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Spectral Gating via Damped Oscillations for Adaptive Implicit Neural Representations

Conference: ECCV 2026
arXiv: 2606.23129
Code: https://alex-costanzino.github.io/fdho/
Area: 3D Vision / Neural Representations
Keywords: Implicit Neural Representations, Spectral Gating, Damped Oscillator, Activation Function Design, Coarse-to-Fine Learning

TL;DR

Starting from the steady-state response of a forced damped harmonic oscillator, this paper derives a physically driven activation function, FDHO, which couples the activation amplitude and frequency through a second-order transfer function. This coupling produces an implicit spectral gating mechanism: the self-interaction term of the damping factor gradient constantly closes the spectral gate, whereas the signal term opens the gate only when the target displays coherent frequency components. This enables automatic distinction between signal and noise without any explicit regularization or task-specific hyperparameter tuning, achieving state-of-the-art (SOTA) or competitive performance across tasks like signal fitting, denoising, CT reconstruction, and super-resolution.

Background & Motivation

Background: Implicit Neural Representations (INRs) encode continuous signals via coordinate networks and have become a core paradigm for tasks such as novel view synthesis, 3D shape representation, and audio reconstruction. The core of INRs lies in the choice of activation function, which directly determines the spectral expressivity of the network, i.e., whether it can simultaneously represent smooth global structures and fine local high-frequency details.

Limitations of Prior Work: Periodic activations (such as SIREN's sinusoidal function) can express arbitrarily high frequencies but act essentially as all-pass filters, fitting noise indiscriminately and lacking inherent spatial localization and frequency-domain regularization. Non-periodic activations (such as Gaussian or the Gabor wavelet in WIRE) offer spatial compactness and implicit regularization but introduce a low-frequency bias, making it difficult to capture oscillatory or fine-grained content. Compromise solutions (e.g., FINER, MIRE, Fourier Reparametrisation) modify these properties at the cost of computational efficiency, hyperparameter sensitivity, or expressivity. None of the existing methods offer a mechanism to dynamically and adaptively adjust the bandwidth during training based on the spectral characteristics of the signal and noise.

Key Challenge: This presents a "spectral dilemma"—expressing high-frequency details requires high bandwidth, but high bandwidth means noise will also be fitted indiscriminately. Existing methods essentially make a static trade-off between expressivity and robustness, lacking the ability to dynamically adjust along with the training process.

Goal: Design an activation function that enables the network to automatically recognize during training which frequency components are genuine signals (to be reconstructed) and which are noise (to be ignored), operating in a manner free of task-specific hyperparameter tuning.

Key Insight: The authors draw inspiration from classical mechanics—the steady-state response of a forced damped harmonic oscillator naturally contains an amplitude transfer function determined by the damping factor, natural frequency, and excitation frequency, which acts as a well-known second-order bandpass filter. If this physical structure is embedded into each neuron and the oscillator parameters are made learnable, can spectral selectivity emerge spontaneously during the optimization process?

Core Idea: Model the activation of each neuron as the steady-state response of a forced damped harmonic oscillator. Use its amplitude transfer function as an implicit spectral gate. The gradient of the damping factor naturally decomposes into a "closing force" (self-interaction term) and an "opening force" (signal term). The competition between these two forces determines when the network expands its bandwidth and when it suppresses noise, thereby achieving adaptive spectral selection without introducing any explicit regularization terms.

Method

Overall Architecture

The core innovation of FDHO lies in the physical derivation of the activation function rather than architectural modifications. Given a standard coordinate MLP (taking coordinates as input and outputting signal values), the activation function of each hidden layer is replaced with FDHO, while other structural aspects (depth, width) remain identical to SIREN. Each layer shares a set of learnable oscillator parameters \(\theta = (\omega, \omega_n, \xi, \varphi)\), representing the forced frequency, natural frequency, damping factor, and phase, respectively. These parameters are co-optimized with the network weights \(\mathbf{W}, \mathbf{b}\) via gradient descent.

