Content-Aware Frequency Encoding for Implicit Neural Representations with Fourier-Chebyshev Features

Junbo Ke; Yangyang Xu; You-Wei Wen; Chao Wang

cs.CVMar 1, 2026

Content-Aware Frequency Encoding for Implicit Neural Representations with Fourier-Chebyshev Features

Authors: Junbo Ke, Yangyang Xu, You-Wei Wen, Chao Wang

Organizations: School of Mathematics and Statistics, Hunan Normal University · Department of Statistics and Data Science, Southern University of Science and Technology

Abstract

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.

Figures & tables

Panel of Figure 1 — Figure 1: Effect of the total number of linear layers on the image fitting task in terms of PSNR. For the RFF baselines, linear layers exist only in the MLP. For our methods (CAFE and CAFE+), the total number of linear layers includes those in both the encoding stage and the MLP. Any increase in this total arises from additional layers in the encoding stage, while the MLP depth remains fixed. The legend indicates the number of hidden neurons per layer.

Panel of Figure 2 — Figure 2: Qualitative comparison of RFF, CAFE, and CAFE+ under the same number of parameters for the image fitting task. High- and low-frequency regions are zoomed in, and corresponding error maps are provided for a detailed evaluation.

Panel of Figure 3 — Figure 3: NTK matrix for the CAFE and RFF. The network width and the number of sampled points were both set to 1024.

Figure 4: Overall framework of CAFE+. CAFE+ maps the input coordinates separately to Chebyshev and Fourier features, which are then concatenated to form the inputs of multiple parallel linear layers. The outputs of these layers are combined via the Hadamard product to obtain the CAFE+ encoded features, which are subsequently fed into the backbone MLP to produce the corresponding values.

Panel of Figure 5 — Figure 5: Illustration of the complementary roles of Chebyshev and Fourier features. The network is first trained with the CAFE+, and during inference, we deactivate the neurons corresponding to either $\Phi_{\text{CF}}(\mathbf{x})$ or $\Phi_{\text{FF}}(\mathbf{x})$ by setting them to zero to isolate their individual effects. CF-only corresponds to using $[\,\mathbf{0},\,\Phi_{\text{CF}}(\mathbf{x})\,]$ while FF-only corresponds to $[\,\Phi_{\text{FF}}(\mathbf{x}),\,\mathbf{0}\,]$ .

Panel of Figure 6 — Figure 6: Visualization results of the 2D image fitting task on the D2K4 dataset.

PSNR (dB)								Param	Time(s)
Method	D2K0	D2K1	D2K2	D2K3	D2K5	D2K6	D2K7
SIREN	33.48	40.08	36.68	40.39	34.41	38.78	37.58	0.20M	148.16
WIRE	27.99	34.44	31.12	36.06	28.80	32.50	33.50	0.07M	373.60
SCONE	35.32	41.88	38.56	42.27	36.87	41.78	38.92	0.19M	380.49
FINER	35.93	42.33	39.44	41.99	36.78	41.08	39.73	0.20M	203.51
SL 2 A	36.22	41.91	39.87	41.87	37.03	41.11	40.23	0.46M	270.50

Table 1: Quantitative comparison of different methods on the 2D image fitting task in terms of PSNR (dB), parameter count, and training time.

IoU (%)						Time (s)
Method	Thai statue	Lucy	Armadillo	Dragon	Buddha
SIREN	0.9797	0.9866	0.9933	0.9940	0.9934	826
WIRE	0.9887	0.9930	0.9861	0.9934	0.9945	1,824
SCONE	0.9875	0.9905	0.9930	0.9918	0.9924	1,662
FINER	0.9893	0.9929	0.9950	0.9959	0.9958	1,056
SL 2 A	0.9987	0.9992	0.9993	0.9991	0.9994	1,282

Table 2: Quantitative comparison of different methods on the 3D shape representation task in terms of IoU (%) and training time.

Panel of Figure 7 — Figure 7: Visualization of the 3D shape representation task.

PSNR (dB)					Time (s)
Method	Ship	Drums	Lego	Hotdog
Gauss	21.24	22.46	26.40	31.28	685.2
SIREN	21.79	25.55	29.94	33.91	403.5
WIRE	21.33	24.57	29.22	32.78	973.8
FINER	22.02	25.66	30.78	33.80	437.0
Ours	23.12	25.59	31.86	34.65	363.2

Table 3: Quantitative comparison of different methods on the NeRF task in terms of PSNR (dB) and training time.

CAFE	Chebyshev	D2K0	D2K1	D2K2	D2K3	D2K4
✗	✗	$26.18$	$30.19$	$30.98$	$29.46$	$30.92$
✗	✓	$33.00$	$37.23$	$35.60$	$36.97$	$35.91$
✓	✗	$34.87$	$37.36$	$40.42$	$40.15$	$40.52$
✓	✓	39.47	44.34	44.18	44.69	44.97

Table 4: Ablation study on the effect of CAFE and Chebyshev features components. PSNR (dB) is reported on five 2D test images.

Panel of Figure 8 — Figure 8: Effect of the number of parallel linear layers on PSNR and training time for the 2D image fitting task.

Panel of Figure 9 — Figure 9: Visualization of the neural radiance fields task.

Panel of Figure 10 — Figure 10: PSNR variations across different Fourier features scales (top) and polynomial orders in the Chebyshev features (bottom).

MLP Layers	D2K5	D2K6	D2K7	Avg.
0	31.23	36.78	38.66	35.56
1	37.78	42.31	42.99	41.03
2	39.15	42.63	45.01	42.26
3	39.36	42.49	44.67	42.17
4	39.60	42.56	45.40	42.52

Table 5: Ablation study on the effect of backbone MLP depth. PSNR (dB) is reported for three test images.

