Compression and Querying of Arbitrary Geodesic Distances

Aiello, R.; Banterle, Francesco; Pietroni, Nico; Malomo, Luigi; Cignoni, Paolo; Scopigno, Roberto

doi:10.1007/978-3-319-23231-7_26

Cited by 4 publications

(3 citation statements)

References 22 publications

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“…For most cases with mediocre resolution they report slightly better results for the mean curvature than for Belkin et al's discretization, becoming (mostly) better with increasing resolutions. Because of the only small advantage at practical resolutions and because it is non-trivial and often expensive to compute the geodesic distance between points on general triangular meshes [105], we have not yet attempted its implementation.…”

Section: Methods Dmentioning

confidence: 99%

On the bending algorithms for soft objects in flows

Guckenberger

Schraml

Chen

et al. 2016

Computer Physics Communications

112

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International audienceOne of the most challenging aspects in the accurate simulation of three-dimensional soft objects such as vesicles or biological cells is the computation of membrane bending forces. The origin of this difficulty stems from the need to numerically evaluate a fourth order derivative on the discretized surface geometry. Here we investigate six different algorithms to compute membrane bending forces, including regularly used methods as well as novel ones. All are based on the same physical model (due to Canham and Helfrich) and start from a surface discretization with flat triangles. At the same time, they differ substantially in their numerical approach. We start by comparing the numerically obtained mean curvature, the Laplace-Beltrami operator of the mean curvature and finally the surface force density to analytical results for the discocyte resting shape of a red blood cell. We find that none of the considered algorithms converges to zero error at all nodes and that for some algorithms the error even diverges. There is furthermore a pronounced influence of the mesh structure: Discretizations with more irregular triangles and node connectivity present serious difficulties for most investigated methods. To assess the behavior of the algorithms in a realistic physical application, we investigate the deformation of an initially spherical capsule in a linear shear flow at small Reynolds numbers. To exclude any influence of the flow solver, two conceptually very different solvers are employed: the Lattice-Boltzmann and the Boundary Integral Method. Despite the largely different quality of the bending algorithms when applied to the static red blood cell, we find that in the actual flow situation most algorithms give consistent results for both hydrodynamic solvers. Even so, a short review of earlier works reveals a wide scattering of reported results for, e.g., the Taylor deformation parameter. Besides the presented application to biofluidic systems, the investigated algorithms are also of high relevance to the computer graphics and numerical mathematics communities

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Section: Methods Dmentioning

confidence: 99%

On the bending algorithms for soft objects in flows

Guckenberger

Schraml

Chen

et al. 2016

Computer Physics Communications

112

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“…Aiello et al [2] adopt a hierarchical approach to support the computation of geodesic distances (not explicit paths) between any pair of vertices. Similarly to SVG and DGG, they aim at building a graph that contains many shortcuts between far vertices.…”

Section: Graph-based Methodsmentioning

confidence: 99%

“…The distance between a pair of vertices is found by a Dijkstra search that is pruned by visiting the graph in a hierarchical manner: search starts from the patch containing the source at the lowest level of the hierarchy, until it meet its boundary; then it traverses either its sibling patch beyond the boundary, or shortcuts from patches in the upper levels, until a patch containing the target is found; the path is concluded by moving down the hierarchy until the patch at the lowest level, which contains the target, is found. See [2] for details. The authors report quite slow preprocessing times for building the graph, but fast performances and empirically good approximations during queries.…”

Section: Graph-based Methodsmentioning

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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