In this article, 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.

In this paper, we propose a parallel and scalable approach for geodesic distance computation on triangle meshes. Our key observation is that the recovery of geodesic distance with the heat method [1] can be reformulated as optimization of its gradients subject to integrability, which can be solved using an efficient first-order method that requires no linear system solving and converges quickly. Afterward, the geodesic distance is efficiently recovered by parallel integration of the optimized gradients in breadth-first order. Moreover, we employ a similar breadth-first strategy to derive a parallel Gauss-Seidel solver for the diffusion step in the heat method. To further lower the memory consumption from gradient optimization on faces, we also propose a formulation that optimizes the projected gradients on edges, which reduces the memory footprint by about 50 percent. Our approach is trivially parallelizable, with a low memory footprint that grows linearly with respect to the model size. This makes it particularly suitable for handling large models. Experimental results show that it can efficiently compute geodesic distance on meshes with more than 200 million vertices on a desktop PC with 128 GB RAM, outperforming the original heat method and other state-of-the-art geodesic distance solvers.

This paper develops a new method for constructing Discrete Geodesic Graph (DGG)—an undirected, sparse graph for computing discrete geodesic distances and paths on triangle meshes. Based on a novel accuracy aware window propagation scheme, our method is able to compute the graph edges in a direct and efficient manner. Given a triangle mesh with n vertices and a user-specified accuracy parameter ɛ, our method produces a DGG with O ( n \√ɛ) edges in empirical O ( n \ɛ 0.75 log 1\ɛ) time, which greatly improves the time complexity O ( n \ɛ log 1\ɛ) of the existing method. Extensive evaluation on a large-scale 3D shape repository shows that our method is efficient and can produce high-quality geodesic distances with predictable accuracy and guaranteed true distance metric. In particular, our method has a great advantage over the existing approximate methods on meshes with high degree of anisotropy. The source code is available at https://github.com/GeodesicGraph.

This paper presents Poisson vector graphics (PVG), an extension of the popular diffusion curves (DC), for generating smooth-shaded images. Armed with two new types of primitives, called Poisson curves and Poisson regions, PVG can easily produce photorealistic effects such as specular highlights, core shadows, translucency and halos. Within the PVG framework, the users specify color as the Dirichlet boundary condition of diffusion curves and control tone by offsetting the Laplacian of colors, where both controls are simply done by mouse click and slider dragging. PVG distinguishes itself from other diffusion based vector graphics for 3 unique features: 1) explicit separation of colors and tones, which follows the basic drawing principle and eases editing; 2) native support of seamless cloning in the sense that PCs and PRs can automatically fit into the target background; and 3) allowed intersecting primitives (except for DC-DC intersection) so that users can create layers. Through extensive experiments and a preliminary user study, we demonstrate that PVG is a simple yet powerful authoring tool that can produce photo-realistic vector graphics from scratch.

We present a new graph-based method, called discrete geodesic graph (DGG), to compute discrete geodesics in a divide-and-conquer manner. Let M be a manifold triangle mesh with n vertices and 𝜀 > 0 the given accuracy parameter. Assume the vertices are uniformly distributed on the input mesh. We show that the DGG associated to M has 𝑂 ⁡ ( 𝑛 √ 𝜀 ) edges and the shortest path distances on the graph approximate geodesic distances on M with relative error 𝑂 ⁡ ( 𝜀 ) . Computational results show that the actual error is less than 0.6 ε on common models. Taking advantage of DGG's unique features, we develop a DGG-tailored label-correcting algorithm that computes geodesic distances in empirically linear time. With DGG, we can guarantee the computed distances are true distance metrics, which is highly desired in many applications. We observe that DGG significantly outperforms saddle vertex graph (SVG) – another graph based method for discrete geodesics – in terms of graph size, accuracy control and runtime performance.