An optimization-driven approach for computing geodesic paths on triangle meshes

Liu, Bangquan; Chen, Shuangmin; Xin, Shiqing; He, Ying; Liu, Zhen; Zhao, Jieyu

doi:10.1016/j.cad.2017.05.022

Cited by 18 publications

(8 citation statements)

References 34 publications

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“…The slow performance of iterative relaxation might be attributed to the fact that the amount of progress made on each iteration is not bounded away from zero ( Figure 5), whereas each FlipOut iteration decreases the size of a discrete set of possible states (Theorem 4.2). Acceleration via L-BFGS can help, but yields only approximate geodesics [Liu et al 2017a, Figure 1] and very similar performance to FlipOut (compare speedup relative to Martínez et al [2005] in Figure 18 with Liu et al [2017a, Table 2]). On the other hand, optimization-based schemes like these nicely support user-defined penalties like anisotropic metrics.…”

Section: Performance 9ms 2m Facesmentioning

confidence: 70%

“…Such methods are critical in applications which do not seek shortest paths, as explored in Section 6. Lagrangian methods represent curves via vertices that can move freely along the surface or in R 3 [Hass and Scott 1994;Martínez et al 2005;Xin and Wang 2007;Appleboim et al 2009;Xin et al 2011;Han et al 2017;Liu et al 2017a;Remešíková et al 2019;, whereas Eulerian methods encode curves as level sets of a scalar function [Sethian 1989;Zhang et al 2010]. Many of these methods seek to numerically approximate geodesics, whereas our method considers a discrete configuration space (the flip graph of a triangulation) that includes the exact solution.…”

Section: Curve Shorteningmentioning

confidence: 99%

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You can find geodesic paths in triangle meshes by just flipping edges

Sharp

Crane

2020

ACM Trans. Graph.

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This paper introduces a new approach to computing geodesics on polyhedral surfaces-the basic idea is to iteratively perform edge flips, in the same spirit as the classic Delaunay flip algorithm. This process also produces a triangulation conforming to the output geodesics, which is immediately useful for tasks in geometry processing and numerical simulation. More precisely, our FlipOut algorithm transforms a given sequence of edges into a locally shortest geodesic while avoiding self-crossings (formally: it finds a geodesic in the same isotopy class). The algorithm is guaranteed to terminate in a finite number of operations; practical runtimes are on the order of a few milliseconds, even for meshes with millions of triangles. The same approach is easily applied to curves beyond simple paths, including closed loops, curve networks, and multiply-covered curves. We explore how the method facilitates tasks such as straightening cuts and segmentation boundaries, computing geodesic Bézier curves, extending the notion of constrained Delaunay triangulations (CDT) to curved surfaces, and providing accurate boundary conditions for partial differential equations (PDEs). Evaluation on challenging datasets such as Thingi10k indicates that the method is both robust and efficient, even for low-quality triangulations. CCS Concepts: • Computing methodologies → Shape modeling.

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Section: Performance 9ms 2m Facesmentioning

confidence: 70%

Section: Curve Shorteningmentioning

confidence: 99%

You can find geodesic paths in triangle meshes by just flipping edges

Sharp

Crane

2020

ACM Trans. Graph.

View full text Add to dashboard Cite

show abstract

“…However, there is a price to pay for this. In particular, it must be observed that straightest geodesics do not converge to geodesic paths on smooth surfaces under mesh refinement [53,58], and that locally straightest distances do not satisfy the triangular inequality, therefore the straightest geodesic distance is not a metric [61]. Last but not least, we remind the reader that these local geodesic criteria apply only to the Euclidean embedding, and do not extend to alternative metrics (e.g., weighted or anisotropic), thus limiting the applicability of the algorithms that are based on them.…”

Section: 21mentioning

confidence: 99%

A Survey of Algorithms for Geodesic Paths and Distances

Crane¹,

Livesu²,

Puppo³

et al. 2020

Preprint

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Numerical computation of shortest paths or geodesics on curved domains, as well as the associated geodesic distance, arises in a broad range of applications across digital geometry processing, scientific computing, computer graphics, and computer vision. Relative to Euclidean distance computation, these tasks are complicated by the influence of curvature on the behavior of shortest paths, as well as the fact that the representation of the domain may itself be approximate. In spite of the difficulty of this problem, recent literature has developed a wide variety of sophisticated methods that enable rapid queries of geodesic information, even on relatively large models. This survey reviews the major categories of approaches to the computation of geodesic paths and distances, highlighting common themes and opportunities for future improvement.

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“…Let us assume the mesh with obstacles have different density with the other base mesh. We use the optimization-driven approach [27] to compute geodesic paths on triangle meshes with nonuniform density setting. Let us assume V (1) and V (1) are the endpoints of the edge , as shown in Figure 1; the initial path (shown in red) through the face sequence Γ can be described by ( , 1 , 2 , .…”

Section: Geodesic Path With Non-uniform Densitymentioning

confidence: 99%

An Evacuation Route Model of Crowd Based on Emotion and Geodesic

Liu

Sun

et al. 2018

Mathematical Problems in Engineering

Self Cite

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Making unconventional emergent plan for dense crowd is one of the critical issues of evacuation simulations. In order to make the behavior of crowd more believable, we present a real-time evacuation route approach based on emotion and geodesic under the influence of individual emotion and multi-hazard circumstances. The proposed emotion model can reflect the dynamic process of individual in group on three factors: individual emotion, perilous field, and crowd emotion. Specifically, we first convert the evacuation scene to Delaunay triangulation representations. Then, we use the optimization-driven geodesic approach to calculate the best evacuation path with user-specified geometric constraints, such as crowd density, obstacle information, and perilous field. Finally, the Smooth Particle Hydrodynamics method is used for local avoidance of collisions with nearby agents in real-time simulation. Extensive experimental results show that our algorithm is efficient and well suited for real-time simulations of crowd evacuation.

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An optimization-driven approach for computing geodesic paths on triangle meshes

Cited by 18 publications

References 34 publications

You can find geodesic paths in triangle meshes by just flipping edges

You can find geodesic paths in triangle meshes by just flipping edges

A Survey of Algorithms for Geodesic Paths and Distances

An Evacuation Route Model of Crowd Based on Emotion and Geodesic

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