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Method	0873	0873 ( $\times 2$ )	0878	0878 ( $\times 2$ )	0891	0891 ( $\times 2$ )	Params (M)	Avg. Time (s)
SIREN	24.91	31.56	32.42	36.94	27.20	32.48	0.45	292.50
WIRE	23.45	28.98	29.12	32.04	23.99	27.23	0.18	707.48
SCONE	26.10	33.14	34.09	39.12	27.68	33.19	0.45	647.06
FINER	25.28	31.59	32.66	36.86	26.77	32.07	0.45	375.28
SL 2 A	25.95	34.28	34.00	38.59	26.82	32.53	0.84	456.12
Ours	29.65	37.18	35.79	40.82	30.79	36.46	0.74	326.95

Appendix

Table 6: PSNR comparison on DIV2K images 0873, 0878, and 0891 under full-resolution and

\times 2

downsampling settings. The number of parameters and average inference time per image are also reported.

Panel of Figure 11 — Figure 11: Visualization results on the full-resolution DIV2K image 0873, where we zoom in on two selected regions.

Panel of Figure 12 — Figure 12: Visualization of image denoising, inpainting, and single-image super-resolution tasks (top to bottom). For denoising, Gaussian noise with standard deviation 0.2 is added. Inpainting randomly removes 50% of pixels. Super-resolution uses a scale factor of 4.

Panel of Figure 13 — Figure 13: Visualization of the gigapixel Pluto image fitting results. Subfigure (a) shows the original $8000\times 8000$ image, (b) presents the $512\times 512$ patches fitted by our method and PE, along with the corresponding residual maps (c) displays the training loss curve.

Method	Lego	Ship	Drums	Hotdog
SIREN	26.94	20.62	23.09	30.89
FINER	28.00	20.88	23.65	31.64
Ours	30.26	21.92	23.51	32.35

Appendix

Table 7: Quantitative comparison on NeRF scenes using all training images at the original resolution. PSNR is reported in dB.

Method	D2K0	D2K1	D2K2	D2K3	D2K4	D2K5	D2K6	D2K7	Params	Time (s)
MFN	35.6	40.1	36.4	41.9	38.4	35.2	39.9	38.9	0.33M	279.1
BACON	34.0	39.0	37.6	39.6	39.5	35.7	38.2	36.0	0.33M	245.6
Ours	39.5	44.3	44.2	44.7	45.0	39.2	42.6	45.0	0.33M	149.8

Appendix

Table 8: Comparison with BACON and MFN on Image Fitting

Figure 14: Comparison of RFF and CF in function fitting. CF excels at capturing smooth, low-frequency components but struggles with high-frequency variations, whereas RFF effectively represents high-frequency content yet introduces oscillatory errors on low-frequency functions.

Panel of Figure 15 — Figure 15: Visualization of the constructed noisy images under different noise ratios.

Panel of Figure 16 — Figure 16: PSNR comparison of CAFE and CAFE+ across different noise ratios.

Figure 17: NTK eigenvalues visualization.

Figure 18: Training loss curves of our method and SIREN. Our method converges faster and reaches a lower loss value.

Panel of Figure 19 — Figure 19: Visualization of the 2D image fitting task. PSNR values are shown at the upper-left corner of each image.

Panel of Figure 20 — Figure 20: Visualization of the 3D shape representation task.

Panel of Figure 21 — Figure 21: Visualization of the neural radiance fields task.

Panel of Figure 22 — Figure 22: Visualization of the full-resolution representation results on the DIV2K images 0878 and 0891.

Implicit Neural Representations

Implicit neural representations encode signals and fields as continuous functions that can be queried at arbitrary coordinates and resolutions.

See all 186 papers

Adaptive Local Frequency Filtering for Fourier-Encoded Implicit Neural Representations
Apr 3, 2026Ligen Shi, Jun Qiu, Yuhang Zheng +2Implicit Neural Representations Fourier Feature Embeddings
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
Spatial-Frequency-Aware Implicit Neural Representation of Multidimensional Signals via MLP-KAN Fusion
Oct 8, 2026Wen Yan, Ligen Shi, Jun Qiu +3Implicit Neural Representations Spectral Bias
Institute of Computational Imaging, Beijing Information Science and Technology University, Beijing 102206, China · College of Computer Science (College of Software), Inner Mongolia University, Hohhot 010021, China · Research Center for Spatiotemporal Intelligence, Inner Mongolia University, Hohhot 010021, China
Spectral Gating via Damped Oscillations for Adaptive Implicit Neural Representations
Jun 22, 2026Alex Costanzino, Pierluigi Zama Ramirez, Giuseppe Lisanti +1Implicit Neural Representations Frequency-Domain Feature Learning
CVLab, University of Bologna · Ca’ Foscari University of Venice

Apr 3, 2026J/K move · Enter open · S save

Adaptive Local Frequency Filtering for Fourier-Encoded Implicit Neural Representations

Ligen Shi, Jun Qiu, Yuhang Zheng, Zengyu Pang+1

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

Implicit Neural Representations Fourier Feature Embeddings Frequency-Domain Feature Learning

Open paper AskPDF arXiv

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(\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

α(\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.

Abstract

Implicit Neural Representations

Explore similar work

Adaptive Local Frequency Filtering for Fourier-Encoded Implicit Neural Representations

Spatial-Frequency-Aware Implicit Neural Representation of Multidimensional Signals via MLP-KAN Fusion

Spectral Gating via Damped Oscillations for Adaptive Implicit Neural Representations