Our 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. Quadrilateral 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.
Then they used a convex mixed-integer quadratic programming formulation REF to generate reliable quad mesh, which could achieve high-quality coarse quad layout by globally searching.
Integer-grid maps for reliable quad meshing
TOGS
Computer Science