Recurrent Sinusoidal INRs for Efficient High-Fidelity Representation

AI summary

Overview

Research area: Implicit neural representations (INRs) / coordinate-based neural networks for 2D image and 3D signal representation, spanning image fitting, super-resolution, neural radiance fields (NeRF), and signed distance functions (SDF).

Technical level: Advanced. The paper combines a Fourier-analysis argument (Jacobi–Anger expansion, integer-frequency closure) with architecture design and empirical spectral diagnostics.

Scope: The paper proposes a weight-tied recurrent sinusoidal decoder for INRs, argues via harmonic line-spectrum analysis why repeated sinusoidal blocks enrich effective spectral support, and validates the design on image fitting, super-resolution, NeRF, and SDF tasks.

What This Paper Is About

Coordinate-based MLPs suffer from spectral bias: they fit low-frequency structure before fine detail. Prior fixes typically add parameters, encodings, or training complexity. This paper asks whether the latent transformation of an INR can be made more effective at resolving fine-scale structure without adding independently parameterized depth, and answers by repeatedly applying one shared sinusoidal block.

Key Contributions

1. Weight-tied sinusoidal refinement for INRs. The authors formulate a decoder that increases effective depth through finite recurrent unrolling of a single shared sinusoidal block, adding no independently parameterized layers. The model is written compactly as a composition of R applications of (σ ∘ W^(rec)) between an input projection W^(in) and an output projection W^(out), all bias-free, with σ the elementwise sine.

2. A harmonic line-spectrum analysis of sinusoidal transformations. They show that the sinusoidal encoder acts as a learnable Fourier feature map and that a hidden sine layer creates new spectral lines at integer combinations of the encoder frequencies (sums, differences, higher-order harmonics), rather than merely reweighting existing ones. Using Euler's identity and the Jacobi–Anger identity, they derive coefficients c_{j,k} = ∏ J_{k_i}(α_{j,i}) and the idealized closure relation Ω^(ℓ+1) ⊆ span_Z(Ω^(ℓ)).

3. A spectral account of recurrent versus feed-forward and equilibrium designs. Intermediate features are analyzed with per-channel 2-D DFTs, summarized by spectral support S_ℓ^τ and a DC-normalized upper-band measure HF_ℓ, comparing sinusoidal recurrent models against feed-forward INRs, non-sinusoidal recurrent variants, and an equilibrium-style sinusoidal baseline.

4. A bias-free recurrent layer and binarized supervision. They argue recurrent bias terms reapply the same phase shift at every unrolled step and empirically degrade fitting, so the recurrent block is bias-free. They also add a bipolar (Gray-coded) cosine-alignment objective that permits exact, bit-error-zero reconstruction of quantized targets in an adaptive number of iterations.

Main Findings

• Input bandwidth controls high-frequency reconstruction. Table 1 reports frequency-separated PSNR as Full/LF/HF. At 200 iterations, increasing the encoder frequency scale ω_in over {32, 64, 128, 256} raises HF PSNR from 23.51 to 36.45 dB (the paper highlights this 23.51 → 36.45 dB shift).

• Recurrent unrolling raises fidelity at a fixed parameter budget. Table 2 (optimization step 500, 593.7K parameters throughout) reports PSNR of 39.724 dB for a single feed-forward pass, then 46.362 (+6.6), 54.772 (+15.0), 60.843 (+21.1), and 63.378 (+23.7) dB for R = 2 to 5, with monotonic gains.

• Spectral support expands with repeated sinusoidal refinement at initialization. Table 3(a), untrained: SIREN's S_ℓ^τ grows 30.9, 26.5, 34.3, 43.7, 52.0, 61.3, 72.3, 85.0, 94.6 across ℓ = 0–8, with HF_ℓ rising from 0.022 to 0.101. FINER goes 51.4, 53.6, 72.4, 89.3, then 100 for the remaining depths, with HF_ℓ 0.048 → 0.232.

• After training, sinusoidal recurrence keeps usable support; the equilibrium baseline does not move. Table 3(b), trained on Set5: SIREN reaches PSNR 78.35 with S and HF values that vary with depth (e.g., 31.0 → 61.7, HF 0.020 → 0.011), and FINER reaches 85.62 with S reaching 100 (HF 0.044 → 0.016). The equilibrium-style iSIREN is stationary at 33.5 for every depth with HF 0.008 → 0.008 and PSNR 47.16. Feed-forward spectral baselines at L = 3 peak early and then drop: GaborNet 40.6, 58.2, 73.0, 35.2 (PSNR 44.48), FourierNet 41.4, 58.3, 66.6, 36.0 (PSNR 38.38), BACON 0.9, 6.3, 14.0, 15.1 (PSNR 27.88).

