Theoretical and Empirical Analysis of a Fast Algorithm for Extracting Polygons from Signed Distance Bounds

TL;DR

This paper introduces a fast, theoretically grounded algorithm that converts signed distance bounds into polygon meshes, combining sphere tracing with traditional polygonization methods, suitable for 3D shape modeling and compression.

Contribution

It presents a novel asymptotically fast algorithm for polygonizing signed distance bounds, with theoretical analysis and practical testing on neural network-generated shapes.

Findings

Algorithm has $O(N^2\log N)$ complexity for $N^3$ grid cells.

Effective on primitive shapes and neural network generated bounds.

Offers a simple, portable method for shape modeling and compression.

Abstract

Recently there has been renewed interest in signed distance bound representations due to their unique properties for 3D shape modelling. This is especially the case for deep learning-based bounds. However, it is beneficial to work with polygons in most computer-graphics applications. Thus, in this paper we introduce and investigate an asymptotically fast method for transforming signed distance bounds into polygon meshes. This is achieved by combining the principles of sphere tracing (or ray marching) with traditional polygonization techniques, such as Marching Cubes. We provide theoretical and experimental evidence that this approach is of the $O(N^2\log N)$ computational complexity for a polygonization grid with $N^3$ cells. The algorithm is tested on both a set of primitive shapes as well as signed distance bounds generated from point clouds by machine learning (and represented as neural networks). Given its speed, implementation simplicity and portability, we argue that it could prove useful during the modelling stage as well as in shape compression for storage. The code is available here: https://github.com/nenadmarkus/gridhopping