CWF: Consolidating Weak Features in High-quality Mesh Simplification

TL;DR

This work tackles the long-standing tradeoff between accuracy, triangle quality, and feature alignment in mesh simplification by introducing a unified smooth functional that combines a normal anisotropy term with a CVT energy term. A decaying weight balances these two terms, enabling simultaneous preservation of strong features and consolidation of weak features while maintaining high-quality triangulations. The method operates on movable surface points and relies on Restricted Voronoi Diagrams, including a practical technique to handle thin-plate models. Across 100 CAD models and 21 organic models, the approach demonstrates superior feature preservation, especially for weak features, and promising potential for CAD lightweighting and shape understanding.

Abstract

In mesh simplification, common requirements like accuracy, triangle quality, and feature alignment are often considered as a trade-off. Existing algorithms concentrate on just one or a few specific aspects of these requirements. For example, the well-known Quadric Error Metrics (QEM) approach prioritizes accuracy and can preserve strong feature lines/points as well but falls short in ensuring high triangle quality and may degrade weak features that are not as distinctive as strong ones. In this paper, we propose a smooth functional that simultaneously considers all of these requirements. The functional comprises a normal anisotropy term and a Centroidal Voronoi Tessellation (CVT) energy term, with the variables being a set of movable points lying on the surface. The former inherits the spirit of QEM but operates in a continuous setting, while the latter encourages even point distribution, allowing various surface metrics. We further introduce a decaying weight to automatically balance the two terms. We selected 100 CAD models from the ABC dataset, along with 21 organic models, to compare the existing mesh simplification algorithms with ours. Experimental results reveal an important observation: the introduction of a decaying weight effectively reduces the conflict between the two terms and enables the alignment of weak features. This distinctive feature sets our approach apart from most existing mesh simplification methods and demonstrates significant potential in shape understanding.

Figures (22)

• Figure 1: We present a functional that integrates the demands of accuracy, triangle quality, and feature alignment. The impact of the CVT energy gradually diminishes, thanks to the decaying weight, achieving a harmonious balance between the two terms. It is noted that a Restricted Voronoi Diagram (RVD) must be computed during each iteration. In this example, a total of 50 iterations activated our termination condition.
• Figure 2: The RVD after optimizing the positions by minimizing our objective function, points near the features are naturally drawn toward the nearby points/lines. In this figure, $\mathbf{x}_1$ (red) controls a planar region and remains static, $\mathbf{x}_2$ (green) dominates an area across the feature line and is shifted to the feature line, and $\mathbf{x}_3$ (blue) governs a corner point and is relocated to the corner.
• Figure 3: By setting the target number of vertices to 400, we compare the ability to consolidate weak features among QEM, LpCVT, and our method. It's noted that the original Mobius-ring model has a smoothly transitioning shape with 7000 vertices.
• Figure 4: Resulting triangle mesh surfaces. For all three approaches, we maintain a constant total of 100 iterations and monitor changes in the normal anisotropy term (d) and the CVT energy term (e). The plots for these approaches are colored yellow, blue, and red, respectively. Please note that the vertical axis of the normal anisotropy term is displayed on a log10 scale, whereas the vertical axis for the CVT energy term uses a linear scale.
• Figure 5: A simple yet effective technique for fixing RVDs by introducing an additional set of biased points. (a) RVDs may fail on a thin-plate model with a set of sparse sites. (b) Rather than fix the problematic RVD, we bias each point at a negligible distance along the inward normal direction. (c) We need to find the contributing sites for each yellow region that is dominated by a biased site. (d) After the yellow regions are partitioned and assigned to the contributing sites, we get a valid surface decomposition.