Integer-grid maps for reliable quad meshing

Abstract

Greedy Rounding + StiffeningOur Reliable Approach Figure 1: (left) State-of-the-art parametrization based quad mesh generators, working with greedy rounding and stiffening, perform well if the target element sizing is chosen conservatively w.r.t. the distance of singularities but fail otherwise. Degeneracies in the map that prevent the iso-lines from stitching to a valid quad mesh -which mostly cannot be repaired locally -are highlighted in red. (right) Our novel reliable algorithm produces a valid output for any target sizing and thus in addition to ordinary quad-remeshing can be applied to coarse quad layout generation as well. The target edge length, indicated by bars, is identical for the left and the right triple. AbstractQuadrilateral remeshing approaches based on global parametrization enable many desirable mesh properties. Two of the most important ones are (1) high regularity due to explicit control over irregular vertices and (2) smooth distribution of distortion achieved by convex variational formulations. Apart from these strengths, state-of-the-art techniques suffer from limited reliability on realworld input data, i.e. the determined map might have degeneracies like (local) non-injectivities and consequently often cannot be used directly to generate a quadrilateral mesh. In this paper we propose a novel convex Mixed-Integer Quadratic Programming (MIQP) formulation which ensures by construction that the resulting map is within the class of so called Integer-Grid Maps that are guaranteed to imply a quad mesh. In order to overcome the NP-hardness of MIQP and to be able to remesh typical input geometries in acceptable time we propose two additional problem specific optimizations: a complexity reduction algorithm and singularity separating conditions. While the former decouples the dimension of the MIQP search space from the input complexity of the triangle mesh and thus is able to dramatically speed up the computation without inducing inaccuracies, the latter improves the continuous relaxation, which is crucial for the success of modern MIQP optimizers. Our experiments show that the reliability of the resulting algorithm does not only annihilate the main drawback of parametrization based quad-remeshing but moreover enables the global search for high-quality coarse quad layouts -a difficult task solely tackled by greedy methodologies before.

Robust field-aligned global parametrization

“…[Bommes et al 2009] described a stiffening method that modifies the energy used to compute parametrization to eliminate flipped triangles. More recently, [Bommes et al 2013] proposed a method based on introducing convex constraints ensuring bijectivity. While the method of [Lipman 2012] was not applied to global parametrization, it easily extends to the global parametrization setting as we show in Section 8.…”

“…The resulting global parametrization can be used either on its own if the application does not require integer parametric cone positions (e.g., constructing T-spline based approximations), or requires minimal rounding (tiling the surface with textures). Alternatively, it can be a robust starting point for a recent method [Bommes et al 2013].…”

Enhanced medial-axis-based block-structured meshing in 2-D

“…In principle, these methods could be applied in the present method (with some modifications to clean up the cross-fields) but it would eliminate the niche that the medial axis decomposition methods fit into. Medial-axisbased methods should be 'lightweight', cheap, fast and robust, otherwise they offer no advantage over more sophisticated methods [20,21,22] that can handle more general problems with target size and orientation fields. To fulfil these objectives undemanding techniques are called for to split the geometry using the medial axis in a similar manner to the T&A and TopMaker methods, but that can produce enhanced decompositions.…”

“…The newest most advanced algorithms for quad mesh generation include Morse-Smale complex methods [18] and cross-field methods [19,20,21,22]. Both approaches solve scalar and or vector fields on the surface from which quadrangular charts are extracted and meshed by parametrisation methods.…”

Advanced Automatic Hexahedral Mesh Generation from Surface Quad Meshes

“…At a first glance, this appears to be a serious restriction of the class of hex-meshable surface meshes. Nevertheless, recent approaches to quad meshing significantly increased the quality of quadrangulations of surfaces while still offering a decent amount of control over the parameters [5,7,10,15,36,6]. Fig.…”