A Constrained Resampling Strategy for Mesh Improvement

Abstract

In many geometry processing applications, it is required to improve an initial mesh in terms of multiple quality objectives. Despite the availability of several mesh generation algorithms with provable guarantees, such generated meshes may only satisfy a subset of the objectives. The conflicting nature of such objectives makes it challenging to establish similar guarantees for each combination, e.g., angle bounds and vertex count. In this paper, we describe a versatile strategy for mesh improvement by interpreting quality objectives as spatial constraints on resampling and develop a toolbox of local operators to improve the mesh while preserving desirable properties. Our strategy judiciously combines smoothing and transformation techniques allowing increased flexibility to practically achieve multiple objectives simultaneously. We apply our strategy to both planar and surface meshes demonstrating how to simplify Delaunay meshes while preserving element quality, eliminate all obtuse angles in a complex mesh, and maximize the shortest edge length in a Voronoi tessellation far better than the state-of-the-art.

Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm

“…In addition, the minimum edge length in a Voronoi cell can be a limiting factor in certain numerical solvers. Post-processing by mesh optimization techniques [ 5 , 53 ] can help eliminate short Voronoi edges away from the surface. Finally, we expect that the abstract algorithm analyzed in this paper can be extended to higher dimensions.…”

Surface Remeshing: A Systematic Literature Review of Methods and Research Directions

“…An algorithm that reaches a target quality limit with minimal point insertion is preferred over the algorithm that requires more point insertions. Sifting ratio α [56] is another metric used to represent the percentage reduction of vertices during mesh simplification.…”

“…However, the triangular approximation is lacking optimality, which is a future challenge. Constraint re-sampling [56] is another method that uses different re-sampling operators (including vertex translation, vertex removal, and vertex insertion in a feasible way) to generate highquality meshes. The advantage of constraint re-sampling is that it can simplify Delaunay meshes with preservation of element quality and elimination of obtuse angles and short edges.…”

“…[1], [3], [5], [11], [12], [30], [32], [33], [44], [60], [65], [66], [67], [68], [69], [70], [71], [72], [73], [74], [75], [76], [77], [78], [79], [80], [81], [82], [83] θmin 45 [3], [5], [11], [12], [30], [33], [36], [54], [56], [59], [60], [65], [66], [67], [68], [70], [71], [72], [80], [84], [85], [86], [87], [88], [89],…”

Isotropic Surface Remeshing without Large and Small Angles

“…Although geodesic-based methods can generate meshes with the same quality as that of RVD-based methods, they involve frequent geodesic path computations, which drastically slow down the remeshing process. Blue-noise sampling has also been introduced for surface remeshing, including capacity-constrained blue-noise sampling [44], maximal Poisson-disk sampling (MPS) [29], [45], [46], [47], farthest point optimization (FPO) [48], and the simple push-pull (SPP) [9] approaches. The main goal of this category of approaches is to produce point samples with the socalled "blue-noise" property.…”