PEPS: AMD Paper Treats Positional Encoding as Lissajous Sampling Points for INR

Forum topic · ✨步子哥 · 2026-07-20

Summary

A 2026 ACM CCGT paper from AMD, PEPS (Positional Encoding Projected Sampling), reframes sinusoidal positional encoding geometrically: each (sin, cos) pair maps an input coordinate to a unique point on a Lissajous curve (Proposition 2.1 proves uniqueness). Instead of concatenating PE values into an MLP, PEPS uses these projected points as query locations for a shared learned grid encoder, yielding 2L+1 independent latent vectors per input. Pink-PEPS further allocates latent dimensions per frequency following the natural-image 1/f power-spectrum law. On Kodak image representation (48.07 dB PSNR vs ~45 for baselines), 4K neural texture compression, and 512³ SDF representation, PEPS beats state-of-the-art methods like NTC and InstantNGP hash grids, often with 25% fewer parameters. It notably fixes hard cases, raising Pitted Stonefish IoU from 0.39 to 0.47 and lifting hash-grid IoU from 0.547 to 0.731. Inference overhead on RDNA4 is modest (Pink-PEPS: +12.5%), making it practical for real-time graphics.

Key points

Background: Plain MLPs exhibit spectral bias — fitting a 1024×1024 image with a 3-layer MLP yields ~18 dB PSNR. Adding sinusoidal positional encoding (PE, as in NeRF / Tancik et al.) pushes this to ~45 dB, but PE is usually treated as just a dimension-lifting trick.
The paper: *PEPS: Positional Encoding Projected Sampling* by Guillaume Perez, Janarbek Matai, Takahiro Harada (AMD), Proc. ACM Comput. Graph. Interact. Tech. 9(1), Article 8, May 2026, DOI: 10.1145/3806062.
Core geometric insight: For 2D inputs, PE generates (sin xφᵢ, cos xφᵢ, sin yφᵢ, cos yφᵢ) per frequency — two circles whose combined trajectory is a Lissajous curve. Proposition 2.1 shows distinct inputs always produce distinct trajectories, giving every coordinate a unique geometric embedding.
PEPS method: Treat each PE projection point Sᵢ(x) = (1+sin(xφᵢ))/2 and Cᵢ(x) = (1+cos(xφᵢ))/2 as an independent query into a *shared* grid encoder, collecting 2L+1 latent vectors that feed the MLP. Each frequency acts as an independent "evidence window" at a different observation scale, effectively spreading a grid's resolution across a larger feature space.
Pink-PEPS: Natural images follow a 1/f^α power spectrum (α ≈ 1–2), so Pink-PEPS allocates d/2ⁿ dimensions to frequency n (minimum 1), with an offset slice start to preserve gradient flow — cutting compute while retaining quality.

Results

Kodak image representation: Grid-PEPS 47.72 dB / Grid-PinkPEPS 47.83 dB / NTC_PinkPEPS 48.07 dB PSNR, vs ~45.3 for BI Grid, LPE, and NTC baselines. With 25% fewer parameters, Grid-PinkPEPS still reaches 44.89 dB — matching SOTA.
4K neural texture compression (18 polyhaven texture sets, 14–18x compression): NTC_PinkPEPS averages 41.89 dB PSNR, beating all baselines. Inference on RDNA4 (Radeon RX 9070 XT): BI grid 4.32 ms, Grid-PEPS 5.47 ms (+26.6%), Grid-PinkPEPS 4.86 ms (+12.5%).
SDF representation (512³, Lucy / Pitted Stonefish / Thai Statue / Armadillo, 227x compression): global IoU 0.816 vs 0.799 (grid), 0.793 (LPE), 0.770 (PE). On the hard Pitted Stonefish instance, Grid-PEPS reaches 0.470 IoU vs 0.390 for grid and 0.242 for hash grid; with L1 loss it matches baselines that need 8x the resolution.
Hash grid synergy: Hash-PEPS raises global IoU from 0.547 to 0.731, likely by giving hash buckets additional independent query points that mitigate hash collisions — PEPS works as a natural extension of hash grids rather than a replacement.
Shared-grid design: A single shared grid (Appendix A) matters for gradient flow: independent per-frequency grids would starve high-frequency cells of gradient; the shared grid couples high-frequency queries to low-frequency traffic.

Open questions raised

1. Pink-PEPS assumes a 1/f^α spectrum; SDFs may not follow it (step-like edges concentrate energy at high frequencies), and the paper doesn't report SDF PSDs. 2. Most experiments use L = 3–4 frequencies; Pink-PEPS is slightly slower at L=4 (5.0 ms vs 4.86 ms), and a more principled frequency-pyramid organization remains open. 3. Why merging per-frequency grid queries outperforms single queries lacks a rigorous quantitative spectral analysis — Proposition 2.1 explains uniqueness, not representation quality.

Takeaway

PEPS shows that a tool used for years without explanation (positional encoding) hides an overlooked geometry: PE draws each coordinate onto unique Lissajous trajectories, which can be exploited as natural sampling locations for grid lookups. Combined with a 1/f compute allocator, this yields a simple, interpretable, and SOTA-beating INR encoder validated on real RDNA4 hardware.

References

1. Perez, G., Matai, J., Harada, T. (2026). PEPS: Positional Encoding Projected Sampling. Proc. ACM Comput. Graph. Interact. Tech. 9, 1, Article 8. DOI: 10.1145/3806062 2. Mildenhall et al. (2021). NeRF. Communications of the ACM 65(1) 3. Müller et al. (2022). Instant Neural Graphics Primitives. ACM TOG 41(4) 4. Rahaman et al. (2019). On the Spectral Bias of Neural Networks. ICML 5. Tancik et al. (2020). Fourier Features Let Networks Learn High Frequency Functions. NeurIPS 6. Vaidyanathan et al. (2023). Random-Access Neural Compression of Material Textures. ACM TOG 42 7. Fujieda, Yoshimura, Harada (2023). Local Positional Encoding for MLPs. Pacific Graphics 8. Vaswani et al. (2017). Attention Is All You Need. NeurIPS 9. Lissajous, J. A. (1857). Mémoire sur l'étude optique des mouvements vibratoires 10. Field, D. J. (1987). Relations between the statistics of natural images and cortical cells. JOSA A 4(12)