Fast spherical quasiconformal parameterization of genus-$0$ closed surfaces with application to adaptive remeshing

Choi, Gilwoo; Man, Mandy Hiu-Ying; Lui, Lok Ming

doi:10.4310/gic.2016.v3.n1.a1

Cited by 18 publications

(7 citation statements)

References 52 publications

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“…For future work, we plan to explore the possibility of extending our method for quasiconformal parameterizations and mappings [43][44][45]. More specifically, note that the partial welding step in our proposed method is conformal, and the quasi-conformal dilatation of a map is preserved under the composition with conformal maps.…”

Section: Discussionmentioning

confidence: 99%

Parallelizable global conformal parameterization of simply-connected surfaces via partial welding

Choi,

Leung-Liu,

et al. 2019

Preprint

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Conformal surface parameterization is useful in graphics, imaging and visualization, with applications to texture mapping, atlas construction, registration, remeshing and so on. With the increasing capability in scanning and storing data, dense 3D surface meshes are common nowadays. While meshes with higher resolution better resemble smooth surfaces, they pose computational difficulties for the existing parameterization algorithms. In this work, we propose a novel parallelizable algorithm for computing the global conformal parameterization of simply-connected surfaces via partial welding maps. A given simply-connected surface is first partitioned into smaller subdomains. The local conformal parameterizations of all subdomains are then computed in parallel. The boundaries of the parameterized subdomains are subsequently integrated consistently using a novel technique called partial welding, which is developed based on conformal welding theory. Finally, by solving the Laplace equation for each subdomain using the updated boundary conditions, we obtain a global conformal parameterization of the given surface, with bijectivity guaranteed by quasi-conformal theory. By including additional shape constraints, our method can be easily extended to achieve disk conformal parameterization for simply-connected open surfaces and spherical conformal parameterization for genus-0 closed surfaces. Experimental results are presented to demonstrate the effectiveness of our proposed algorithm. When compared to the state-of-the-art conformal parameterization methods, our method achieves a significant improvement in both computational time and accuracy.

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Section: Discussionmentioning

confidence: 99%

Parallelizable global conformal parameterization of simply-connected surfaces via partial welding

Choi,

Leung-Liu,

et al. 2019

Preprint

Self Cite

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“…Our proposed VDERM method can also be applied for remeshing, which aims at improving the discretization of meshes [25,26]. More specifically, suppose our goal is to construct a tetrahedral mesh for a genus-0 closed surface.…”

Section: Adaptive Remeshingmentioning

confidence: 99%

Volumetric density-equalizing reference map with applications

Choi,

Rycroft

2020

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The density-equalizing map, a technique developed for cartogram creation, has been widely applied to data visualization but only for 2D applications. In this work, we propose a novel method called the volumetric density-equalizing reference map (VDERM) for computing density-equalizing map for volumetric domains. Given a prescribed density distribution in a volumetric domain in R 3 , the proposed method continuously deforms the domain, with different volume elements enlarged or shrunk according to the density distribution. With the aid of the proposed method, medical and sociological data can be visualized via deformations of 3D objects. The method can also be applied to adaptive remeshing and shape modeling. Furthermore, by exploiting the time-dependent nature of the proposed method, applications to shape morphing can be easily achieved. Experimental results are presented to demonstrate the effectiveness of the proposed method.

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“…Area information has also been utilized in the computation of optimal mass transport maps [21,22,50] and density-equalizing maps [10,16,17]. More recently, quasi-conformal mappings have become increasingly popular for the development of non-rigid image registration [31,47] and surface mapping methods [12,13,19,32,37], with applications to geometry processing [7,11,14], biological shape analysis [8,15] and medical visualization [9,18]. Specifically, quasi-conformal theory allows one to ensure the bijectivity and reduce the local geometric distortion of the mappings.…”

Section: Introductionmentioning

confidence: 99%

A unifying framework for $n$-dimensional quasi-conformal mappings

Zhang,

Choi,

Zhang

et al. 2021

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With the advancement of computer technology, there is a surge of interest in effective mapping methods for objects in higher-dimensional spaces. To establish a one-to-one correspondence between objects, higher-dimensional quasi-conformal theory can be utilized for ensuring the bijectivity of the mappings. In addition, it is often desirable for the mappings to satisfy certain prescribed geometric constraints and possess low distortion in conformality or volume. In this work, we develop a unifying framework for computing n-dimensional quasi-conformal mappings. More specifically, we propose a variational model that integrates quasi-conformal distortion, volumetric distortion, landmark correspondence, intensity mismatch and volume prior information to handle a large variety of deformation problems. We further prove the existence of a minimizer for the proposed model and devise efficient numerical methods to solve the optimization problem. We demonstrate the effectiveness of the proposed framework using various experiments in two-and three-dimensions, with applications to medical image registration, adaptive remeshing and shape modeling.

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Fast spherical quasiconformal parameterization of genus-$0$ closed surfaces with application to adaptive remeshing

Cited by 18 publications

References 52 publications

Parallelizable global conformal parameterization of simply-connected surfaces via partial welding

Parallelizable global conformal parameterization of simply-connected surfaces via partial welding

Volumetric density-equalizing reference map with applications

A unifying framework for $n$-dimensional quasi-conformal mappings

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