Periodic global parameterization

Abstract

We present a new globally smooth parameterization method for triangulated surfaces of arbitrary topology. Given two orthogonal piecewise linear vector elds dened over the input mesh (typically the estimated principal curvature directions), our method computes two piecewise linear periodic functions, aligned with the input vector elds, by minimizing an objective function. The bivariate function they dene is a smooth parameterization almost everywhere on the surface, except in the vicinity of singular vertices, edges and triangles, where the derivatives of the parameterization vanish. We extract a quadrilateral chart layout from the parameterization function and propose an automatic procedure to detect the singularities, and x them by splitting and re-parameterizing the containing charts. Our method can construct both quasi-conformal (angle preserving) and quasi-isometric (angle and area preserving) parameterizations. The more restrictive class of quasi-isometric parameterizations is constructed at the expense of introducing more singularities. The constructed parameterizations can be used for a variety of geometry processing applications. Since we can align the parameterization with the principal curvature directions, our result is particularly suitable for surface tting and remeshing.

Isotropic 2D Quadrangle Meshing with Size and Orientation Control

“…Including methods devised to generate quadrangle surface tilings, the rich literature on this topic contains approaches which proceed by quadrangulation [9,3], square packing [28], advancing front [22], conversion [8], decimation [18], Morse-Smale complexes [14,33,17], clustering [4,20], local and global operators [23,19], whisker weaving [32], medial axis [24], streamlining [1,21] and parameterization [25,29,7].…”

Feature aligned quad dominant remeshing using iterative local updates

“…A zippering operation is performed to keep the remeshed model water-tight. Periodic global parametrization is proposed in [25] to parameterize the input model so that principal directions are aligned with coordinate axes in the parameter domain. Non-linear optimization is required to achieve this.…”

Mesh Parameterization Methods and Their Applications

“…Ray et al [102] use global periodic parameterization to generate a predominantly quadrilateral mesh directly on the 3D surface. Dong et al [26] use a parameterization induced by the Morse complex to generate a quad only mesh of the surface.…”

“…Recently, several methods [46,62,102,126], have proposed implicitly creating a parameterization by solving for differential one-forms. Instead of associating (u,v) coordinates with vertices as in typical planar parameterization, they associate planar vectors (du, dv) with the edges of the mesh, computing a gradient field, or one-form.…”

“…Motivated by geometry processing applications, Ray et al [102] incorporate more geometric information into the problem setting ( Figure 5.8(b)). Given the curvature tensor of the surface, their method computes two piecewise linear periodic functions, aligned with the minimal and maximal curvatures.…”