NeuroGF: A Neural Representation for Fast Geodesic Distance and Path Queries

Geodesics are essential in many geometry processing applications. However, traditional algorithms for computing geodesic distances and paths on 3D mesh models are often inefficient and slow. This makes them impractical for scenarios that require extensive querying of arbitrary point-to-point geodesics. Although neural implicit representations have emerged as a popular way of representing 3D shape geometries, there is still no research on representing geodesics with deep implicit functions. To bridge this gap, this paper presents the first attempt to represent geodesics on 3D mesh models using neural implicit functions. Specifically, we introduce neural geodesic fields (NeuroGFs), which are learned to represent the all-pairs geodesics of a given mesh. By using NeuroGFs, we can efficiently and accurately answer queries of arbitrary point-to-point geodesic distances and paths, overcoming the limitations of traditional algorithms. Evaluations on common 3D models show that NeuroGFs exhibit exceptional performance in solving the single-source all-destination (SSAD) and point-to-point geodesics, and achieve high accuracy consistently. Moreover, NeuroGFs offer the unique advantage of encoding both 3D geometry and geodesics in a unified representation. Code is made available at https://github.com/keeganhk/NeuroGF/tree/master

Fast construction of discrete geodesic graph as generalized discrete geodesic approximation algorithms

Discrete geodesic graph (DGG) is an emerging technique for computing geodesic distances and paths on polyhedral surfaces. DGG has many advantages, such as information reuse, accuracy control, ease of parallelization, good scalability, and guaranteed metric, fast linear time SSAD search. To construct DGG, the existing method by Wang et al. firstly finds a large graph with the user-specified accuracy, and then iteratively prunes the redundant edges from it. Finally, it adds pseudo vertices and edges to the graph for the poorly sampled regions. Computational results show that for real-world meshes, more than 80\% of the candidate edges do not contribute to the final graph and are hereby deleted. Moreover, for anisotropic meshes, a large amount of pseudo vertices and edges (up to 40 times the original vertex number) in order to maintain the graph quality, which significantly increases the graph complexity by up to 400x larger. As a result, the existing indirect approach is conservative and far from optimal. This paper aims at improving the performance of DGG construction on general meshes models. Towards this goal, we develop a novel accuracy aware window propagation scheme, allowing us to compute DGG edges in a direct manner. We prove that the new method greatly improving the time complexity of Wang et al's approach. Our method is memory efficient and it scales very well. Through extensive evaluation, we demonstrate that our method produces DGGs with comparable or better accuracy than the existing method, but running up to 2 orders of magnitude faster on common meshes with millions of vertices.In sharp contrast to Wang et al's algorithm, our algorithm can handle anisotropic meshes well without adding pseudo vertices or edges, thanks to its better understanding of the theoretical underpinning of Discrete Geodesic Graph. This effectively generalize DGG to other highly anisotropic and non-uniform cases of polyhedron with a far better time complexity. This novel improvements also brings DGG to be a better performing algorithm than all existing practical approaches - such as SVG or heat method, not only in term of its SSAD solving time, but also in its initial construction time.Bachelor of Engineering (Computer Science

An accuracy controllable and memory efficient method for computing high-quality geodesic distances on triangle meshes

This paper presents a new method for computing approximate geodesic distances and paths on triangle meshes. Our method combines two state-of-the-art discrete geodesic methods, which are discrete geodesic graphs (DGG) and vertex-oriented triangle propagation (VTP), so that it allows the user to specify the desired accuracy using a single parameter ɛ. The method, called DGG-VTP, extends the conventional window propagation framework by monitoring the accuracy of the computed distances so that propagation can terminate immediately when the desired accuracy is reached. It is worth noting that for robustness consideration, tiny windows with length less than a threshold (usually, between 10−7 and 10−6) are discarded in the implementation of the existing exact algorithms, such as the Mitchel–Mount–Papadimitriou (MMP) algorithm, the Chen–Han (CH) algorithm and their many variants. By setting the accuracy parameter ɛ∈[10−7,10−6], our method can produce results with comparable accuracy to VTP, while being 3–40 times faster and consuming much less memory. Furthermore, the performance of our method is less sensitive to mesh tessellation than what VTP does. Our method empirically produces [Formula presented] windows and scales well to deal with large-scale models. Though the parameter ɛ in DGG-VTP is not a guaranteed error bound, it acts as an intuitive guide for the user to set the desired accuracy. Extensive evaluations demonstrate the effectiveness of our accuracy control: given a parameter ɛ∈[10−7,10−4], 99% of the computed distances have error less than the accuracy parameter. The features of predicable accuracy and computational efficiency distinguish DGG-VTP from the existing approximation methods, and make it an alternative to exact methods in computing accurate geodesic distances on large-scale mesh models. We also develop a parallel version of DGG-VTP on multi-core CPUs, which runs up to 60× faster than the existing parallel VTP algorithm with comparable accuracy under single floating point precision setting. The source code is available at https://github.com/GeodesicGraph/DGG-VTP.Ministry of Education (MOE)We thank the anonymous reviewers for their constructive comments. This project was partially supported by the Ministry of Education, Singapore, under its Academic Research Fund Tier 1 (RG20/20) and Tier 2 (MOE-T2EP20220-0005)