Phase Transitions, Distance Functions, and Implicit Neural Representations

TL;DR

This paper introduces a novel loss function inspired by phase transition theory for training Implicit Neural Representations, enabling simultaneous learning of occupancy and distance functions with improved surface reconstruction quality.

Contribution

The paper proposes a new loss function for INRs that ensures convergence to both occupancy and distance functions, with theoretical analysis and state-of-the-art experimental results.

Findings

Loss converges to proper occupancy and distance functions.

Theoretical analysis shows minimal surface perimeter bias.

Achieves state-of-the-art surface reconstructions.

Abstract

Representing surfaces as zero level sets of neural networks recently emerged as a powerful modeling paradigm, named Implicit Neural Representations (INRs), serving numerous downstream applications in geometric deep learning and 3D vision. Training INRs previously required choosing between occupancy and distance function representation and different losses with unknown limit behavior and/or bias. In this paper we draw inspiration from the theory of phase transitions of fluids and suggest a loss for training INRs that learns a density function that converges to a proper occupancy function, while its log transform converges to a distance function. Furthermore, we analyze the limit minimizer of this loss showing it satisfies the reconstruction constraints and has minimal surface perimeter, a desirable inductive bias for surface reconstruction. Training INRs with this new loss leads to state-of-the-art reconstructions on a standard benchmark.