During the forward pass, for an input \(z = \mathbf{w}^\top \mathbf{x} + b\), the activation output is given by \(\sigma(z; \theta) = A(\omega, \omega_n, \xi) \sin(\omega z + \varphi)\), where the amplitude \(A\) is the transfer function of a classical second-order system:

\[A(\omega, \omega_n, \xi) = \frac{\omega_n^2}{\sqrt{(\omega_n^2 - \omega^2)^2 + (2\xi\omega_n\omega)^2}}\]

The behavior of this amplitude function is precisely controlled by three physical parameters: the peak occurs at the resonant frequency \(\omega_r = \omega_n\sqrt{1-2\xi^2}\) (when \(\xi < 1/\sqrt{2}\)), with a peak amplitude of \(A_{\text{peak}} = (2\xi\sqrt{1-\xi^2})^{-1}\); when \(\xi \geq 1/\sqrt{2}\), the amplitude decreases monotonically starting from \(A(0)=1\). Training starts with a stopband initialization (\(\omega_0 < \omega_{n,0}\), \(\xi_0 = 1/\sqrt{2}\) corresponding to the Butterworth maximally flat response), where the amplitude is suppressed and the network bandwidth is extremely narrow. As training progresses, when the loss landscape demands richer frequency content, the damping factor \(\xi\) decreases, the amplitude increases, and the bandwidth gradually expands, producing a natural coarse-to-fine learning process.

%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400}}}%%
flowchart TD
    A["Input Coordinate x"] --> B["Linear Layer<br/>z = Wx + b"]
    B --> C["FDHO Activation<br/>σ(z) = A(ω,ωn,ξ)·sin(ωz+φ)"]
    C --> D["Multi-layer Stack<br/>Modulation Index β∝∏Ak"]
    D --> E["Output Signal Value"]
    C --> F["Spectral Gating<br/>Tself Close vs Tsignal Open"]

Key understanding: During forward inference, the amplitude \(A\) is a scalar constant; the network does not perform spatially-varying filtering on different frequency components. The inductive bias of spectral gating operates entirely through the gradient dynamics during training—whether a specific frequency can be expressed by the network depends on the interaction between that frequency component and the damping factor during gradient descent.

Key Designs

1. Physically Coupled Activation Function: Replacing Free Amplitude Parameters with Oscillator Transfer Functions

A critical limitation of sinusoidal activation (SIREN) in INRs is that frequency and amplitude are decoupled: the network can assign arbitrarily large amplitudes to any frequency, including those driven by noise. An intuitive remedy would be making the amplitude a learnable parameter as well (Adaptive SIREN), but the authors prove through control experiments that this approach is ineffective—the decoupled parameter space contains a massive number of degenerate solutions (multiple combinations of \((a, \omega, \varphi)\) yielding the same output), causing gradient descent to drift infinitely along flat directions (experiments show that the convergence PSNR is 77 dB lower than FDHO, with a 10\(\times\) increase in variance).

The key constraint of FDHO is that the amplitude \(A(\omega, \omega_n, \xi)\) is not a free parameter but is uniquely determined by the forced frequency, natural frequency, and damping factor through the physical transfer function. This coupling eliminates degeneracy in the parameter space: once a frequency component is sufficiently fitted, \(\xi\) reaches an equilibrium (Corollary 1(a)) and ceases to drift. The authors provide three formal guarantees: smoothness (\(C^\infty\), ensuring gradient optimization is viable), Lipschitz continuity (Lipschitz constant \(L_z = |\omega|A\), tying the capacity for high-frequency representation to the amplitude), and bounded output (\(|\sigma| \leq A\), where the activation range is directly controlled by the parameters).

2. Gradient Decomposition of Spectral Gating: \(\partial\mathcal{L}/\partial\xi = T_{\text{self}} + T_{\text{signal}}\)

This is the core theoretical contribution of FDHO. For the MSE loss \(\mathcal{L} = \frac{1}{N}\sum_i (\sigma(z_i;\theta) - y_i)^2\), the gradient of the loss with respect to the damping factor \(\xi\) can be strictly decomposed into two terms:

\[\frac{\partial\mathcal{L}}{\partial\xi} = \underbrace{\frac{\partial A}{\partial\xi}\frac{2A}{N}\sum_i \sin^2(\omega z_i + \varphi)}_{T_{\text{self}}} + \underbrace{\left(-\frac{\partial A}{\partial\xi}\right)\frac{2}{N}\sum_i y_i \sin(\omega z_i + \varphi)}_{T_{\text{signal}}}\]

\(T_{\text{self}}\) (self-interaction term): Theorem 3.1 guarantees that \(\partial A/\partial\xi < 0\) (increasing damping always decreases amplitude). Thus, in the large-sample limit, \(T_{\text{self}} \approx \partial A/\partial\xi \cdot A < 0\), which constantly pushes \(\xi\) upward (closing the spectral gate) and acts as an unconditional regularizing force.

