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Recurrent Sinusoidal INRs for Efficient High-Fidelity Representation

Hyunmin Cho, Jaejun Yoo, Kyong Hwan Jin

arXiv:2607.21485v1cs.CV

TL;DR

Coordinate-based INRs struggle to recover fine detail without adding independently parameterized depth. This paper uses weight-tied sinusoidal recurrence for harmonic spectral enrichment, achieving higher-fidelity image reconstruction with fewer parameters and optimization steps while transferring favorably to super-resolution, NeRF, and SDF tasks.

  • Problem

    Coordinate-based INRs exhibit spectral bias, and improving fine-scale reconstruction without increasing independently parameterized depth remains an open question.

  • Method

    The paper uses a shared sinusoidal block recurrently to refine latent INR representations and interprets the resulting harmonic line spectrum as expanded effective spectral support.

  • Results

    The method achieves higher-fidelity image reconstruction with fewer parameters and optimization steps than feed-forward baselines, while transferring favorably to super-resolution, NeRF, and SDF tasks.

  • Takeaways & Limitations

    Sinusoidal recurrence provides a parameter-efficient spectral mechanism for refining INR representations across image and continuous 3D representation tasks.

  • Takeaways & Limitations

    The method’s gains are not consistently observed across all model sizes and may be limited under severe capacity constraints.

Abstract

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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.

1 Introduction

The introduction frames INRs as continuous coordinate-based signal representations whose fidelity is limited by spectral bias and costly capacity increases. It motivates weight-tied sinusoidal recurrence as a parameter-efficient route to harmonic spectral enrichment and higher-fidelity reconstruction.

  • INRs represent continuous signals from spatial or spatio-temporal coordinates and interpolate values at unseen locations, supporting applications including 3D reconstruction and novel view synthesis.
  • Spectral bias makes coordinate-based MLPs fit low-frequency structure more readily than high-frequency detail, while many remedies increase model size, auxiliary parameters, or training complexity.
  • Increasing latent depth or width improves fine-scale resolution but adds independently parameterized layers and computation, motivating more effective refinement without independently parameterized depth.
  • Finite unrolling of a shared recurrent update increases effective depth while reusing parameters, providing a parameter-efficient alternative to independently parameterized depth.
  • Sinusoidal recurrence is formulated as harmonic spectral enrichment: repeated shared sinusoidal refinement induces harmonic line spectra and enriches effective spectral support without adding independently parameterized depth.
  • The formulation achieves higher-fidelity image reconstruction with fewer parameters and optimization steps than feed-forward baselines, while its decoder transfers favorably to continuous representation tasks.

2 Preliminaries: Design Strategies for High-Fidelity Implicit Signal Representation

This section frames implicit neural representations as continuous coordinate-based neural functions whose main challenge is spectral bias. It reviews two strategies for improving high-frequency representation: periodic or adaptive activations and multi-frequency coordinate encodings.

  • Implicit Neural Representations and Spectral Bias: Implicit neural representations model signals as continuous coordinate-based functions, enabling continuous sampling, interpolation, and resolution-independent evaluation.The formulation maps coordinates to signal values through a neural network.
  • Implicit Neural Representations and Spectral Bias: Coordinate MLPs exhibit spectral bias by fitting low-frequency structure before fine-scale detail, motivating architectural choices for high-fidelity representation.This low-frequency-first behavior is identified as a central challenge for implicit signal representation.
  • Enhancing Representational Capacity through Activation Functions: Activation-based methods improve high-frequency representation by using periodic sine activations or adapting sinusoidal activations across broader frequency ranges.SIREN uses periodic sine activations, while FINER targets broader frequency coverage and reconstruction stability.
  • Enriching Coordinate Features with Multi-Frequency Encodings: Encoding-based methods enrich coordinates with multi-frequency features through positional encoding, sampled sinusoidal projections, or learned grid and hash-based representations.NeRF uses positional encoding, RFF uses sampled sinusoidal projections, and multiresolution encodings augment coordinate features.

