Multi-Class Boundary Extraction from Implicit Representations

TL;DR

This work lays the groundwork by introducing a 2D boundary extraction algorithm for the multi-class case focusing on topological consistency and water-tightness, which also allows for setting minimum detail restraint on the approximation.

Abstract

Surface extraction from implicit neural representations modelling a single class surface is a well-known task. However, there exist no surface extraction methods from an implicit representation of multiple classes that guarantee topological correctness and no holes. In this work, we lay the groundwork by introducing a 2D boundary extraction algorithm for the multi-class case focusing on topological consistency and water-tightness, which also allows for setting minimum detail restraint on the approximation. Finally, we evaluate our algorithm using geological modelling data, showcasing its adaptiveness and ability to honour complex topology.

Figures (7)

• Figure 1: The top part shows the output of a Neural Network for all in $\{ C_1, C_2, \dots, C_k \}$, where $k=14$, densely sampled across $\mathbb{R}^1$. The bottom figure displays the results of the 1D Root-Finder, i.e. the precise intervals where each class with the maximum probability exists.
• Figure 2: The graph depicts the behaviour of classes $C_1$, $C_2$, and $C_3$ on a 1D axis in the interval $[-4, 4]$. The domain transition occurs in the interval $[0, 2]$, which the 1D algorithm eventually localises using the topological bracketing method.
• Figure 3: Illustration of the gradient-based localisation approach for root finding. The figure demonstrates the scenario where $g(x)$ exhibits multiple domain transitions, with tangents plotted at $x_1$ and $x_2$. The regions $[x_1 - \delta, x_1]$ and $[x_2, x_2 + \delta]$ are shown in yellow, while the interval $[x_1, x_2]$ is in green, indicating the interval under consideration for root localisation. The plot captures a potential root classification since the signs of $g(x)$ change at $x_1$ and $x_2$, and not both projected intersections $x_1^*$ and $x_2^*$ lie in the green area.
• Figure 4: This figure showcases all possible permutations of the Polygonisable, Three Domains Meeting and Ambiguous cases of a 2D rectangle.
• Figure 5: Illustration of method imposing geometric criteria on a naive edge represented in yellow, which is trying to approximate the boundary in purple.