\(T_{\text{signal}}\) (signal term): Its sign depends entirely on the empirical inner product between the target \(y\) and the carrier wave \(\sin(\omega\cdot + \varphi)\). Only when the target possesses coherent energy at this carrier frequency will \(T_{\text{signal}} > 0\), driving \(\xi\) downward (opening the spectral gate).

Corollary 1 gives the precise gating condition—the spectral gate opens if and only if \(\langle y, \sin(\omega\cdot + \varphi)\rangle_N > A/2\). Three cases correspond to three distinct behaviors: (a) If the target contains a coherent component of frequency \(\omega\), the gate opens until \(A\) approaches the target amplitude \(A_y\), reaching equilibrium; (b) If the target is zero-mean i.i.d. noise, the signal term is of order \(O_p(N^{-1/2})\), the self-action term dominates, and the gate remains closed; (c) If the target is a pure tone with a mismatched frequency, the inner product approaches zero in large samples, and the gate likewise remains closed. Consequently, FDHO possesses an optimization-level intrinsic bias toward fitting coherent signal components and rejecting random noise.

3. Multi-layer Frequency Synthesis and Coarse-to-Fine Learning Curriculum

Single-layer analysis explains the gradient behavior for the damping factor, but the expressive power of INRs stems from interlayer interactions. Proposition 4 proves that the composition of two FDHO layers is equivalent to a frequency-modulated signal: \(h_2(x) = A_2 \sin(\beta \sin(\alpha(x)) + c)\), where the modulation index is \(\beta = \omega_2 w_2 A_1\). Via the Jacobi-Anger expansion, this signal can be decomposed into a sum of integer harmonics of the frequency \(\alpha(x)\), with harmonic amplitudes controlled by the Bessel functions \(J_n(\beta)\), and the number of active harmonics is approximately \(\lfloor\beta\rfloor + 1\).

Extending to \(L\) layers (Proposition 5), the effective modulation index for layer \(\ell\) is \(\beta_\ell = \omega_\ell w_\ell \prod_{k=1}^{\ell-1} A_k\). This means that the total harmonic bandwidth is exponentially sensitive to the product of individual layer amplitudes. Initially, each layer's amplitude is \(A_k \approx A_0 < 1\), so \(\beta_\ell \propto A_0^{\ell-1}\) decays geometrically with depth—the deepest layers contribute virtually no frequency support at the beginning, causing the bandwidth to expand gradually outward from the input layer. This multiplicative structure means that even a minor decrease in \(\xi_k\) at each layer can yield a massive change in the network's overall spectral bandwidth after multi-layer composition. The authors supplement this perspective with Neural Tangent Kernel (NTK) analysis: the kernel amplitude scales with \(A^2\), and the network "bootstraps" its convergence rate by progressively opening the spectral gate—at the early stage, \(A\) is small and learning is slow, prioritizing the capture of low-frequency structures; as \(A\) grows, high frequencies are progressively incorporated.

Loss & Training

FDHO uses standard MSE loss without any explicit regularization terms. Key training details are as follows:

