Topology-driven Surface Mappings with Robust Feature Alignment

Garner, Corey; Jin, Miao; Gu, Xianfeng; Qin, Hong

doi:10.1109/vis.2005.107

Cited by 8 publications

(2 citation statements)

References 39 publications

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“…A successful strategy to further enlarge the family of domains that can be considered is map composition. Methods in this category internally use the Tutte embedding, generating mappings to strictly convex domains that are then combined to obtain more general mappings to manifolds of arbitrary genus [Liv20c, LBG*08, GJGQ05, KS04, SAPH04], grid spaces for quad remeshing [Liv23], and even non‐convex planar polygons that may also partially overlap [WZ14]. Being internally based on Tutte, all these methods inherit the same theoretical guarantees of correctness.…”

Section: Related Workmentioning

confidence: 99%

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: Related Workmentioning

confidence: 99%

Advancing Front Surface Mapping

Livesu

2024

Computer Graphics Forum

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“…If there is only one singularity p 0 , the cut graph can be constructed using a set of canonical homology basis passing through the singularity as introduced in [Carner et al 2005]. The cut graph has one node and 2g edges.…”

Section: Update Vertex Radiimentioning

confidence: 99%

Manifold splines with single extraordinary point

Jin

et al. 2007

Proceedings of the 2007 ACM Symposium on Solid and Physical Modeling

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This paper develops a novel computational technique to define and construct powerful manifold splines with only one singular point by employing the rigorous mathematical theory of Ricci flow. The central idea and new computational paradigm of manifold splines are to systematically extend the algorithmic pipeline of spline surface construction from any planar domain to arbitrary topology. As a result, manifold splines can unify planar spline representations as their special cases. Despite their earlier success, the existing manifold spline framework is plagued by the topology-dependent, large number of singular points (i.e., |2g − 2| for any genus-g surface), where the analysis of surface behaviors such as continuity remains extremely difficult. The unique theoretical contribution of this paper is that we devise new mathematical tools so that manifold splines can now be constructed with only one singular point, reaching their theoretic lower bound of singularity for real-world applications. Our new algorithm is founded upon the concept of discrete Ricci flow and associated techniques. First, Ricci flow is employed to compute a special metric of any manifold domain (serving as a parametric domain for manifold splines), such that the metric becomes flat everywhere except at one point. Then, the metric naturally induces an affine atlas covering the entire manifold except this singular point. Finally, manifold splines are defined over this affine atlas. The Ricci flow method is theoretically sound, and practically simple and efficient. We conduct various shape experiments and our new theoretical and algorithmic results alleviate the modeling difficulty of manifold splines, and hence, promising to promote the widespread use of manifold splines in surface and solid modeling, geometric design, and reverse engineering.

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Computing general geometric structures on surfaces using Ricci flow

Jin

Luo

2007

Computer-Aided Design

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Topology-driven Surface Mappings with Robust Feature Alignment

Cited by 8 publications

References 39 publications

Advancing Front Surface Mapping

Advancing Front Surface Mapping

Manifold splines with single extraordinary point

Computing general geometric structures on surfaces using Ricci flow

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