StEik: Stabilizing the Optimization of Neural Signed Distance Functions and Finer Shape Representation

TL;DR

This paper introduces StEik, a new regularization and network design for neural implicit shape representations that stabilize optimization, enabling finer detail capture and improved topology in reconstructed shapes.

Contribution

It provides a PDE-based analysis of eikonal loss instability and proposes a novel regularization and quadratic-layer network to enhance shape detail and stability.

Findings

Enhanced shape detail and topology in reconstructions

Stability in optimization through PDE-inspired regularization

Superior performance on benchmark datasets

Abstract

We present new insights and a novel paradigm (StEik) for learning implicit neural representations (INR) of shapes. In particular, we shed light on the popular eikonal loss used for imposing a signed distance function constraint in INR. We show analytically that as the representation power of the network increases, the optimization approaches a partial differential equation (PDE) in the continuum limit that is unstable. We show that this instability can manifest in existing network optimization, leading to irregularities in the reconstructed surface and/or convergence to sub-optimal local minima, and thus fails to capture fine geometric and topological structure. We show analytically how other terms added to the loss, currently used in the literature for other purposes, can actually eliminate these instabilities. However, such terms can over-regularize the surface, preventing the representation of fine shape detail. Based on a similar PDE theory for the continuum limit, we introduce a new regularization term that still counteracts the eikonal instability but without over-regularizing. Furthermore, since stability is now guaranteed in the continuum limit, this stabilization also allows for considering new network structures that are able to represent finer shape detail. We introduce such a structure based on quadratic layers. Experiments on multiple benchmark data sets show that our new regularization and network are able to capture more precise shape details and more accurate topology than existing state-of-the-art.