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Improved Implicit Neural Representation with Fourier Reparameterized Training
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Proposes Fourier reparameterization (FR), which trains a coefficient matrix over fixed Fourier bases to form MLP weights, to alleviate spectral bias and improve implicit neural representation accuracy without changing architecture or inference.
Problem
Vanilla MLPs suffer from spectral (low-frequency) bias, which limits the accuracy of implicit neural representations (INR) for images, shapes, and radiance fields.
Approach
Reparameterize each MLP weight matrix as a trainable coefficient matrix times a fixed Fourier basis matrix (cosine bases with varied frequencies and phases), theoretically argued to rebalance gradient magnitudes across frequencies and NTK eigenvalues; the bases are merged into ordinary weights after training so inference is unchanged.
Key results
- On Kodak 2D image approximation, average PSNR improves from 21.33 to 22.44 for MLP+ReLU, 27.18 to 30.59 for MLP+ReLU+PE, and 35.31 to 36.34 for MLP+Sin when adding FR.
- Adding FR to MLP+ReLU in the 1D toy task reduces the gap between low- and high-frequency error and balances NTK eigenvalues; training time per epoch rises only from 2.89 to 3.31 ms (about 17%).
- Random basis (RR) and randomly initialized reparameterization (RIR) underperform Fourier bases in the ablation on 2D fitting (e.g., MLP+ReLU average 23.83 and 24.46 vs 24.71 for FR on three images).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add Fourier reparameterization | MLP + ReLU · Kodak 24 (first 8 images), average | PSNR: 21.33 -> 22.44 | Helped |
| Add Fourier reparameterization | MLP + ReLU + PE · Kodak 24 (first 8 images), average | PSNR: 27.18 -> 30.59 | Helped |
| Add Fourier reparameterization | MLP + Sin · Kodak 24 (first 8 images), average | PSNR: 35.31 -> 36.34 | Helped |
| Use random bases instead of fixed Fourier bases (ablation) | MLP + ReLU reparameterization · Kodak 24 (first 3 images), average | PSNR: 24.71 -> 23.83 | Hurt |
| Train randomly initialized bases instead of fixed Fourier bases (ablation) | MLP + ReLU reparameterization · Kodak 24 (first 3 images), average | PSNR: 24.71 -> 24.46 | Hurt |
| Use random bases instead of fixed Fourier bases (ablation) | MLP + ReLU + PE reparameterization · Kodak 24 (first 3 images), average | PSNR: 32.00 -> 30.86 | Hurt |
| Train randomly initialized bases instead of fixed Fourier bases (ablation) | MLP + ReLU + PE reparameterization · Kodak 24 (first 3 images), average | PSNR: 32.00 -> 30.60 | Hurt |
| Use random bases instead of fixed Fourier bases (ablation) | MLP + Sin reparameterization · Kodak 24 (first 3 images), average | PSNR: 37.24 -> 34.20 | Hurt |
| Train randomly initialized bases instead of fixed Fourier bases (ablation) | MLP + Sin reparameterization · Kodak 24 (first 3 images), average | PSNR: 37.24 -> 26.97 | Hurt |
| Set sampling interval to 0.1×Tmax (ablation) | Fourier reparameterization · Kodak 24 (first 3 images), average | PSNR: 24.71 -> 23.34 | Hurt |
| Set sampling interval to 0.25×Tmax (ablation) | Fourier reparameterization · Kodak 24 (first 3 images), average | PSNR: 24.71 -> 24.07 | Hurt |
| Set sampling interval to 0.5×Tmax (ablation) | Fourier reparameterization · Kodak 24 (first 3 images), average | PSNR: 24.71 -> 24.38 | Hurt |
Surprising findings
- Random fixed bases (RR) sometimes outperform Fourier FR on specific images (e.g., MLP+Sin+RR on Kodim 01), and RIR is far worse than FR for MLP+Sin (26.97 vs 37.24 average).
Limitations
- Theoretical result (Theorem 2) is an existence argument and relies on extending a one-hidden-layer analysis to deeper MLPs by assumption.
- Optimal basis choice depends on training data and is not determined; fixed Fourier bases are used empirically.
- Sampling interval and frequency/phase hyperparameters (F, P) must be tuned per setting.
Experimental setup
Datasets: Kodak 24 Blender dataset synthetic 1D rounded sine function 3D shape voxel grid (following occupancy/SDF setting of [34])
Benchmarks: Kodak 24 Blender dataset
Metrics: PSNR IOU NTK eigenvalue distribution frequency-specific relative error training time per epoch
Baselines: MLP+ReLU MLP+ReLU+PE MLP+Sin (SIREN) random reparameterization (RR) randomly initialized reparameterization (RIR) InstantNGP DVGO NeRF
Builds on: positional encoding SIREN Xu frequency-principle analysis (low-frequency bias) NTK spectral bias analysis (Tancik et al.) RepVGG-style structural reparameterization
Closest papers by meaning
- Inductive Gradient Adjustment for Spectral Bias in Implicit Neural Representations (ICML 2025 · similarity 0.88)
- FINER: Flexible Spectral-bias Tuning in Implicit NEural Representation by Variable-periodic Activation Functions (CVPR 2024 · similarity 0.86)
- VI^3NR: Variance Informed Initialization for Implicit Neural Representations (CVPR 2025 · similarity 0.86)
- Implicit Neural Representations and the Algebra of Complex Wavelets (ICLR 2024 · similarity 0.85)
- Learning Spatially Collaged Fourier Bases for Implicit Neural Representation (AAAI 2024 · similarity 0.85)
- Implicit Representations via Operator Learning (ICML 2024 · similarity 0.84)
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