HotSpot: Signed Distance Function Optimization with an Asymptotically Sufficient Condition

TL;DR

HotSpot introduces a novel loss function for neural signed distance functions that guarantees convergence to a true distance function, ensuring stability and improved surface reconstruction in 2D and 3D datasets.

Contribution

The paper presents a new loss based on a screened Poisson equation that provides an asymptotically sufficient condition for true distance function recovery, addressing limitations of existing methods.

Findings

Achieves more accurate surface reconstruction.

Ensures convergence to true distance functions.

Provides stable optimization process.

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.