On Optimal Sampling for Learning SDF Using MLPs Equipped With Positional Encoding.

Area of Science:

• Computer Vision
• Geometric Deep Learning
• Signal Processing

Background:

• Neural implicit fields, like neural signed distance fields (SDFs), are crucial for 3D shape representation and analysis.
• Multi-layer Perceptrons (MLPs) with positional encoding (PE) are commonly used but can produce noisy artifacts in learned fields.
• Increasing sampling rates can mitigate artifacts, but an optimal rate is not well-defined.

Purpose of the Study:

• To analyze the cause of noisy artifacts in PE-equipped MLPs using Fourier analysis.
• To develop a method for determining the appropriate sampling rate for training neural implicit fields.
• To improve the accuracy and efficiency of learning neural implicit representations.

Main Methods:

• Applied Fourier analysis to understand the behavior of PE-equipped MLPs.
• Proposed a method to estimate the intrinsic frequency of a network based on its responses.
• Utilized the Nyquist-Shannon sampling theorem to derive an optimal training sampling rate.
• Empirically validated the method in the context of SDF fitting.

Main Results:

• Identified that PE-equipped MLPs possess an intrinsic frequency significantly higher than the highest PE frequency component.
• Demonstrated that sampling according to the estimated intrinsic frequency prevents undesirable artifacts.
• Showcased that the recommended sampling rate achieves accurate SDF fitting without further gains from oversampling.
• Achieved superior performance compared to existing methods by employing the proposed sampling strategy.

Conclusions:

• The proposed Fourier analysis-based sampling strategy effectively mitigates artifacts in neural implicit fields.
• This method provides a principled way to set training sampling rates, enhancing accuracy and efficiency.
• The findings offer a valuable tool for researchers and practitioners working with neural implicit representations.