Mixed-integer quadrangulation

Abstract

We present a novel method for quadrangulating a given triangle mesh. After constructing an as smooth as possible symmetric cross field satisfying a sparse set of directional constraints (to capture the geometric structure of the surface), the mesh is cut open in order to enable a low distortion unfolding. Then a seamless globally smooth parametrization is computed whose iso-parameter lines follow the cross field directions. In contrast to previous methods, sparsely distributed directional constraints are sufficient to automatically determine the appropriate number, type and position of singularities in the quadrangulation. Both steps of the algorithm (cross field and parametrization) can be formulated as a mixed-integer problem which we solve very efficiently by an adaptive greedy solver. We show several complex examples where high quality quad meshes are generated in a fully automatic manner.

CubeCover– Parameterization of 3D Volumes

“…The parameterization problem is formulated as finding least squares fitting to the given field. The MIQ-algorithm [BZK09] is based on a similar formulation, but exchanges the rounding strategy of QUADCOVER by using a mixed integer solver. Additionally, a method for generating input frame-fields from a sparse set of vectors is proposed.…”

All-hex meshing using singularity-restricted field

“…Quadrilateral meshing has gained great success by using surface parametrization techniques [Tong et al 2006;Kälberer et al 2007;Bommes et al 2009]. Both Kalberer et al [2007] and Bommes et al [2009] present theoretical and practical approaches to find a proper surface parametrization based on a cross field defined on the triangular surface.…”

“…Both Kalberer et al [2007] and Bommes et al [2009] present theoretical and practical approaches to find a proper surface parametrization based on a cross field defined on the triangular surface. It is natural to consider how to extend the quad meshing technique to hex meshing.…”

“…Unlike the cross field on surface [Ray et al 2008;Bommes et al 2009;Crane et al 2010], the smoothness of a 3D frame field cannot be encoded by only one parameter. Mathematically, the smoothness of two frames Fs, Ft should reflect the closeness from the transformation between Fs and Ft to the chiral cubical symmetry group G. We extend the permutation approach in [Liu et al 2011] to measure the smoothness of a 3D frame field.…”

“…We propose to solve the volume parametrization via an adaptive rounding method similar to [Bommes et al 2009] so as to reduce the distortion and avoid non-matched integer gaps.…”