  • Parameterization: \(\omega\) and \(\omega_n\) are constrained to be positive via softplus; \(\xi\) is constrained to \((0, 1)\) via sigmoid (since \(\xi = 0\) at \(\omega = \omega_n\) would lead to a zero denominator, corresponding to divergent undamped resonance).
  • Initialization Strategy: The forced frequency is initialized below the natural frequency (\(\omega_0 = 45\), \(\omega_{n,0} = 50\)) to keep the system far from the resonance zone, suppressing the initial amplitude. The damping factor is initialized to \(\xi_0 = 1/\sqrt{2}\), corresponding to the Butterworth maximally flat response, providing moderate amplitude but no resonance peak. Phase is initialized to the physical phase lag \(\varphi_0 = -\arctan(2\xi_0\omega_{n,0}\omega_0 / (\omega_{n,0}^2 - \omega_0^2))\) and is allowed to evolve freely thereafter. Network weights are initialized following the SIREN scheme but scaled according to the initial natural frequency: first layer \(w \sim \mathcal{U}(-1/m, 1/m)\), and hidden layers \(w \sim \mathcal{U}(-\sqrt{6/m}/\omega_n, \sqrt{6/m}/\omega_n)\).
  • Optimizer & Learning Rate: Adam is used with a learning rate for network weights of \(\eta_w = 10^{-4}\) and for oscillator parameters of \(\eta_\theta = 10^{-2}\) (the two orders of magnitude difference reflects that oscillator parameters directly impact global spectral behavior); this is paired with a ReduceLROnPlateau scheduler (factor=0.1, patience=500, min_lr=\(10^{-6}\)).
  • Architecture: A uniform configuration is used across all tasks with \(m=256\) hidden dimensions and \(L=5\) hidden layers, without any task-specific adjustments—conversely, each baseline requires separate hyperparameter configurations for different tasks.

Key Experimental Results

Main Results

FDHO is evaluated under three main categories of observation models: signal representation (full supervision), recovery from corrupted observations (denoising, CT reconstruction), and recovery from incomplete observations (inpainting, super-resolution), and is compared against 9 INR baselines (SIREN, Gauss, WIRE, BACON, FINER, MFN, Fourier Features, FR). The table below selects image fitting (on 5 natural images) as representative main experimental results.

INR Tiger (Final/Peak) Bikers (Final/Peak) Butterfly (Final/Peak) Knot (Final/Peak)
FDHO 63.79 / 63.79 57.06 / 57.25 57.85 / 57.96 59.21 / 64.05
SIREN 53.69 / 58.52 43.11 / 55.92 48.81 / 55.83 51.90 / 59.72
FINER 56.58 / 57.09 47.46 / 53.53 53.65 / 55.10 53.39 / 59.72
Gauss 44.10 / 46.46 41.45 / 42.59 43.19 / 45.01 43.81 / 47.08
WIRE 38.64 / 39.20 36.71 / 36.99 36.78 / 37.10 40.32 / 40.80

FDHO achieves the best Final and Peak PSNR on 4 out of the 5 images, leading the second-best method by up to ~10 dB (57.06 vs FINER's 47.46 on Bikers). The only exception is the Tiles image (highly regular, repetitive textures), where the variable-period activation of FINER fits the highly concentrated spectral characteristics of the image better. The most notable pattern is SIREN's peak-to-final degradation: on Bikers, its peak of 55.92 dB drops to only 43.11 dB at convergence (a decrease of ~13 dB), reflecting post-overshoot drift caused by the lack of a stabilization mechanism in the all-pass activation. FDHO achieves identical Final and Peak PSNR on Tiger (63.79 dB), and across all images, the gap between the two does not exceed 5 dB, verifying that the equilibrium condition of Corollary 1(a) indeed enforces convergence stability.

Noisy scenarios (image denoising, with \(\sigma=0.1\) Gaussian noise) provide the most direct validation of spectral gating. FDHO outperforms all others on all 5 images, leading the runner-up MFN by approximately 2-3 dB. SIREN degrades from its high peak in clean fitting to 26.2 dB (comparable to Gauss and WIRE), confirming that its all-pass nature leaves it entirely defenseless against noise. CT reconstruction (sparse Radon projection + measurement noise) serves as the most comprehensive test, where FDHO reaches 39.97 dB, leading the runner-up MFN by 2.8 dB—the largest absolute margin in the entire experiment.

