Generalized volumetric foliation from inverted viscous flow

Cohen, David B.; Ben‐Chen, Mirela

doi:10.1016/j.cag.2019.05.015

Cited by 9 publications

(2 citation statements)

References 25 publications

(38 reference statements)

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“…Since its introduction in 2016, we are not aware of any method that internally uses it to compute volume maps. Another foliation algorithm for volumetric mapping to spheres was presented in [CBC19]. However, in this case the approach is numerical, hence it does not provide guarantees of correctness.…”

Section: Existing Mapping Methodologiesmentioning

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: Existing Mapping Methodologiesmentioning

confidence: 99%

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

Cherchi

Livesu

2023

Computer Graphics Forum

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“…Their algorithm requires the domain to be a topological ball whose tetrahedral mesh is bishellable. Cohen and Ben-Chen [2019] describe an alternative method to compute foliations of more-general volumetric domains using a Hele-Shaw flow along a potential function from a Möbius inversion of the domain boundary to a sphere. Unlike these methods that decompose the domain into surfaces, our method does end-to-end optimization of a mapping over an entire volume at once.…”

Section: Related Workmentioning

confidence: 99%

Symmetric Volume Maps: Order-Invariant Volumetric Mesh Correspondence with Free Boundary

Abulnaga¹,

Stein²,

Golland³

et al. 2022

Preprint

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Although shape correspondence is a central problem in geometry processing, most methods for this task apply only to two-dimensional surfaces. The neglected task of volumetric correspondence-a natural extension relevant to shapes extracted from simulation, medical imaging, volume rendering, and even improving surface maps of boundary representations-presents unique challenges that do not appear in the two-dimensional case. In this work, we propose a method for mapping between volumes represented as tetrahedral meshes. Our formulation minimizes a distortion energy designed to extract maps symmetrically, i.e., without dependence on the ordering of the source and target domains. We accompany our method with theoretical discussion describing the consequences of this symmetry assumption, leading us to select a symmetrized ARAP energy that favors isometric correspondences. Our final formulation optimizes for near-isometry while matching the boundary. We demonstrate our method on a diverse geometric dataset, producing low-distortion matchings that align to the boundary.

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Foreword to the Special Section on Shape Modelling International 2019

Chaine

Patanè²

2019

Computers & Graphics

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Generalized volumetric foliation from inverted viscous flow

Cited by 9 publications

References 25 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

Symmetric Volume Maps: Order-Invariant Volumetric Mesh Correspondence with Free Boundary

Foreword to the Special Section on Shape Modelling International 2019

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