GeodesicEmbedding (GE): A High-Dimensional Embedding Approach for Fast Geodesic Distance Queries

Abstract

In this paper, we develop a novel method for fast geodesic distance queries. The key idea is to embed the mesh into a high-dimensional space, such that the Euclidean distance in the high-dimensional space can induce the geodesic distance in the original manifold surface. However, directly solving the high-dimensional embedding problem is not feasible due to the large number of variables and the fact that the embedding problem is highly nonlinear. We overcome the challenges with two novel ideas. First, instead of taking all vertices as variables, we embed only the saddle vertices, which greatly reduces the problem complexity. We then compute a local embedding for each non-saddle vertex. Second, to reduce the large approximation error resulting from the purely Euclidean embedding, we propose a cascaded optimization approach that repeatedly introduces additional embedding coordinates with a non-Euclidean function to reduce the approximation residual. Using the precomputation data, our approach can determine the geodesic distance between any two vertices in near-constant time. Computational testing results show that our method is more desirable than previous geodesic distance queries methods.

Learning the Geodesic Embedding with Graph Neural Networks

“…Although plenty of methods [Adikusuma et al 2020;Crane et al 2013b;Mitchell et al 1987;Ying et al 2013] have been proposed for computing singlesource-all-destinations geodesic distances, and some of them can even run empirically in linear time [Crane et al 2013b;Tao et al 2019;Ying et al 2013], it is still too expensive to leverage these methods for GDQs. Consequently, a few dedicated methods [Gotsman and Hormann 2022;Panozzo et al 2013;Xia et al 2021;Xin et al 2012] have been proposed to ensure that the computation complexity of GDQ is constant to meet the huge requirements of frequent GDQs in interactive applications.…”

“…However, these methods either have low accuracy or require long precomputation time, which severely limits their applicability to large-scale meshes. Specifically, in the precomputation stage, these methods typically need to compute the exact geodesic distances between a large number of vertex pairs [Gotsman and Hormann 2022;Panozzo et al 2013;Xia et al 2021;Xin et al 2012], incurring at least quadratic space and computation complexity; then some methods employ nonlinear optimization [Panozzo et al 2013;Rustamov et al 2009] like Metric Multidimensional Scaling (MDS) [Carroll and Arabie 1998] or cascaded optimization [Xia et al 2021] to embed the input mesh into a high-dimensional space and approximate the geodesic distance with Euclidean distance in the high-dimensional space, where the optimization process is also time-consuming and costs several minutes, up to hours, even for meshes with tens of thousands of vertices. Additionally, these methods are highly dependent on the quality of input meshes, severely limiting their ability to deal with noisy or incomplete meshes.…”

“…Our key idea is to train a graph neural network (GNN) to embed an input mesh into a high-dimensional feature space and compute the geodesic distance between any pair of vertices with the corresponding feature distance defined by a learnable mapping function. The high-dimensional feature space is referred to as a geodesic embedding of a mesh [Panozzo et al 2013;Xia et al 2021]. Therefore, our method can be regarded as learning the geodesic embedding with a GNN, instead of relying on costly optimization procedures [Panozzo et al 2013;Rustamov et al 2009;Xia et al 2021]; thus, we name our method as GeGnn.…”

“…The high-dimensional feature space is referred to as a geodesic embedding of a mesh [Panozzo et al 2013;Xia et al 2021]. Therefore, our method can be regarded as learning the geodesic embedding with a GNN, instead of relying on costly optimization procedures [Panozzo et al 2013;Rustamov et al 2009;Xia et al 2021]; thus, we name our method as GeGnn. After training, our GeGnn can predict the geodesic embedding by just one forward pass of the network on GPUs, which is significantly more efficient than previous optimization-based methods [Panozzo et al 2013;Rustamov et al 2009;Xia et al 2021].…”

“…Therefore, our method can be regarded as learning the geodesic embedding with a GNN, instead of relying on costly optimization procedures [Panozzo et al 2013;Rustamov et al 2009;Xia et al 2021]; thus, we name our method as GeGnn. After training, our GeGnn can predict the geodesic embedding by just one forward pass of the network on GPUs, which is significantly more efficient than previous optimization-based methods [Panozzo et al 2013;Rustamov et al 2009;Xia et al 2021]. Our GeGnn also learns shape priors for geodesic distances during the training process, which makes it robust to the quality of input meshes.…”