Ablation Study

Configuration Image Fitting (Tiger) Final/Peak CT Reconstruction PSNR Description
FDHO (Full Model) 63.27 / 63.27 40.16 Uniform configuration, no task-specifc tuning
Adaptive SIREN 54.65 / 58.21 32.44 Learnable amplitude + frequency, but decoupled
SIREN (Standard) 53.37 / 58.19 33.28 Fixed sinusoidal activation

All three use the exact same architecture (\(m=256, L=5\)), optimizer, and training protocol to isolate scheduler effects and other confounding factors. The peak PSNR of Adaptive SIREN is close to FDHO (58.21 vs 63.27, a gap much smaller than their final convergence gap), demonstrating that its capacity is sufficient to represent the target. However, the 9 dB final gap and the substantially increased variance (3.97 vs 1.30) substantiate the hypothesis that decoupled parameters lack a stabilizing mechanism. Restricting the amplitude-frequency relationship to a single path through physical coupling eliminates parameter degeneracy planes, which is the foundational source of FDHO's stability.

Key Findings

  • Spectral gating benefits noisy environments with coexisting high-frequency components the most: The advantages in denoising (+3-5 dB over MFN) and CT reconstruction (+2.8 dB over MFN) are the most pronounced. This precisely matches the prediction of Corollary 1(b)—the gating condition is violated for noise but satisfied for the ground-truth signal.
  • The coarse-to-fine curriculum is evident in super-resolution: FDHO locks onto observed low-frequency structures before expanding to high frequencies, achieving optimal results on Tiger, Bikers, Butterfly, and Knot images. However, the margin of improvement is narrower than in denoising (with a ~1 dB gap to FINER/SIREN), as constraints in super-resolution come from missing information rather than active noise.
  • Poisson reconstruction is a weak spot: When supervision is applied on the Laplacian (second derivative), the \(\omega^2\) scaling factor causes high-frequency components to be weighted by \(\propto \omega^4\) in the loss. Consequently, the delay in high-frequency recovery caused by stopband initialization becomes a bottleneck. Relaxing the initial bandwidth by changing \(\omega_0=45\) to \(15\) improved the PSNR on Tiger from 26.13 dB to 30.11 dB, verifying this analysis.
  • FDHO is the only method stable across all tasks: MFN excels at denoising but is weak at fitting, SIREN excels at fitting but is weak at denoising, and FINER is strong in super-resolution but unstable in CT (with a variance of 6.06 dB). FDHO consistently ranks in the top two across all tasks with controlled variance.
  • Moderate computational overhead: Parameters increase by only 24 (an increase of 0.007% compared to SIREN) and the inference cost is identical. Training time is 191s (SIREN: 134s, ~1.4\(\times\)), and training memory is 2908MB (SIREN: 1205MB, ~2.4\(\times\)), primarily due to backpropagation of parameters shared across per-layer settings and dual-learning-rate optimization—both of which are within an acceptable range.

Highlights & Insights

  • Replacing hand-crafted activation functions with the transfer functions of physical systems: This is not a mere stack of tricks, but a conceptual elevation. Instead of asking "what shape should the activation function be," it asks "what physical system possesses the spectral selectivity we desire," and derives the result. The mature theory of the damped harmonic oscillator as a second-order linear system (including classic results like Butterworth filters) provides a solid mathematical foundation for FDHO.
  • Gradient decomposition is the "soul" of this paper: The decomposition into \(T_{\text{self}} + T_{\text{signal}}\) not only elegantly explains why the network rejects noise, but also yields precise gating conditions—"the gate opens only when the inner product of the signal exceeds \(A/2\)." This analyzes the INR from the perspective of the implicit bias in training dynamics, rather than a static perspective of representation capacity, complementing traditional paths of NTK spectral analysis.
  • Stopband initialization naturally yields a coarse-to-fine curriculum: This is a "zero-cost" curriculum learning—requiring no schedulers or phase switches. It achieves natural progression from low to high frequencies purely through the selection of initial parameters. Implementing curriculum learning through optimization dynamics rather than explicit control is highly transferable to other tasks involving spectrally selective learning, such as NeRF training or diffusion model denoising schedules.
  • Parameter coupling as a source of stability: The failure of Adaptive SIREN clearly demonstrates that "more degrees of freedom is not necessarily better." Physical constraints eliminate degenerate directions in the parameter space, stabilizing convergence. This strongly supports the design philosophy in deep learning of intentionally introducing inductive biases rather than merely scaling model capacity.

