α(x)) are reported to rule out instabilities that could cancel the gains, as flagged by the weakest assumption.
minor comments (2)
  1. [Tables/Figures] Table 1 and Figure 4 captions should explicitly state the number of runs and standard deviations for the reported PSNR/IOU metrics to allow assessment of statistical significance.
  2. [§4] The NTK derivation in §4 assumes the modulation acts as a multiplicative filter on the feature map; an explicit equation showing the modified kernel K_α(x,x') would clarify the spectrum-reshaping argument.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the insightful comments. Below we provide detailed responses to each major comment and indicate the revisions that will be incorporated in the updated manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (method description): The claim of 'smooth transition' among filtering behaviors relies on α(x) being learned jointly with the INR, but no smoothness regularizer, Lipschitz constraint, or post-processing is specified to bound spatial gradients of α(x). This leaves open the possibility of discontinuous effective Fourier kernels, which would invalidate the NTK spectrum-reshaping interpretation in §4.

    Authors: The referee correctly notes that the manuscript does not specify an explicit smoothness regularizer or constraint on α(x). While the joint training with the INR tends to produce smooth α(x) in practice—as can be seen from the α(x) visualizations provided in the paper—the possibility of discontinuities cannot be entirely ruled out without additional measures. The NTK interpretation in §4 assumes a locally applied filter, and given that α(x) is generated by a continuous neural network, the effective spectrum reshaping remains valid at each point. To address this concern directly, we will revise §3 to include a brief discussion of the smoothness properties observed in experiments and add an ablation study enforcing a total variation penalty on α(x) to demonstrate that the performance gains persist under smoothness constraints. revision: partial

  2. Referee: [§5] §5 (experiments, sparse reconstruction): The central claim of improved quality and faster convergence on sparse data is load-bearing, yet no quantitative diagnostics (e.g., gradient norms of α(x), frequency of high-frequency ringing, or ablation with enforced smoothness on α(x)) are reported to rule out instabilities that could cancel the gains, as flagged by the weakest assumption.

    Authors: We acknowledge the importance of providing quantitative evidence to support the stability of the method on sparse data. Although our experiments showed consistent improvements without visible artifacts or ringing, we did not report specific diagnostics such as gradient norms of α(x). In the revised manuscript, we will augment §5 with these metrics, including average ||∇α(x)|| and comparisons of reconstruction quality with and without smoothness enforcement on α(x). This will confirm that the reported gains are not due to instabilities. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new parameter and NTK interpretation are independent of fitted outputs

full rationale

The paper introduces an explicit new spatially varying parameter α(x) to modulate Fourier features and analyzes its effect via standard NTK machinery (not derived from the present fitted values). Experimental gains are reported against fixed-frequency baselines on image fitting, 3D shapes, and sparse reconstruction; these comparisons do not reduce to a quantity defined solely by α(x) itself. No self-citations, uniqueness theorems, or ansatzes from prior author work are invoked as load-bearing steps. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The approach rests on standard Fourier feature mapping and neural tangent kernel theory for INRs, with the novel element being the introduction of the position-dependent modulator α(x) whose effect is interpreted through the existing NTK lens.

free parameters (1)
  • α(x)
    Spatially varying scalar field learned during optimization to control local frequency filtering behavior.
axioms (1)
  • domain assumption Neural tangent kernel analysis remains valid when Fourier features are modulated by a spatially varying α(x)
    The paper states it provides an NTK-inspired interpretation of how the filter reshapes the effective kernel spectrum.
invented entities (1)
  • α(x) no independent evidence
    purpose: Spatially varying modulator that enables local low-pass, band-pass, or high-pass behavior on Fourier components
    New parameter introduced by the method; no independent evidence of its existence outside the optimization is provided.

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Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

  1. [1]

    Where do we stand with implicit neural representations? a technical and performance survey,

    A. Essakine, Y. Cheng, C.-W. Cheng, L. Zhang, Z. Deng, L. Zhu, C.- B. Schönlieb, A. I. Aviles-Rivero, Where do we stand with implicit neural representations? a technical and performance survey, arXiv preprint arXiv:2411.03688 (2024)

    work page arXiv 2024

  2. [2]

    8628–8638

    Y.Chen,S.Liu,X.Wang, Learningcontinuousimagerepresentation with local implicit image function, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2021, pp. 8628–8638

    work page 2021

  3. [3]

    Z. Chen, H. Zhang, Learning implicit fields for generative shape modeling,in:ProceedingsoftheIEEE/CVFConferenceonComputer Vision and Pattern Recognition, 2019, pp. 5939–5948

    work page 2019

  4. [4]

    Sitzmann, S

    V. Sitzmann, S. Rezchikov, B. Freeman, J. Tenenbaum, F. Du- rand, Light field networks: Neural scene representations with single- evaluation rendering, volume 34, 2021, pp. 19313–19325

    work page 2021

  5. [5]

    Z. Li, L. Song, C. Liu, J. Yuan, Y. Xu, Neulf: Efficient novel view synthesiswithneural4dlightfield, arXivpreprintarXiv:2105.07112 (2021)

    work page arXiv 2021

  6. [6]

    Rahaman, A

    N. Rahaman, A. Baratin, D. Arpit, F. Draxler, M. Lin, F. Hamprecht, Y. Bengio, A. Courville, On the spectral bias of neural networks, in:InternationalConferenceonMachineLearning,PMLR,2019,pp. 5301–5310

    work page 2019

  7. [7]

    Singhal, R

    M.Tancik,P.Srinivasan,B.Mildenhall,S.Fridovich-Keil,N.Ragha- van, U. Singhal, R. Ramamoorthi, J. Barron, R. Ng, Fourier features let networks learn high frequency functions in low dimensional do- mains,AdvancesinNeuralInformationProcessingSystems33(2020) 7537–7547

    work page 2020

  8. [8]