3 Latent Refinement in INRs

This section examines how latent transformations in INRs are parameterized and evaluated across depth. It contrasts independently parameterized feed-forward layers and equilibrium fixed-point computation with an explicitly unrolled refinement trajectory that exposes intermediate latent states.

  • The section focuses on latent transformation F as a design axis for parameterizing and evaluating latent computation across depth.
  • Feed-forward refinement applies independently parameterized affine maps and nonlinearities, with each layer using its own parameters (Wℓ, bℓ).Increasing depth L can enrich the transformation but generally requires additional independently parameterized layers.
  • Frequency-aware feed-forward designs control frequency processing across depth through band-limited sinusoidal filtering or frequency-aware transformations.BACON uses band-limited sinusoidal filtering, whereas FourierNet and GaborNet construct frequency-aware feed-forward transformations.
  • Equilibrium-style INRs define the latent representation implicitly as a fixed point, enabling constant-memory backpropagation through implicit differentiation and amortized fixed-point solving.
  • The proposed perspective instead considers a finite, explicitly unrolled refinement trajectory whose intermediate states reveal how repeated shared-map transformations alter the latent representation.Fixed-point formulations characterize computation only at equilibrium, making successive-map evolution less directly visible.

4 Harmonic Enrichment by Sinusoidal Recurrence

Sinusoidal layers generate harmonic line spectra by creating integer combinations of encoder frequencies, so deeper or weight-tied recurrent application progressively enriches spectral support under a fixed parameter budget. Empirically, recurrent refinement substantially improves reconstruction quality, while sinusoidal recurrence expands spectral support unlike equilibrium-style and non-sinusoidal alternatives.

  • Frequency-controlled reconstruction: 23.51 →36.45 dB: increasing ωin improves high-pass reconstruction at 200 iterations while also improving full-image PSNR.The trend persists across all training stages in the frequency-separated reconstruction results.
  • Harmonic line-spectrum mechanism: Sinusoidal hidden layers create new spectral lines at integer combinations of encoder frequencies, including sums, differences, and higher-order harmonics.The resulting frequencies have the form Ω' = Σ_i k_iΩ_i for k ∈ Z^m.
  • Harmonic line-spectrum mechanism: Increasing depth enables progressively higher-order harmonic interactions from the same coordinate lifting.Repeated application yields integer-combination closure over the prominent frequency vectors.
  • Recurrent refinement: 63.378 dB: at optimization step 500, R = 5 improves PSNR from 39.724 dB for a feed-forward pass, with monotonic gains at intermediate steps under the same parameter count.Weight-tied unrolling realizes iterative harmonic refinement under a fixed parameter budget.
  • Spectral-support dynamics: Repeated sinusoidal transformations rapidly expand measured spectral support and upper-band content across refinement steps at random initialization.After training, the trajectory reflects task-dependent reweighting; iSIREN remains stationary, while non-sinusoidal recurrent controls lose support.

5 Method

The method uses a shared sinusoidal recurrent block to iteratively refine latent representations, expanding harmonic interactions without increasing parameter count. It supervises quantized outputs as bipolar codes using cosine-similarity alignment, with Gray coding preserving local adjacency.

  • Recurrent architecture: A shared sinusoidal block is repeatedly applied to refine latent features, enriching harmonic interactions while preserving the parameter count.The design replaces independently parameterized depth with recurrent unrolling under a fixed parameter budget.
  • Recurrent architecture: Given coordinate x, the model initializes a latent state, applies the shared recurrent transformation for R steps, and maps the final state through an output projection.The recurrent map is bias-free, weight-tied across steps, and uses elementwise sinusoidal activation.
  • Binarized supervision: Quantized targets are encoded as B-bit bipolar codes in {-1,+1}^B rather than directly regressing intensity or amplitude values.Bipolar coding is zero-centered and matches the symmetric range of sinusoidal activations; all codewords have identical norm.
  • Binarized supervision: The network predicts real-valued code vectors and optimizes cosine similarity between predictions and bipolar targets over sampled coordinates.Because target norms are constant, cosine alignment captures the target-dependent term while reducing sensitivity to prediction scale.
  • Binarized supervision: Gray coding preserves local adjacency in quantized signal space because neighboring quantization levels differ by only one bit.The implementation uses a Gray encoder instead of standard binary coding.

