Technical SectionLow budget and high fidelity relaxed 567-remeshing
Graphical abstract
A 567-remeshing framework that is capable of a low vertex budget while preserving a high fidelity to the input mesh surface.
Introduction
Non-regular triangular meshes are common place and their low and high valence vertices raise several issues. For instance, Aghdaii et al. [1] mention that valence-3 vertices may cause an edge collapse to generate a non-manifold mesh and that high valence vertices can lead to visible artifacts during mesh subdivision (e.g. for the butterfly scheme [2]). In addition, irregular valences, especially high ones, result in irregular sampling when applying either Laplacian or angle-based smoothing [3], [4]. In particular, Surazhsky and Gotsman [3] noticed that edges whose two vertices have a valence greater than 7 (resp. smaller than 5) are made longer (resp. shorter) after an angle-based smoothing. The same authors [4] also mention that angle-based smoothing produces less inverted elements when the mesh is close to regular, which effectively happens for 567-meshes. 567-meshes are closed and 2-manifold triangular meshes, whose vertex valence is either 5, 6 or 7 [1]. In the remainder of this paper, and denote a vertex with a valence, respectively, strictly less than k and strictly greater than l.
A regular mesh has faces only of the same degree and vertices only of the same valence. Completely regular (closed) meshes with only valence-6 vertices exist only for genus g=1 [5]. It is not possible to get a completely regular closed manifold of genus 0 [6]. In addition, either regular or semi-regular remeshing algorithms usually need a costly 2D (global) parametrization [7] which generally introduces some distortion.
Highly regular meshes are necessary for engineers performing numerical simulations [3]. Many authors [1], [3], [8], [9] have developed (complex) strategies to either drastically reduce the number of non-regular vertices or at least avoid too irregular ones (5− and 7+). 567-remeshing algorithm [1] falls in the later category and has gain particular attention since it can resolve all issues mentioned in the first paragraph. In particular, a 567-remeshing step applied before a mesh smoothing greatly increases the final quality of the remeshed model. Note that 567-remeshed models could benefit from a specific data structure for representing adjacency relationships with constant-size buffers instead of linked-lists.
However, there is actually a costly counterpart with 567-remeshing: the decrease in the amplitude of valence irregularity is offset by an increase in the number of irregular and regular vertices. That is due by essence to vertex split operations that add vertices and to a theorem [1] that states that a valence vertex can be replaced by valence-7 vertices plus a valence 5 or 6 or 7 vertex, while incrementing the valence of one-ring vertices by at most one. The current reference 567-remeshing algorithm [1] does a 1–9 triangle subdivision (see Fig. 1 (b)) before applying vertex splits to avoid creating a new valence-7+ vertex. The resulting 567-remeshed models have their number of faces multiplied by about 10 before the algorithm׳ vertex removal step. And because the vertex removal step must preserve the 567 property and the fidelity to the initial 567-remeshed model, it rarely counterbalances all added vertices, which thus becomes a severe issue when processing large meshes.
This paper addresses the vertex budget issue of the 567-remeshing introduced in [1] and proposes a framework also capable of controlling the geometric error and preserving mesh features (e.g. feature edges) during all the 567-remeshing operations. Our main contributions are the following
- •
New local strategies for the removal of valence 5− and7+ vertices: We only consider local remeshing strategies (no global triangle subdivision) and we minimize the valence potential increase (see Eq. (1)) when we remove irregular vertices, by greedily selecting the local configuration associated with less valence potential increase.
- •
Better control of the mesh quality: Additionally to the vertex valences, the vertex budget and the fidelity to the original surface are monitored during the removal of valence 5− or 7+ vertices. An additional criterion, the triangle equilateralness, is also taken into account during the final mesh enhancement steps.
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Preservation of lines of feature edges during all 567-remeshing operations: Our framework offers the possibility to preserve a set of feature edges during 567-remeshing. It considerably improves the fidelity to the original surface, even for non-CAD models.
Section snippets
Related work
This section presents some works related to vertex valence optimization and the Aghdaii et al. [1] 567-remeshing algorithm published recently.
Overview of our relaxed 567-remeshing algorithm
Our 567-remeshing framework, presented in Fig. 3, contains the following steps:
- 1.
optional: feature edge detection;
- 2.
elimination of valence-5− vertices without introducing geometrical error;
- 3.
elimination of valence-7+ vertices: the choice to introduce either some geometric error or some degenerated triangles is set as a parameter of our framework;
- 4.
final mesh quality improvement: local remeshing operations are applied both to attempt to reach the initial vertex budget and to improve the quality of mesh
Mesh enhancement
The mesh enhancement step is needed both to reach the initial vertex budget and to improve the vertex positioning. Indeed, even if we monitor the vertex budget, the removal of irregular vertices usually adds many new vertices and introduces many ill-positioned vertices due to vertex split operations. Therefore, we propose a mesh enhancement step, which alternates vertex decimation and vertex relaxation up to no more decimation is neither needed nor possible.
Up to this section, the used criteria
Experimental results
In order to demonstrate the efficiency of our 567-remeshing framework, we applied it to several mesh models with irregular vertices, which contain both smooth parts and sharp features. Results are presented in Figs. 7, 9 and 10 and in Table 2. The presented results have been obtained on an Intel Core I7-2760QM (2.4 GHz) with 8 GB RAM .
Applications
Our 567-remeshing framework might be a good preprocessing step for triangular mesh regularization and for connectivity-based mesh compression.
Conclusion and future work
In this paper, we demonstrate that 567-remeshing can be conducted without the need of a 1–9 global subdivision step, which is a severe issue of state-of-the-art for large meshes because the algorithm can be run out of memory. We also showed that the decimation step, needed to balance added vertices during 567-remeshing with edge collapses, could be done by also taking into account triangle equilateralness; by doing so, the mesh quality is improved at the end of the decimation process, and thus,
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