    Mildenhall, P

    B. Mildenhall, P. P. Srinivasan, M. Tancik, J. T. Barron, R. Ra- mamoorthi,R.Ng,Nerf:Representingscenesasneuralradiancefields for view synthesis, Communications of the ACM 65 (2021) 99–106

    work page 2021

  9. [9]

    Singh, A

    R. Singh, A. Shukla, P. Turaga, Polynomial implicit neural represen- tations for large diverse datasets, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2023, pp. 2041–2051

    work page 2023

  10. [10]

    14153–14163

    Z.Wu,Y.Jin,K.M.Yi, Neuralfourierfilterbank, in:Proceedingsof theIEEE/CVFConferenceonComputerVisionandPatternRecogni- tion, 2023, pp. 14153–14163

    work page 2023

  11. [11]

    V.Sitzmann,J.Martel,A.Bergman,D.Lindell,G.Wetzstein,Implicit neural representations with periodic activation functions, Advances in Neural Information Processing Systems 33 (2020) 7462–7473

    work page 2020

  12. [12]

    Ramasinghe, S

    S. Ramasinghe, S. Lucey, Beyond periodicity: Towards a unifying framework for activations in coordinate-mlps, in: European Confer- ence on Computer Vision, Springer, 2022, pp. 142–158

    work page 2022

  13. [13]

    Saragadam, D

    V. Saragadam, D. LeJeune, J. Tan, G. Balakrishnan, A. Veeraragha- van, R. G. Baraniuk, Wire: Wavelet implicit neural representations, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2023, pp. 18507–18516

    work page 2023

  14. [14]

    D.Serrano,J.Szymkowiak,P.Musialski, Hosc:Aperiodicactivation function for preserving sharp features in implicit neural representa- tions, arXiv preprint arXiv:2401.10967 (2024)

    work page arXiv 2024

  15. [15]

    H.Saratchandran,S.Ramasinghe,V.Shevchenko,A.Long,S.Lucey, A sampling theory perspective on activations for implicit neural representations, arXiv preprint arXiv:2402.05427 (2024)

    work page arXiv 2024

  16. [16]

    Z. Liu, H. Zhu, Q. Zhang, J. Fu, W. Deng, Z. Ma, Y. Guo, X. Cao, Finer: Flexible spectral-bias tuning in implicit neural representation by variable-periodic activation functions, in: Proceedings of the IEEE/CVFConferenceonComputerVisionandPatternRecognition, 2024, pp. 2713–2722

    work page 2024

  17. [17]

    R.Fathony,A.K.Sahu,D.Willmott,J.Z.Kolter, Multiplicativefilter networks, in: International Conference on Learning Representations, 2020, pp. 1–15

    work page 2020

  18. [18]

    D. B. Lindell, D. Van Veen, J. J. Park, G. Wetzstein, Bacon: Band- limited coordinate networks for multiscale scene representation, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2022, pp. 16252–16262

    work page 2022

  19. [19]

    Takikawa, J

    T. Takikawa, J. Litalien, K. Yin, K. Kreis, C. Loop, D. Nowrouzezahrai, A. Jacobson, M. McGuire, S. Fidler, Neural geometric level of detail: Real-time rendering with implicit 3d shapes, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2021, pp. 11358–11367

    work page 2021

  20. [20]

    J. N. Martel, D. B. Lindell, C. Z. Lin, E. R. Chan, M. Monteiro, G. Wetzstein, Acorn: Adaptive coordinate networks for neural scene representation, 2021

    work page 2021

  21. [21]

    Müller, A

    T. Müller, A. Evans, C. Schied, A. Keller, Instant neural graphics primitives with a multiresolution hash encoding, ACM transactions on graphics (TOG) 41 (2022) 1–15

    work page 2022

  22. [22]

    22386–22397

    T.Xie,Y.Wang,Y.Zhou,X.Zhang,Y.Zhang,Y.Zhang,J.Yu,Diner: Disorder-invariant implicit neural representation, in: Proceedings of theIEEE/CVFConferenceonComputerVisionandPatternRecogni- tion, 2023, pp. 22386–22397

    work page 2023

  23. [23]

    A.Jacot,F.Gabriel,C.Hongler, Neuraltangentkernel:Convergence and generalization in neural networks, Advances in Neural Informa- tion Processing Systems 31 (2018)

    work page 2018

  24. [24]

    Agustsson, R

    E. Agustsson, R. Timofte, Ntire 2017 challenge on single image super-resolution: Dataset and study, in: Proceedings of the IEEE ConferenceonComputerVisionandPatternRecognitionWorkshops, 2017, pp. 126–135

    work page 2017

  25. [25]

    Zhang, P

    R. Zhang, P. Isola, A. A. Efros, K. He, J.-Y. Zhu, T. L. Berg, The unreasonable effectiveness of deep features in visual recognition, ProceedingsoftheIEEEConferenceonComputerVisionandPattern Recognition (2018) 586–597

    work page 2018

  26. [26]

    Kinga, J

    D. Kinga, J. B. Adam, et al., A method for stochastic optimization, in: International Conference on Learning Representations (ICLR), volume 5, California;, 2015

    work page 2015

  27. [27]

    M. W. Jones, J. A. Baerentzen, M. Sramek, 3d distance fields: A survey of techniques and applications, IEEE Transactions on Visualization and Computer Graphics 12 (2006) 581–599

    work page 2006

  28. [28]

    S. G. Laboratory, The stanford 3d scanning repository,https:// graphics.stanford.edu/data/3Dscanrep/, 1996. Accessed: 2025. Shi et al.:Preprint submitted to ElsevierPage 12 of 12

    work page 1996