Poisson Coordinates

Li, Xianying; Hu, Shi‐Min

doi:10.1109/tvcg.2012.109

Cited by 36 publications

(3 citation statements)

References 43 publications

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“…They are positive inside the kernel of star‐shaped polygons, satisfy the Lagrange property, and are smooth everywhere in the plane, except at the vertices of the polygon, where they are only C 0 continuous. Other closed‐form GBC that are well‐defined for concave polygons are metric [MLD05; SM06], moving least squares [MS10], Poisson [LH13], cubic mean value [LJH13], and Gordon–Wixom [GW74; Bel06]. There even exists a whole family of coordinates that are well‐defined for degenerate polygons [YS19], but all these constructions can be negative inside the domain.…”

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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“…This family also includes mean value coordinates [Flo03, HF06], which come with the advantage of being well‐defined for non‐convex polygons and polyhedra, too [FKR05, JSW05]. Whole families of barycentric coordinates for non‐convex polygons and polyhedra were constructed by [BLTD16] and [YS19], but just like metric [MLD05], Poisson [LH13], and Gordon–Wixom coordinates [Bel06], they may take on negative values at certain v ∈ Ω. Some constructions guarantee the non‐negativity of the coordinates, but at the price of not depending smoothly on either v ∈ Ω [LKCOL07,MLS11] or the vertices v i [APH17].…”

Section: Introductionmentioning

confidence: 99%

Maximum Likelihood Coordinates

Chang

Deng

Hormann

2023

Computer Graphics Forum

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Any point inside a d‐dimensional simplex can be expressed in a unique way as a convex combination of the simplex's vertices, and the coefficients of this combination are called the barycentric coordinates of the point. The idea of barycentric coordinates extends to general polytopes with n vertices, but they are no longer unique if n > d+1. Several constructions of such generalized barycentric coordinates have been proposed, in particular for polygons and polyhedra, but most approaches cannot guarantee the non‐negativity of the coordinates, which is important for applications like image warping and mesh deformation. We present a novel construction of non‐negative and smooth generalized barycentric coordinates for arbitrary simple polygons, which extends to higher dimensions and can include isolated interior points. Our approach is inspired by maximum entropy coordinates, as it also uses a statistical model to define coordinates for convex polygons, but our generalization to non‐convex shapes is different and based instead on the project‐and‐smooth idea of iterative coordinates. We show that our coordinates and their gradients can be evaluated efficiently and provide several examples that illustrate their advantages over previous constructions.

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“…Moreover, we compare the approximation qualities (precision/convergence) of these partition-of-unity, generalized barycentric finite elements, through several numerical experiments, and using various barycentric coordinates. These are Wachspress coordinates [47], mean value coordinates (MVCs) [20], natural neighbor coordinates (also called as Sibson’s coordinates) [40], and Poisson coordinates [26]. Two greedy algorithms to generate Voronoi meshes for adaptive functional/scattered data approximation.…”

Section: Introductionmentioning

confidence: 99%

Functional data approximation on bounded domains using polygonal finite elements

Cao

Xiao

Chen

et al. 2018

Computer Aided Geometric Design

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We construct and analyze piecewise approximations of functional data on arbitrary 2D bounded domains using generalized barycentric finite elements, and particularly quadratic serendipity elements for planar polygons. We compare approximation qualities (precision/convergence) of these partition-of-unity finite elements through numerical experiments, using Wachspress coordinates, natural neighbor coordinates, Poisson coordinates, mean value coordinates, and quadratic serendipity bases over polygonal meshes on the domain. For a convex -sided polygon, the quadratic serendipity elements have 2 basis functions, associated in a Lagrange-like fashion to each vertex and each edge midpoint, rather than the usual ( + 1)/2 basis functions to achieve quadratic convergence. Two greedy algorithms are proposed to generate Voronoi meshes for adaptive functional/scattered data approximations. Experimental results show space/accuracy advantages for these quadratic serendipity finite elements on polygonal domains versus traditional finite elements over simplicial meshes. Polygonal meshes and parameter coefficients of the quadratic serendipity finite elements obtained by our greedy algorithms can be further refined using an -optimization to improve the piecewise functional approximation. We conduct several experiments to demonstrate the efficacy of our algorithm for modeling features/discontinuities in functional data/image approximation.

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Poisson Coordinates

Cited by 36 publications

References 43 publications

A Survey on Cage‐based Deformation of 3D Models

A Survey on Cage‐based Deformation of 3D Models

Maximum Likelihood Coordinates

Functional data approximation on bounded domains using polygonal finite elements

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