6 Experiments

Experiments evaluate recurrent sinusoidal decoding for quantized image fitting and transfer to super-resolution and 3D representation tasks. The method improves fitting efficiency and fidelity under matched budgets, with favorable results in super-resolution and NeRF settings.

  • Experimental setup: The evaluation covers quantized 2D image fitting, super-resolution, and 3D NeRF pipelines, using four image datasets resized to 256 × 256.The datasets are Set5, Kodak24, DIV2K, and FFHQ; experiments generally use five recurrent steps and matched parameter counts.
  • Image fitting: 609K parameters and 100 iterations outperform 791K-parameter baselines trained for 1,000 iterations on reconstruction fidelity.The model can also reach exact quantized reconstruction with an adaptive number of iterations.
  • Super-resolution: Ours produces sharper per-image ×2 super-resolution results than SIREN and bilinear interpolation, while achieving the lowest Kodak24 LPIPS with fewer average optimization iterations.In pretrained arbitrary-scale LIIF super-resolution, only the coordinate MLP is replaced under a matched approximately 50K-parameter decoder budget.
  • Neural radiance field: Ours consistently improves PSNR, SSIM, and LPIPS over NeRF on the forward-facing LLFF dataset under a comparable parameter budget.The recurrent sinusoidal decoder replaces NeRF’s standard RGB decoder while leaving the rest of the rendering pipeline unchanged.
  • Coding ablation: Gray coding improves reconstruction at the training resolution and generalization to unseen coordinates over natural binary coding under the same architecture and parameter budget.This ablation concerns 2D image ×2 super-resolution.

7 Discussion and Limitations

The discussion shows that Gray coding reduces code-space sensitivity, while bit-plane reweighting, recurrent refinement, and binarized supervision improve fidelity and reconstruction efficiency. It also notes a computational tradeoff: recurrent unrolling increases per-step computation despite faster wall-clock convergence.

  • Gray-code analysis: Gray coding flips exactly one bit between adjacent levels, reducing worst-case local sensitivity by a factor of n.Under natural binary coding, adjacent levels can differ in as many as n bits.
  • Gray-code analysis: Gray coding improves unseen-coordinate generalization for Kodak fitting and super-resolution, and improves PSNR and SSIM while reducing LPIPS on LLFF NeRF.The largest NeRF gains are observed for Ours.
  • Bit-plane reweighting: Bit-plane reweighting improves reconstruction quality under the same parameter budget by prioritizing high-significance bits and shifting residual errors to less significant planes.Fig. 7 uses 375K trainable parameters and 1,000 optimization steps.
  • Wall-clock efficiency: 42.84 dB is reached in 6.52 seconds, 59.12 dB in 22.45 seconds, and exact reconstruction after 636 iterations in 34.17 seconds.These wall-clock results are reported for Kodak24, while recurrent unrolling increases computation at each optimization step.
  • Ablation: Recurrence provides the primary fidelity improvement over the feed-forward baseline, while binarized supervision removes the remaining mismatch and enables exact quantized reconstruction.Table 12 separates the contributions of recurrence and binarized supervision.

8 Conclusion, Limitations, and Implications

The paper presents finite sinusoidal recurrence as a parameter-efficient mechanism for enriching INR spectral support, while noting that its benefits depend on model size. It identifies mitigating this dependence as future work.

  • Conclusion: Finite sinusoidal recurrence enriches the spectral support of intermediate INR features through a harmonic line-spectrum interpretation.The perspective links sinusoidal features to Fourier structure and explains spectral expansion without increasing parameter count.
  • Limitations: The method remains beneficial at moderate-to-large parameter budgets but does not consistently improve performance across all model sizes.Its effectiveness may be limited under severe capacity constraints.
  • Implications: Recurrence is proposed as a simple, modular alternative to increasing independently parameterized depth.Understanding and mitigating recurrence’s model-size dependence are identified as directions for future work.
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