A Local/Global Approach to Mesh Parameterization

Abstract

We present a novel approach to parameterize a mesh with disk topology to the plane in a shapeWe present also a more general "hybrid" parameterization model which provides a continuous spectrum of possibilities, controlled by a single parameter. The two cases described above lie at the two ends of the spectrum. We generalize our local/global algorithm to compute these parameterizations. The local phase may also be accelerated by parallelizing the independent computations per triangle.

Global parametrization by incremental flattening

“…As it is shown in [Liu et al 2008], the integrand can be expressed directly in terms of signed singular values σ1 and σ2 of ∇f , as…”

“…Our work builds on ideas from four sources: (1) featurealigned parametrization and quadrangulation, based on fitting parametrization gradients to a field, primarily [Bommes et al 2009], but also [Ray et al 2006;Kälberer et al 2007]; (2) asrigid-as-possible parametrization [Liu et al 2008]; (3) conformal parametrization/metric computation [Jin et al 2004;Ben-Chen et al 2008;Springborn et al 2008]; and (4) cross-field and Nsymmetry/rotational symmetry field and connection constructions [Hertzmann and Zorin 2000;Palacios and Zhang 2007;Ray et al 2008;Ray et al 2009;Lai et al 2009;Crane et al 2010]. We discuss the relationship in Sections 5 and 6 in more detail.…”

Anisotropic deformation for local shape control

“…We show that the directional scaling can be applied to the parameterization method introduced by Ref. [13]. Since our directional scaling is invariant to the surface elements (triangles or fans) used, the method can easily be used to also enrich the above formulation by adding anisotropy (see Fig.…”

“…Parameterizing meshes using ARAP methods was introduced by Liu et al [13], extended for length conservation by Zhang and Wang [14] and further extended by Smith and Schaefer [15]. When developing functional garments based on pattern shrinkage during compression [16], directed geometry manipulation is crucial.…”

As-Rigid-As-Possible molecular interpolation paths

“…It is this latter application that inspired the present work. Some alternatives to ARAP are As-Isometric-As-Possible (AIAP) methods that preserve distances within the mesh [39,40], the As-Similar-As-Possible method (ASAP) which enforces the preservation of the mesh tetrahedrals [41], path deformation techniques based on physics [42] or possible dynamics [43], and interpolation methods based on specific surface representations [44][45][46][47][48]. The reader is referred to [49] for a survey of mesh deformation techniques.…”