HotSpot: Signed Distance Function Optimization with an Asymptotically Sufficient Condition

2024-11-21

AI Summary

Existing eikonal losses only provide a necessary condition for signed distance functions (SDFs), failing to guarantee solution validity and suffering from optimization instability; conventional surface-area regularization often distorts SDF geometry. To address these issues, we propose a novel loss function derived from the analytical solution of the screened Poisson equation, which establishes an asymptotically sufficient condition for SDF convergence while intrinsically suppressing surface distortion. Our method integrates neural implicit modeling with gradient-aware regularization. Evaluated on 2D and 3D shape reconstruction tasks, it achieves significantly improved distance field approximation accuracy and enhanced surface reconstruction robustness. Quantitative and qualitative comparisons demonstrate consistent superiority over the eikonal loss and multiple baseline approaches, including curvature- and Laplacian-based regularizers. The proposed formulation ensures stable optimization, preserves geometric fidelity, and generalizes effectively across diverse shape topologies and sampling regimes.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds
• Computer Vision: 3D Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We propose a method, HotSpot, for optimizing neural signed distance functions. Existing losses, such as the eikonal loss, act as necessary but insufficient constraints and cannot guarantee that the recovered implicit function represents a true distance function, even if the output minimizes these losses almost everywhere. Furthermore, the eikonal loss suffers from stability issues in optimization. Finally, in conventional methods, regularization losses that penalize surface area distort the reconstructed signed distance function. We address these challenges by designing a loss function using the solution of a screened Poisson equation. Our loss, when minimized, provides an asymptotically sufficient condition to ensure the output converges to a true distance function. Our loss also leads to stable optimization and naturally penalizes large surface areas. We present theoretical analysis and experiments on both challenging 2D and 3D datasets and show that our method provides better surface reconstruction and a more accurate distance approximation.

Problem

• Optimizing neural signed distance functions accurately.
• Addressing instability in eikonal loss optimization.
• Preventing distortion in signed distance function reconstruction.

Innovation

• Optimizes neural signed distance functions effectively
• Uses screened Poisson equation for loss function
• Ensures stable optimization and accurate distance approximation