Recognizing Shrinkable Complexes Is NP-Complete

Attali, Dominique; Devillers, Olivier; Glisse, Marc; Lazard, Sylvain

doi:10.1007/978-3-662-44777-2_7

Cited by 2 publications

(2 citation statements)

References 18 publications

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“…Despite the main ingredients of this algorithm (edge collapse, vertex split) are well defined also for tetrahedral meshes, extending Progressive Embeddings to the generation of volume maps seems overly complex. In fact, differently from triangle meshes, tetrahedral meshes cannot always be collapsed through a sequence of edge collapses, and even deciding whether a given mesh is collapsible is an NP‐complete problem [Tan16, MF08, ADGL16]. Barycentric subdivision ensures collapsibility [AB20] but, even if a valid collapsing sequence of a refined mesh is guaranteed to exist, the size of the search space is exponential w.r.t.…”

Section: Field Overviewmentioning

confidence: 99%

VOLMAP: a Large Scale Benchmark for Volume Mappings to Simple Base Domains

Cherchi

Livesu

2023

Computer Graphics Forum

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Correspondences between geometric domains (mappings) are ubiquitous in computer graphics and engineering, both for a variety of downstream applications and as core building blocks for higher level algorithms. In particular, mapping a shape to a convex or star‐shaped domain with simple geometry is a fundamental module in existing pipelines for mesh generation, solid texturing, generation of shape correspondences, advanced manufacturing etc. For the case of surfaces, computing such a mapping with guarantees of injectivity is a solved problem. Conversely, robust algorithms for the generation of injective volume mappings to simple polytopes are yet to be found, making this a fundamental open problem in volume mesh processing. VOLMAP is a large scale benchmark aimed to support ongoing research in volume mapping algorithms. The dataset contains 4.7K tetrahedral meshes, whose boundary vertices are mapped to a variety of simple domains, either convex or star‐shaped. This data constitutes the input for candidate algorithms, which are then required to position interior vertices in the domain to obtain a volume map. Overall, this yields more than 22K alternative test cases. VOLMAP also comprises tools to process this data, analyze the resulting maps, and extend the dataset with new meshes, boundary maps and base domains. This article provides a brief overview of the field, discussing its importance and the lack of effective techniques. We then introduce both the dataset and its major features. An example of comparative analysis between two existing methods is also present.

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Section: Field Overviewmentioning

confidence: 99%

VOLMAP: a Large Scale Benchmark for Volume Mappings to Simple Base Domains

Cherchi

Livesu

2023

Computer Graphics Forum

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“…The algorithm is inspired by the progressive meshes concept [14], and is based on the ability to deconstruct the topology of a triangle mesh by an ordered sequence of edge collapses, reconstructing the same mesh in another embedding with a sequence of vertex splits in the opposite order. Also this approach does not extend to 3d, the reason being twofold: (i) simplicial complexes in dimensions d ≥ 3 may not be fully collapsible with a sequence of edge collapses, and even deciding whether a tetrahedral mesh is collapsible is NP Complete [15], [16], [17]. Theory says that after a finite set of barycentric subdivisions any simplicial complex becomes collapsible [18], but still one should navigate the exponential space of all possible collapsing sequences to find a valid solution.…”

Section: Introductionmentioning

confidence: 99%

Mapping Surfaces with Earcut

Livesu

2020

Preprint

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Mapping a shape to some parametric domain is a fundamental tool in graphics and scientific computing. In practice, a map between two shapes is commonly represented by two meshes with same connectivity and different embedding. The standard approach is to input a meshing of one of the two domains plus a function that projects its boundary to the other domain, and then solve for the position of the interior vertices. Inspired by basic principles in mesh generation, in this paper we present the reader a novel point of view on mesh parameterization: we consider connectivity as an additional unknown, and assume that our inputs are just two boundaries that enclose the domains we want to connect. We compute the map by simultaneously growing the same mesh inside both shapes.This change in perspective allows us to recast the parameterization problem as a mesh generation problem, granting access to a wide set of mature tools that are typically not used in this setting. Our practical outcome is a provably robust yet trivial to implement algorithm that maps any planar shape with simple topology to a homotopic domain that is weakly visible from an inner convex kernel. Furthermore, we speculate on a possible extension of the proposed ideas to volumetric maps, listing the major challenges that arise. Differently from prior methodsfor which we show that a volumetric extension is not possible -our analysis leaves us reasoneable hopes that the robust generation of volumetric maps via compatible mesh generation could be obtained in the future.

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Recognizing Shrinkable Complexes Is NP-Complete

Cited by 2 publications

References 18 publications

VOLMAP: a Large Scale Benchmark for Volume Mappings to Simple Base Domains

VOLMAP: a Large Scale Benchmark for Volume Mappings to Simple Base Domains

Mapping Surfaces with Earcut

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