Blended barycentric coordinates

Anisimov, Dmitry; Panozzo, Daniele; Hormann, Kai

doi:10.1016/j.cagd.2017.02.007

Cited by 11 publications

(6 citation statements)

References 16 publications

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“…Positive coordinates for arbitrary concave polygons include positive mean value [LKCL07], positive Gordon–Wixom coordinates [MLS11], and power coordinates [BLTD16], but they are not smooth. Blended barycentric coordinates [APH17] overcome this limitation, but they depend on an initial triangulation of the domain. Coordinates that are at least C 1 and non‐negative for arbitrary polygons include harmonic [JMD*07], maximum entropy [HS08], maximum likelihood [CDH23], positive and smooth Gordon–Wixom [WLMD19], iterative [DCH20], and local barycentric coordinates [ZDL*14; TDZ19], but they do not have a closed form and must be approximated by some numerical procedure.…”

Section: Preliminaries and Notationmentioning

confidence: 99%

A Survey on Cage‐based Deformation of 3D Models

Ströter,

Thiery,

Hormann

et al. 2024

Computer Graphics Forum

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Interactive deformation via control handles is essential in computer graphics for the modeling of 3D geometry. Deformation control structures include lattices for free‐form deformation and skeletons for character articulation, but this report focuses on cage‐based deformation. Cages for deformation control are coarse polygonal meshes that encase the to‐be‐deformed geometry, enabling high‐resolution deformation. Cage‐based deformation enables users to quickly manipulate 3D geometry by deforming the cage. Due to their utility, cage‐based deformation techniques increasingly appear in many geometry modeling applications. For this reason, the computer graphics community has invested a great deal of effort in the past decade and beyond into improving automatic cage generation and cage‐based deformation. Recent advances have significantly extended the practical capabilities of cage‐based deformation methods. As a result, there is a large body of research on cage‐based deformation. In this report, we provide a comprehensive overview of the current state of the art in cage‐based deformation of 3D geometry. We discuss current methods in terms of deformation quality, practicality, and precomputation demands. In addition, we highlight potential future research directions that overcome current issues and extend the set of practical applications. In conjunction with this survey, we publish an application to unify the most relevant deformation methods. Our report is intended for computer graphics researchers, developers of interactive geometry modeling applications, and 3D modeling and character animation artists.

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Section: Preliminaries and Notationmentioning

confidence: 99%

A Survey on Cage‐based Deformation of 3D Models

Ströter,

Thiery,

Hormann

et al. 2024

Computer Graphics Forum

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“…The subdivision schemes proposed by Anisimov et al (2016) can be applied to refine a coarse result from Zhang et al (2014) while maintaining the locality. Recently, Anisimov et al (2017) also presented a new closed-form construction of generalized barycentric coordinates that are non-negative, smooth, and locally supported.…”

Section: Generalized Barycentric Coordinatesmentioning

confidence: 99%

“…This can lead to some undesirable consequence in practice, including lack of local control for shape deformation, and excessive storage for the coordinates on densely discretized domains. To address these issues, some recent studies have developed GBC schemes with local support for the coordinate functions Landreneau and Schaefer (2010); Zhang et al (2014); Anisimov et al (2017). In particular, the Local Barycentric Coordinates (LBC) proposed in Zhang et al (2014) are computed by minimizing the total variation of the coordinate functions to induce their locality, subject to a set of constraints that ensure desired properties such as partition of unity, reproduction, and non-negativity.…”

Section: Introductionmentioning

confidence: 99%

A fast numerical solver for local barycentric coordinates

Tao

Deng

Zhang

2019

Computer Aided Geometric Design

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The local barycentric coordinates (LBC), proposed in Zhang et al. (2014), demonstrate good locality and can be used for local control on function value interpolation and shape deformation. However, it has no closedform expression and must be computed by solving an optimization problem, which can be time-consuming especially for high-resolution models. In this paper, we propose a new technique to compute LBC efficiently. The new solver is developed based on two key insights. First, we prove that the non-negativity constraints in the original LBC formulation is not necessary, and can be removed without affecting the solution of the optimization problem. Furthermore, the removal of this constraint allows us to reformulate the computation of LBC as a convex constrained optimization for its gradients, followed by a fast integration to recover the coordinate values. The reformulated gradient optimization problem can be solved using ADMM, where each step is trivially parallelizable and does not involve global linear system solving, making it much more scalable and efficient than the original LBC solver. Numerical experiments verify the effectiveness of our technique on a large variety of models.

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“…[LS08] modified mean value coordinates to control derivatives at interpolated points. [APH17] blended mean value coordinates over Delaunay triangulation of the control polygon to generate local, closed form coordinates.…”

Section: Related Workmentioning

confidence: 99%

A Family of Barycentric Coordinates for Co‐Dimension 1 Manifolds with Simplicial Facets

Yan

Schaefer

2019

Computer Graphics Forum

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We construct a family of barycentric coordinates for 2D shapes including non‐convex shapes, shapes with boundaries, and skeletons. Furthermore, we extend these coordinates to 3D and arbitrary dimension. Our approach modifies the construction of the Floater‐Hormann‐Kós family of barycentric coordinates for 2D convex shapes. We show why such coordinates are restricted to convex shapes and show how to modify these coordinates to extend to discrete manifolds of co‐dimension 1 whose boundaries are composed of simplicial facets. Our coordinates are well‐defined everywhere (no poles) and easy to evaluate. While our construction is widely applicable to many domains, we show several examples related to image and mesh deformation.

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Blended barycentric coordinates

Cited by 11 publications

References 16 publications

A Survey on Cage‐based Deformation of 3D Models

A Survey on Cage‐based Deformation of 3D Models

A fast numerical solver for local barycentric coordinates

A Family of Barycentric Coordinates for Co‐Dimension 1 Manifolds with Simplicial Facets

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