Implicit Neural Representations with Periodic Activation Functions (2006.09661v1)

Abstract: Implicitly defined, continuous, differentiable signal representations parameterized by neural networks have emerged as a powerful paradigm, offering many possible benefits over conventional representations. However, current network architectures for such implicit neural representations are incapable of modeling signals with fine detail, and fail to represent a signal's spatial and temporal derivatives, despite the fact that these are essential to many physical signals defined implicitly as the solution to partial differential equations. We propose to leverage periodic activation functions for implicit neural representations and demonstrate that these networks, dubbed sinusoidal representation networks or Sirens, are ideally suited for representing complex natural signals and their derivatives. We analyze Siren activation statistics to propose a principled initialization scheme and demonstrate the representation of images, wavefields, video, sound, and their derivatives. Further, we show how Sirens can be leveraged to solve challenging boundary value problems, such as particular Eikonal equations (yielding signed distance functions), the Poisson equation, and the Helmholtz and wave equations. Lastly, we combine Sirens with hypernetworks to learn priors over the space of Siren functions.

• The paper introduces periodic sine activation functions that capture fine details and signal derivatives more accurately than traditional ReLU networks.
• It presents a principled initialization scheme that stabilizes deep network training and ensures robust convergence for SIRENs.
• The study demonstrates SIRENs’ effectiveness across diverse applications such as image fitting, PDE solving, and inverse problems like image inpainting.

Implicit Neural Representations with Periodic Activation Functions

The paper "Implicit Neural Representations with Periodic Activation Functions" presents a novel approach to representing signals implicitly using neural networks with periodic activation functions, or sinusoidal representation networks (SIRENs). This methodological shift addresses notable limitations in conventional architectures like ReLU-based multilayer perceptrons (MLPs) by improving the representation of fine details and better accounting for signal derivatives.

Novel Contributions

The key contributions of this research lie in several areas:

1. Periodic Activation Functions: Unlike traditional neural architectures that struggle with fine detail and derivative representation, the deployment of sine functions as periodic activations enables the model to represent not only the signals but also their derivatives more accurately. This property is essential for many applications across scientific domains that rely on partial differential equations (PDEs).
2. Principled Initialization Scheme: The paper introduces an initialization scheme that preserves the distribution of activations throughout the network, which is critical for the successful training of deep networks with sine activations. This ensures robust convergence and superior performance.
3. Versatile Applications: The paper demonstrates that SIRENs can effectively represent various types of data, including images, wavefields, video, and sound. Additionally, the model shows proficiency in solving boundary value problems such as Eikonal equations for Signed Distance Functions (SDFs), the Poisson equation, the Helmholtz equation, and the wave equation.
4. Hypernetwork Integration: The combination of SIRENs with hypernetworks is employed to learn priors over the space of functions, showcasing significant potential for applications like image inpainting and video representation.

Key Results

The empirical results are robust, showcasing the strengths of SIRENs in several contexts:

• Image Fitting: SIRENs outperform ReLU-MLPs in terms of higher fidelity to fine details and derivatives of images.
• Video Representation: The method provides a substantial improvement in Peak Signal-to-Noise Ratio (PSNR) compared to conventional architectures.
• Poisson Equation: The network efficiently reconstructs images from gradients or Laplacians, demonstrating its capability to solve inverse problems.
• SDF Representation: SIRENs manage to capture detailed shapes and large-scale scenes more accurately than ReLU-based representations, highlighting their capacity for high-complexity 3D shape representation.
• Helmholtz and Wave Equations: In solving these PDEs and related inverse problems like full-waveform inversion (FWI), SIRENs achieve high-accuracy reconstructions that align closely with traditional grid-based solvers while surpassing other neural network solutions.
• Learning Implicit Function Spaces: The integration of SIRENs with hypernetworks showcases adept generalization across signal classes, outperforming baseline models in image inpainting tasks.

Implications and Future Directions

The implications of this research are twofold: practical and theoretical. Practically, SIRENs provide a powerful toolkit for various applications requiring fine detail and well-behaved derivatives, including 3D modeling, video processing, and PDE solving. Theoretically, they expand the boundaries of neural function representation by incorporating periodic activation functions, which allow for better handling of derivatives crucial for numerous scientific computations.

Future developments may explore further integration with more advanced hypernetworks, non-Euclidean domains, and adaptations for different types of nonlinear PDEs. Additionally, applications in more complex, real-world scenarios, such as seismic imaging, fluid dynamics, and high-resolution medical imaging, could provide fruitful ground for further exploration.

In conclusion, this work significantly advances the field of implicit neural representations, providing a robust framework for a wide range of applications by leveraging the unique properties of periodic activation functions.