Galaxy Maps: Localized Foliations for Bijective Volumetric Mapping

Hinderink, Steffen; Campen, Marcel

doi:10.1145/3592410

Cited by 6 publications

(3 citation statements)

References 49 publications

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“…In the worst case the algorithm took two days to complete. These timings seem compatible with recent exact numerical approaches that suffer from similar issues with cascaded constructions [NCB23, HC23].…”

Section: Resultssupporting

confidence: 86%

“…Last but not least, an appealing property of AFM is its apparent compatibility with a volumetric extension. Provably injective volume mapping is a fundamental yet open problem in the literature ([Liv20b] §2), for which existing robust approaches fall short for their limited applicability [Ale23] or high computational cost and excessive amount of mesh refinement [HC23, NCB23]. Regarding AFM, the generation of the polar mesh in the initialization phase (Section 3.1) and the basic advancing moves by means of triangle splits and edge flips (Section 3.2) have a direct counterpart in tetrahedral meshes, with the only difference that in 3D there are two alternative flip operations to advance the front: a face flip move to conquer a tetrahedron having two faces on the front and an edge flip move to conquer a tetrahedron having three faces on the front (Figure 14).…”

Section: Discussionmentioning

confidence: 99%

“…Fo-liations yield maps that are non linear inside each triangle, which can be transformed into piece-wise linear maps only at the expense of massive mesh refinement, increasing mesh size even by orders of magnitude. A recent article, mostly focused on volumes, significantly improved the original algorithm [HC23], alleviating some of these limitations.…”

Section: Provably Robust Methodsmentioning

confidence: 99%

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Advancing Front Surface Mapping

Livesu

2024

Computer Graphics Forum

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We present Advancing Front Mapping (AFM), a novel algorithm for the computation of injective maps to simple planar domains. AFM is inspired by the advancing front meshing paradigm, which is here revisited to operate on two embeddings at once, becoming a tool for compatible mesh generation. AFM extends the capabilities of existing robust approaches, supporting a broader set of embeddings (star‐shaped polygons) with a direct approach, without resorting to intermediate constructions. Our method only relies on two topological operators (split and flip) and on the computation of segment intersections, thus permitting to compute a valid embedding without solving any numerical problem. AFM is therefore easy to implement, debug and deploy. This article is mainly focused on the presentation of the compatible advancing front idea and on the demonstration that the algorithm provably converges to an injective map. We also complement our theoretical analysis with an extensive practical validation, executing more than one billion advancing front moves on 36K mapping tasks.

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Section: Resultssupporting

confidence: 86%

Section: Discussionmentioning

confidence: 99%

Section: Provably Robust Methodsmentioning

confidence: 99%

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Advancing Front Surface Mapping

Livesu

2024

Computer Graphics Forum

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Expansion Cones: A Progressive Volumetric Mapping Framework

Nigolian,

Campen,

Bommes

2023

ACM Trans. Graph.

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Volumetric mapping is a ubiquitous and difficult problem in Geometry Processing and has been the subject of research in numerous and various directions. While several methods show encouraging results, the field still lacks a general approach with guarantees regarding map bijectivity. Through this work, we aim at opening the door to a new family of methods by providing a novel framework based on the concept of progressive expansion. Starting from an initial map of a tetrahedral mesh whose image may contain degeneracies but no inversions, we incrementally adjust vertex images to expand degenerate elements. By restricting movement to so-called expansion cones , it is done in such a way that the number of degenerate elements decreases in a strictly monotonic manner, without ever introducing any inversion. Adaptive local refinement of the mesh is performed to facilitate this process. We describe a prototype algorithm in the realm of this framework for the computation of maps from ball-topology tetrahedral meshes to convex or star-shaped domains. This algorithm is evaluated and compared to state-of-the-art methods, demonstrating its benefits in terms of bijectivity. We also discuss the associated cost in terms of sometimes significant mesh refinement to obtain the necessary degrees of freedom required for establishing a valid mapping. Our conclusions include that while this algorithm is only of limited immediate practical utility due to efficiency concerns, the general framework has the potential to inspire a range of novel methods improving on the efficiency aspect.

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Collapsing Embedded Cell Complexes for Safer Hexahedral Meshing

Brückler,

Campen

2023

ACM Trans. Graph.

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We present a set of operators to perform modifications, in particular collapses and splits, in volumetric cell complexes which are discretely embedded in a background mesh. Topological integrity and geometric embedding validity are carefully maintained. We apply these operators strategically to volumetric block decompositions, so-called T-meshes or base complexes, in the context of hexahedral mesh generation. This allows circumventing the expensive and unreliable global volumetric remapping step in the versatile meshing pipeline based on 3D integer-grid maps. In essence, we reduce this step to simpler local cube mapping problems, for which reliable solutions are available. As a consequence, the robustness of the mesh generation process is increased, especially when targeting coarse or block-structured hexahedral meshes. We furthermore extend this pipeline to support feature alignment constraints, and systematically respect these throughout, enabling the generation of meshes that align to points, curves, and surfaces of special interest, whether on the boundary or in the interior of the domain.

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Galaxy Maps: Localized Foliations for Bijective Volumetric Mapping

Cited by 6 publications

References 49 publications

Advancing Front Surface Mapping

Advancing Front Surface Mapping

Expansion Cones: A Progressive Volumetric Mapping Framework

Collapsing Embedded Cell Complexes for Safer Hexahedral Meshing

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