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Adaptive Wavelet-Positional Encoding for High-Frequency Information Learning in Implicit Neural Representation
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Proposes Wavelet-Positional Encoding (WPE) with High-Frequency Perception (HFP) to place local Morlet wavelet embeddings at high-frequency locations, combined with global PE, to improve implicit neural representations for 1D, 2D, and 3D tasks.
Problem
Coordinate-based implicit neural representations suffer from spectral bias and fail to capture high-frequency fine details, because global encodings process the whole input space indiscriminately.
Approach
Decompose the target signal with a Laplacian-pyramid/DWT high-frequency decomposition, find grid centers of high-frequency content (HFP), and embed input coordinates with trainable local Morlet wavelet functions at those centers, concatenated with global Fourier positional encoding before the MLP.
Key results
- On LLFF, WPE+HFP reaches mean PSNR 27.89 vs 27.08 for PE baseline (SSIM 0.85, LPIPS 0.097).
- On NeRF-Synthetic, the method reports PSNR 28.81, SSIM 0.94, LPIPS 0.067, versus WIRE at 28.49 and PE at 28.07.
- In 2D image regression, the method reaches 28.29 dB versus 25.59 dB for PE and 28.18 dB for WIRE.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add Wavelet-Positional Encoding with HFP alongside global PE (ablation) | NeRF · LLFF | PSNR: +0.81 over PE (27.08 to 27.89) | Helped |
| Add Wavelet-Positional Encoding with HFP alongside global PE | NeRF · NeRF-Synthetic | PSNR: +0.74 over PE (28.07 to 28.81) | Helped |
| Add uniformly distributed wavelet encoding without HFP (ablation) | NeRF · LLFF | PSNR: +0.23 PSNR, SSIM +0.01, LPIPS -0.008 over PE | Mixed |
| Remove global PE and use only HFP-placed wavelet encoding (ablation) | NeRF · LLFF | PSNR: 27.14 vs 27.89 for full WPE | Hurt |
| Use wavelet encoding alone for 1D signal fitting (no HFP) | 1D signal fitting · Girl With a Pearl Earring row signal | qualitative fit | Helped |
| Replace global PE/FFN/WIRE with WPE in 2D image regression | 2D image regression · Girl With a Pearl Earring | PSNR: WPE 28.29dB vs WIRE 28.18dB, FFN 26.53dB, PE 25.59dB | Helped |
| Add WPE to NeRF on scenes with little high-frequency content (Horns) | NeRF · LLFF Horns | PSNR: 25.49 vs PE 25.35 | No clear effect |
Surprising findings
- Uniformly distributed wavelet encoding without HFP gives little improvement over PE in 3D (27.31 vs 27.08 PSNR in ablation only for wavelet-only variant with PE, while wavelet-only without PE is worse at 27.14).
Limitations
- Gains are small on simple scenes like Horns where high-frequency content is lacking.
- Early-training convergence is slower than WIRE.
- Relies on the number and placement of wavelet centers chosen by HFP heuristics.
Experimental setup
Datasets: Girl With a Pearl Earring (restored image) NeRF-Synthetic LLFF
Benchmarks: NeRF-Synthetic LLFF
Metrics: PSNR SSIM LPIPS
Baselines: PE FFN WIRE
Builds on: NeRF Fourier feature networks (FFN) WIRE WaveNeRF Mip-NeRF
Closest papers by meaning
- Implicit Neural Representations and the Algebra of Complex Wavelets (ICLR 2024 · similarity 0.86)
- A sampling theory perspective on activations for implicit neural representations (ICML 2024 · similarity 0.84)
- FreSh: Frequency Shifting for Accelerated Neural Representation Learning (ICLR 2025 · similarity 0.84)
- Learning High-Frequency Functions Made Easy with Sinusoidal Positional Encoding (ICML 2024 · similarity 0.84)
- Representing Sounds as Neural Amplitude Fields: A Benchmark of Coordinate-MLPs and a Fourier Kolmogorov-Arnold Framework (AAAI 2025 · similarity 0.84)
- Wavelet-based Positional Representation for Long Context (ICLR 2025 · similarity 0.83)
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