Derivation and Analysis of Green Coordinates

Lipman, Yaron; Levin, David

doi:10.1007/bf03321761

Cited by 14 publications

(19 citation statements)

References 6 publications

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“…The question is whether we can find 3D QCMs that introduce a degree of anisotropy sufficiently low so as to be negligible in practice. Here, we employ a recently-developed mapping technique based on Green coordinates (GCs) [30,31] to give a positive answer. GCs use a cage-based representation in which each point within a simplicial surface (cage or mesh made up of triangles) is expressed as a linear combination of the cage vertices and faces normals…”

Section: Methodsmentioning

confidence: 99%

Full three-dimensional isotropic transformation media

et al. 2014

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We present a method that enables the implementation of full three-dimensional (3D) transformation media with minimized anisotropy. It is based on a special kind of shape-preserving mapping and a subsequent optimization process. For sufficiently smooth transformations, the resulting anisotropy can be neglected, paving the way for practically realizable 3D devices. The method is independent of the considered wave phenomenon and can thus be applied to any field for which a transformational technique exists, such as acoustics or thermodynamics. Full 3D isotropy has an additional important implication for optical transformation media, as it eliminates the need for magnetic materials in many situations. To illustrate the potential of the method, we design 3D counterparts of transformation-based electromagnetic squeezers and bends.

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Section: Methodsmentioning

confidence: 99%

Full three-dimensional isotropic transformation media

et al. 2014

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“…Hence, it is enough to show that the mapping (20) is conformal inside and outside the cage, and that it is continuous on the edge t . The conformality inside and outside the cage is proven in [Lipman and Levin 2008]. The continuity across the edge t can be understood from the fact that the new coordinatesφi,ψj (inside and outside) are solutions of the non-singular system of equations (15),(16) which has C ∞ smooth coefficients.…”

Section: Extending To the Cage's Exteriormentioning

confidence: 99%

“…The derivation of the formulas is rather technical, so to keep the fluency of the reading we have attached only the final pseudocodes for calculating the 2D and 3D coordinates for η ∈ P in , see Algorithms 1 and 2 in Appendix A. The detailed derivations are listed in [Lipman and Levin 2008].…”

Section: Quasi-conformalitymentioning

confidence: 99%

“…It can be shown that this system is invertible for any η ∈ R d (see [Lipman and Levin 2008]). Thus, we may view this system as an alternative way to define φi 1 (η), ..., φi d (η), ψ (η).…”

Section: Extending To the Cage's Exteriormentioning

confidence: 99%

“…Finally, regarding the fourth property, in the case of d = 2, the mapping η → F (η; P ) is pure conformal. The proof is rather long and technical and provided fully in [Lipman and Levin 2008]. Generally, the proof is based on …”

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Green Coordinates

Lipman¹,

Levin²,

Cohen–Or³

2008

ACM SIGGRAPH 2008 Papers

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Figure 1: (a-c) Cage-based 2D deformation of a Gecko. (b) Using Green Coordinates induces a pure conformal mapping. (c) The result of Harmonic Coordinates. Note the preservation of shape in the marked square. (d-f) Cage-based 3D articulation of an Ogre. (e) Using Green Coordinates in 3D admits a quasi-conformal deformation. In (f) the result using Mean Value Coordinates is presented. Note how Green Coordinates nicely preserve the shape of the Ogre's head. AbstractWe introduce Green Coordinates for closed polyhedral cages. The coordinates are motivated by Green's third integral identity and respect both the vertices position and faces orientation of the cage. We show that Green Coordinates lead to space deformations with a shape-preserving property. In particular, in 2D they induce conformal mappings, and extend naturally to quasi-conformal mappings in 3D. In both cases we derive closed-form expressions for the coordinates, yielding a simple and fast algorithm for cage-based space deformation. We compare the performance of Green Coordinates with those of Mean Value Coordinates and Harmonic Coordinates and show that the advantage of the shape-preserving property is not achieved at the expense of speed or simplicity. We also show that the new coordinates extend the mapping in a natural analytic manner to the exterior of the cage, allowing the employment of partial cages.

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GreenWarps: A Two-Stage Warping Model for Stitching Images Using Diffeomorphic Meshes and Green Coordinates

Jacob

Das

2019

Lecture Notes in Computer Science

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Derivation and Analysis of Green Coordinates

Cited by 14 publications

References 6 publications

Full three-dimensional isotropic transformation media

Full three-dimensional isotropic transformation media

Green Coordinates

GreenWarps: A Two-Stage Warping Model for Stitching Images Using Diffeomorphic Meshes and Green Coordinates

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