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FINER++: Building a Family of Variable-periodic Functions for Activating Implicit Neural Representation

Hao Zhu, Zhen Liu, Qi Zhang, Jingde Fu, Weibing Deng, Zhan Ma, Yanwen Guo, Xun Cao

TL;DR

This work tackles the persistent spectral-bias and capacity-convergence gaps in implicit neural representations (INRs) by introducing FINER++, a universal framework that extends existing activation functions to variable-periodic forms. By modulating the bias initialization range $k$ and introducing a frequency-overlap parameter $\omega_f$, FINER++ expands the effective frequency set $\mathcal{F}_{\omega}$ that INRs can represent, improving performance across 2D, 3D, and 5D tasks and enabling streamable INR transmission. The approach is validated on 2D image fitting, 3D signed distance fields, neural radiance fields, and streaming scenarios, showing consistent gains over SIREN, Gaussian, Wavelet, and Fourier-feature baselines. The work leverages geometrical and Neural Tangent Kernel analyses to explain how frequency coverage and diagonal NTK grow with bias-range, supporting the observed improvements and robustness across activations.

Abstract

Implicit Neural Representation (INR), which utilizes a neural network to map coordinate inputs to corresponding attributes, is causing a revolution in the field of signal processing. However, current INR techniques suffer from the "frequency"-specified spectral bias and capacity-convergence gap, resulting in imperfect performance when representing complex signals with multiple "frequencies". We have identified that both of these two characteristics could be handled by increasing the utilization of definition domain in current activation functions, for which we propose the FINER++ framework by extending existing periodic/non-periodic activation functions to variable-periodic ones. By initializing the bias of the neural network with different ranges, sub-functions with various frequencies in the variable-periodic function are selected for activation. Consequently, the supported frequency set can be flexibly tuned, leading to improved performance in signal representation. We demonstrate the generalization and capabilities of FINER++ with different activation function backbones (Sine, Gauss. and Wavelet) and various tasks (2D image fitting, 3D signed distance field representation, 5D neural radiance fields optimization and streamable INR transmission), and we show that it improves existing INRs. Project page: {https://liuzhen0212.github.io/finerpp/}

FINER++: Building a Family of Variable-periodic Functions for Activating Implicit Neural Representation

TL;DR

This work tackles the persistent spectral-bias and capacity-convergence gaps in implicit neural representations (INRs) by introducing FINER++, a universal framework that extends existing activation functions to variable-periodic forms. By modulating the bias initialization range and introducing a frequency-overlap parameter , FINER++ expands the effective frequency set that INRs can represent, improving performance across 2D, 3D, and 5D tasks and enabling streamable INR transmission. The approach is validated on 2D image fitting, 3D signed distance fields, neural radiance fields, and streaming scenarios, showing consistent gains over SIREN, Gaussian, Wavelet, and Fourier-feature baselines. The work leverages geometrical and Neural Tangent Kernel analyses to explain how frequency coverage and diagonal NTK grow with bias-range, supporting the observed improvements and robustness across activations.

Abstract

Implicit Neural Representation (INR), which utilizes a neural network to map coordinate inputs to corresponding attributes, is causing a revolution in the field of signal processing. However, current INR techniques suffer from the "frequency"-specified spectral bias and capacity-convergence gap, resulting in imperfect performance when representing complex signals with multiple "frequencies". We have identified that both of these two characteristics could be handled by increasing the utilization of definition domain in current activation functions, for which we propose the FINER++ framework by extending existing periodic/non-periodic activation functions to variable-periodic ones. By initializing the bias of the neural network with different ranges, sub-functions with various frequencies in the variable-periodic function are selected for activation. Consequently, the supported frequency set can be flexibly tuned, leading to improved performance in signal representation. We demonstrate the generalization and capabilities of FINER++ with different activation function backbones (Sine, Gauss. and Wavelet) and various tasks (2D image fitting, 3D signed distance field representation, 5D neural radiance fields optimization and streamable INR transmission), and we show that it improves existing INRs. Project page: {https://liuzhen0212.github.io/finerpp/}
Paper Structure (26 sections, 1 theorem, 13 equations, 13 figures, 3 tables)

This paper contains 26 sections, 1 theorem, 13 equations, 13 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Introduction
  3. Related Work
  4. Implicit Neural Representation
  5. Spectral-bias Problem
  6. INR and its Characteristics
  7. Pipeline of INRs
  8. Under-utilized Definition Domain in Activation functions
  9. "Frequency"-specified Spectral Bias
  10. Capacity-Convergence Gap
  11. FINER++: Variable-periodic Activation Functions for INRs
  12. Variable-periodic Activation Functions
  13. Handling INR's Characteristics
  14. Selection of $\omega_f$
  15. Initialization scheme for bias vector
  16. ...and 11 more sections

Key Result

Proposition 1

The supported frequency set $\mathcal{F}_{p,k}$ of FINER++ increases with the increase of the initialization range of $\vec{b}$, and the supported "frequency" set $\mathcal{F}_{p}$ in previous INRs is a subset of $\mathcal{F}_{\omega_0,k}$ in FINER++.

Figures (13)

  • Figure 1: FINER++ framework for INR. We observe that the supported frequency set in classical INRs is limited due to the under-utilization of activation functions' definition domain, i.e., they mainly employ the central region near the origin point. To overcome this limitation, we propose the FINER++ framework by extending the activation functions from periodic/non-periodic functions to variable-periodic ones. This innovation allows for tuning the supported frequency set by adjusting the initialization range of the bias vector in the neural network. (a) visualizes the selected narrow activation functions in classical activation Sine, Gauss. and Wavelet alongside our proposed variable-periodic ones with different bias settings (purple areas). (b) plots the training curves of previous INRs and FINER++, demonstrating the impact of different initializations applied to the bias vector $\vec{b}$ (see Sec. \ref{['sec:img_diff_bias']} for more details).
  • Figure 2: Visualization of capacity-convergence gap in various INRs for fitting a $2K$ image. The performance of various INRs drops significantly when set the empirical scale parameters as 1 and changing the initialization range of network parameters.
  • Figure 3: Visualizations of FINER++ (Wavelet) with different $\omega_f$. Inappropriate $\omega_f$ setting results in the problems of "degeneration" (left) and "overlap" (right).
  • Figure 4: Comparisons of used activation function $\sin((|x|+1)x)$ under different bias $\vec{b}$. More sub-functions with high-frequency are included when $b$ is set with a larger value.
  • Figure 5: Visualizations of NTKs and the corresponding eigenvalues in FINER++. From top to bottom, the NTKs and NTKs' eigenvalues of FINER++ with sine, Gaussian and Wavelet activation functions are visualized, respectively. From left to right, (a)-(d) visualize the NTKs when $\vec{b}$ is initialized following $\mathcal{U}(-1,1)$, $\mathcal{U}(-5,5)$, $\mathcal{U}(-10,10)$ and $\mathcal{U}(-20,20)$, respectively. (e) plots the corresponding eigenvalues. Because the max eigenvalue is much larger than the smallest one, all eigenvalues are processed by a $\log$ function for visualization.
  • ...and 8 more figures

Theorems & Definitions (1)

  • Proposition 1