• Non-sinusoidal recurrence collapses the spectrum. Table 3(c): Gauss (σ = 30) support goes 1.0, 1.9, 14.0, then 1.2, 0.0, 0.0, 0.0, 0.0, 0.0 with HF_ℓ 2e-4 → 7e-3 and PSNR 12.02; PEMLP (N_f = 10) goes 0.8, 0.9, 0.8, 0.5, 0.1, 0.1, 0.0, 0.0, 0.0 with HF_ℓ 6e-4 → 2e-5 and PSNR 24.09.

• Recurrence helps sinusoidal models but not all activations, and the benefit depends on capacity. Table 4 (capacity-limited PSNR): SIREN without recurrence 33.60, 39.57, 46.23, 48.64 at roughly 100K/200K/400K/600K versus with recurrence 37.26, 46.26, 61.36, 78.35; FINER 38.01, 40.59, 45.56, 48.36 versus 34.21, 44.45, 68.06, 85.62. Gauss degrades sharply (20.48, 26.49, 35.33, 60.31 versus 11.74, 11.77, 11.92, 12.02), and PEMLP shows no persistent gain (28.89, 31.04, 31.80, 34.20 versus 27.48, 29.64, 33.82, 31.04). The paper states recurrence consistently improves SIREN and benefits FINER beyond roughly 200K parameters.

• Recurrent bias should be removed. Table 5 (mean peak PSNR in dB over 1K iterations on Kodak24, ω_0 = 256/45, 792K parameters): bias ON gives 51.16, 60.35, 51.97, 63.57, 78.10 for R = 1 to 5; bias OFF gives 51.41, 83.49, 87.54, 89.59, 90.28, with differences of +0.25, +23.14, +35.57, +26.02, +12.18 dB. The two settings behave similarly at R = 1 but diverge for R ≥ 2.

• Image fitting with fewer parameters and iterations. Table 6: baselines use 791K parameters and 1000 iterations; the proposed model uses 609K parameters and 100 iterations. On Set5 it reports 58.16 PSNR / .9997 SSIM with 0.78K bit errors; Kodak24 53.20 / .9989 with 2.51K; DIV2K-100 46.28 / .9973 with 5.66K; FFHQ-600 61.39 / .9998 with 0.62K. For reference, at 1000 iterations the learnable-frequency baselines report iSIREN 51.80 / .9976 (Set5), GaborNet 50.31 / .9970, FourierNet 45.67 / .9924, and on DIV2K-100 iSIREN 51.43 / .9983 and GaborNet 49.51 / .9976. The paper's claim is that the 100-iteration result exceeds the 1000-iteration results of all learnable-frequency spectral baselines across the datasets tested.

• Exact quantized reconstruction is attainable. With adaptive iteration counts, the method reaches zero bit error at 322 ± 117 iterations (Set5), 447 ± 327 (Kodak24), 537 ± 432 (DIV2K-100), and 440 ± 350 (FFHQ-600). Figure 4 shows an example converging to zero bit error at a 125-iteration checkpoint, where the comparison boxes read 17.85 (Gauss), 23.52 (FINER), 32.88 (FourierNet), 36.60 (iSIREN), and zero bit error for the proposed method.

• Super-resolution. For per-image ×2 fitting (Table 7a), the proposed model (609K parameters, 595 ± 285 iterations) reaches PSNR 26.55 and LPIPS 0.2272, versus SIREN (611.6

Authors’ abstract

We study sinusoidal recurrence as an iterative mechanism for harmonic spectral enrichment in implicit neural representations (INRs). Our analysis reveals that sinusoidal activations induce a harmonic line spectrum, providing a spectral account of how recurrent unrolling enriches the effective spectral support. We realize this principle with a shared sinusoidal block that iteratively refines the latent representation. We empirically validate the resulting spectral behavior against feed-forward INRs, non-sinusoidal recurrent variants, and equilibrium-style sinusoidal models. Complementing this analysis, we evaluate the proposed architecture across image and 3D representation tasks. On RGB image benchmarks, our method achieves higher fidelity than feed-forward baselines with fewer parameters and fewer optimization steps, and it further transfers favorably to super-resolution, NeRF, and SDF tasks.