Adaptive Local Frequency Filtering for Fourier-Encoded Implicit Neural Representations
Authors: Ligen Shi, Jun Qiu, Yuhang Zheng, Zengyu Pang, Chang Liu
Organizations: College of Computer Science (College of Software), Inner Mongolia University, Hohhot, 010021, China · Institute of Computational Imaging, Beijing Information Science and Technology University, Beijing, 102206, China · School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China
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 α(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 α(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.
Figures & tables
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.
Method
Spatially
Explicit Local
Position-
Adaptive
Frequency Control
Dependent
SIREN [ 22 ]
No
No
No
WIRE [ 18 ]
No
No
No
FINER [ 12 ]
Partial
No
Partial
MFN [ 5 ]
No
Partial
No
BACON [ 11 ]
No
Yes
No
Table 1: Comparison of representative INR methods in terms of local frequency control and spatial adaptivity.
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.
Method
Time Complexity
Space Complexity
Fourier Features
O(N(CN+W))
O(W+NCN)
SIREN
O(NW)
O(W+Ndhidden)
DINER
O(N(dfeat+W))
O(Gdfeat+W)
Ours
O(N(CN+W))
O(G+W+Nmax(CN,dhidden))
Table 2: Coarse computational complexity comparison of representative INR methods.
Figure 3: Visualization of the learned adaptive parameter α(x) and reconstruction fidelity. From left to right: ground truth, reconstructed image, learned α(x) map, and absolute difference map.
Figure 4: Empirical NTK analysis. (a) Comparison of normalized eigenspectra. (b) Relative retention ratio of normalized eigenvalues.
Figure 5: Convergence curves of INR models on 2D image fitting. Left: PSNR during the first 100 iterations. Right: PSNR over 5,000 iterations.
Metric
PE-MLP
SIREN
MFN
DINER
Ours-R
Ours-S
PSNR ↑
31.45
36.97
31.23
38.67
40.09
46.27
SSIM ↑
0.8706
0.9739
0.9398
0.9635
0.9719
0.9938
LPIPS ↓
2.54e-1
5.07e-2
6.87e-3
1.56e-2
5.14e-2
2.41e-3
Table 3: Quantitative comparison of 2D image fitting.
Figure 6: Qualitative comparison of the proposed method and baseline approaches on 2D image fitting.
Metric
Method
Armad.
Dragon
Lucy
Thai
Avg.
Chamfer ↓
PE-MLP
1.71e-6
1.24e-6
2.18e-6
2.93e-6
2.02e-6
SIREN
1.26e-6
1.40e-6
1.73e-6
2.82e-6
1.80e-6
BACON
1.08e-6
1.61e-6
2.16e-6
2.88e-6
1.93e-6
Ours
1.08e-6
1.24e-6
1.69e-6
2.90e-6
1.73e-6
IoU ↑
PE-MLP
0.968
0.992
0.985
0.964
0.977
SIREN
0.987
0.982
0.976
0.976
0.980
Table 4: Quantitative comparison of signed distance field representations.
Figure 7: Qualitative comparison of signed distance field representations for the Armadillo model.
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.
Implicit Neural Representations (INRs) have emerged as a powerful paradigm for various signal processing tasks, but their inherent spectral bias limits the ability to capture high-frequency details. Existing methods partially mitigate this issue by using Fourier-based features, which usually rely on fixed frequency bases. This forces multi-layer perceptrons (MLPs) to inefficiently compose the required frequencies, thereby constraining their representational capacity. To address this limitation, we propose Content-Aware Frequency Encoding (CAFE), which builds upon Fourier features through multiple parallel linear layers combined via a Hadamard product. CAFE can explicitly and efficiently synthesize a broader range of frequency bases, while the learned weights enable the selection of task-relevant frequencies. Furthermore, we extend this framework to CAFE+, which incorporates Chebyshev features as a complementary component to Fourier bases. This combination provides a stronger and more stable frequency representation. Extensive experiments across multiple benchmarks validate the effectiveness and efficiency of our approach, consistently achieving superior performance over existing methods. Our code is available at https://github.com/JunboKe0619/CAFE.
Junbo Ke, Yangyang Xu, You-Wei Wen +1
School of Mathematics and Statistics, Hunan Normal University · Department of Statistics and Data Science, Southern University of Science and Technology
Implicit Neural Representations (INRs) have been proven successful in encoding continuous signals through coordinate-based networks, yet facing a spectral dilemma: periodic activations capture fine details but act as all-pass filters that memorise noise, while spatially compact activations regularise effectively but suffer from low-frequency bias. Existing attempts to resolve this trade-off introduce computational overhead or tuning frailty. We propose to model each neuron's activation as the steady-state response of a sinusoidally-forced damped harmonic oscillator, whose amplitude naturally governs the network's spectral selectivity during training. By jointly optimising the oscillator parameters alongside the network weights, our method adapts to the target signal's spectral content without explicit regularisation. Initialised in the stopband, the network exhibits a coarse-to-fine learning curriculum that progressively expands its spectral gate, capturing low-frequency structures first and high-frequency details only when justified by the reconstruction objective. Comprehensive experiments show that our approach consistently achieves state-of-the-art or competitive results against established INRs, while requiring no task-specific tuning of any hyperparameters.
Alex Costanzino, Pierluigi Zama Ramirez, Giuseppe Lisanti +1
CVLab, University of Bologna · Ca’ Foscari University of Venice
Periodic Implicit Neural Representations (INRs) such as SIREN and FINER assign every neuron, the same global frequency, spending the representational budget inefficiently when local signal content varies. We introduce FiRe (Frequency Reparameterization), that accelerates optimization by reparameterizing per-neuron frequency of periodic INRs without changing their underlying activation function. FiRe gives each neuron a bounded, input-dependent frequency via a separate low-rank gating path and is applicable to any periodic activation function. The gate acts as an implicit preconditioner that improves optimization conditioning at initialization via the Neural Tangent Kernel (NTK). This better-conditioned initialization makes optimization converge faster, and the high-frequency content of the reconstruction tracks the target more closely at a fixed computational budget. On 2D image fitting, FiRe increases PSNR over a parameter-matched baseline (up to +1 dB at short training budgets), with gains that vary with resolution and diminish at full convergence. We characterize how performance depends on resolution, rank, and training budget, and give an NTK account that predicts these trends.
Harinandan Shukla, Rajarshi Verma, Jitin Singla
Department of Computer Science and Engineering, IIT Roorkee · Department of Biosciences and Bioengineering, IIT Roorkee