Limitations & Future Work

  • Performance degradation under derivative supervision: The Poisson reconstruction task indicates that when the supervision signal is transformed by differential operators, the stopband initialization of FDHO may no longer be the optimal starting point. Although the authors verified that relaxing the initial bandwidth mitigates the issue, a systematic solution is yet to be established. Learning strategies that automatically adjust initialization based on the spectral properties of the supervisory operator represent a promising direction for future research.
  • Limitations of parameter sharing across layers: In the current design, all neurons within the same hidden layer share \((\omega, \omega_n, \xi, \varphi)\), meaning neuron frequency diversity is uniquely generated by varying scales in weights (\(\|\mathbf{w}_j\|\)). Allowing each neuron to maintain independent oscillator parameters could enhance expressivity but would substantially increase parameter counts and training complexity. This requires a careful balance between expressivity and efficiency.
  • Analysis limited to MSE loss: The theoretical derivation of spectral gating relies on the pointwise nature of the MSE loss; generalizations to other loss functions (such as L1 or perceptual losses) have not yet been discussed. Whether gradient decomposition under different loss functions still yields similar gating behaviors remains an open theoretical question.
  • Relatively limited experimental scale: Although the task coverage is broad (1D signals, audio, 2D images, 3D SDFs), evaluations are primarily conducted on classic INR benchmarks. Real-world applications like larger-scale NeRF scenes or video representations are not explored. While preliminary integration results on FreSh and ResField are positive (yielding a 3-7 dB gain), systematic large-scale validation has yet to be completed.
  • vs SIREN: SIREN’s sinusoidal activation is a purely periodic function with a constant amplitude of 1, operating as an all-pass filter. The key distinction of FDHO is that it couples amplitude with frequency through a physical transfer function, allowing the network to adaptively adjust its bandwidth during training, whereas SIREN grants identical spectral access to both signal and noise.
  • vs WIRE: WIRE utilizes Gabor wavelets for joint spatio-frequency localization, which offers spatial compactness but suffers from low-frequency bias due to static spectral bandwidths. FDHO uses a learnable damping factor to bridge different spectral regimes over a continuous spectrum: as \(\xi \to 0\), it approaches SIREN's unconstrained bandwidth, and as \(\xi\) increases, it transitions smoothly into a spatially compact representation.
  • vs FINER: FINER introduces variable periods into sinusoidal activations, enabling different layers to exhibit distinct frequency characteristics, but lacks the gradient-level noise rejection mechanism of spectral gating. Empirically, FINER excels in structured textures (Tiles) and SDFs by locking onto the dominant period, but under noisy conditions, it degrades to a performance tier comparable to non-periodic activations.
  • vs TUNER: TUNER stabilizes training by restricting the spectral boundaries of the network at initialization, representing a static form of spectral control. In contrast, FDHO's spectral gating is dynamic and evolves during training—whereas TUNER establishes a hard spectral ceiling, FDHO permits the network to discover the appropriate bandwidth on its own.
  • vs Fourier Features / BACON: Fourier Feature mapping and BACON's band-limited hierarchy perform spectral control strictly within the input encoding or output structure. FDHO's philosophy lies in directly encoding spectral selectivity into the differential structure of the activation function, acting as an implicit, optimization-driven inductive bias that is complementary to input/output-centric approaches.

Rating

  • Novelty: ⭐⭐⭐⭐⭐ Introduces the concept of damped harmonic oscillators from classical mechanics to INR activation design, with a rigorous proof of the spectral gating mechanism via gradient decomposition. The concept is highly original and theoretically self-consistent.
  • Experimental Thoroughness: ⭐⭐⭐⭐ Covers 8 tasks spanning 1D/2D/3D/audio + 9 baselines + control experiments + ablation studies + computational overhead analysis, though lacking large-scale NeRF validation.
  • Writing Quality: ⭐⭐⭐⭐⭐ Exceptional clarity in theoretical derivations (from definition to properties, gating, multi-layer composition, and NTK—forming a complete logical chain), analytical rigor in experimental interpretations (tying each setting back to Corollary predictions), and comprehensive appendices.
  • Value: ⭐⭐⭐⭐⭐ Proposes an adaptive spectral control mechanism that does not rely on explicit regularization. The principle is generic, lightweight (adding only 24 parameters), and compatible with existing INR frameworks, granting direct practical value while serving as a methodological reference for physics-informed deep learning.