CWF: Consolidating Weak Features in High-quality Mesh Simplification (2404.15661v1)

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.

• The paper introduces an objective function that integrates a normal anisotropy term and a CVT energy term to achieve balanced weak feature preservation and high triangle quality.
• It uses a decaying weight to gradually shift focus from even point distribution to precise feature alignment, effectively reconciling conflicting simplification demands.
• Experimental results on CAD and organic models demonstrate improved feature fidelity and robust mesh quality, highlighting its applicability in diverse 3D modeling scenarios.

CWF: Consolidating Weak Features in High-quality Mesh Simplification

Introduction

This paper introduces a novel functional for mesh simplification that effectively consolidates weak features while ensuring high-quality triangle mesh outcomes. Traditional approaches such as Quadric Error Metrics (QEM) and Centroidal Voronoi Tessellation (CVT) each address specific aspects of mesh simplification requirements but often fail to balance these adequately. The proposed method combines a normal anisotropy term based on QEM and a CVT energy term to harmonize the requirements of accuracy, triangle quality, and feature alignment. This combination is regulated by a decaying weight to reduce the potential conflict between these terms, enhancing the alignment of weak features.

Methodology

Objective Function

The main contribution is the formulation of a new objective derived from the integration of the normal anisotropy term and the CVT energy term. The objective function is expressed as:

E(x) = λ_NA * E_NA + λ_CVT * E_CVT

where E_NA represents the normal anisotropy and E_CVT represents the Centroidal Voronoi energy. Key innovations include:

• Normal Anisotropy: Inherited from QEM but modified to operate over Voronoi cells formed around each sample point on the mesh surface, it helps in aligning features by minimizing the weighted squared distance between surface points and the mesh representation.
• Centroidal Voronoi Tessellation (CVT): Modified to cooperate with the normal anisotropy term, it promotes even point distribution and thereby enhances the mesh quality.

Balanced Weight Adjustment

A significant enhancement proposed is the use of a decaying weight for the CVT term, allowing this term's influence to gradually reduce over iterations. This strategic balance assists in starting with even distribution and slowly transitioning focus to feature preservation, tackling the commonly observed challenge of balancing accuracy and model fidelity.

Results and Evaluation

Using 100 CAD models and 21 organic models, the results demonstrate a marked improvement in weak feature consolidation compared to existing methods. The functional proves particularly adept at maintaining a balance between the accuracy of strong features and the promotion of mesh quality, typically concluding its optimization within tens of iterations.

Quantitative results also underscore this balance:

• Feature Preservation: Strong and weak features were preserved with higher fidelity compared to traditional methods.
• Mesh Quality: High-quality triangle output was consistent, avoiding common issues such as excessive elongation or size disparity in triangles.

Experiments further show the method's robustness across different types of models, from organic shapes to mechanical parts, indicating wide applicability.

Theoretical and Practical Implications

Theoretically, the introduction of a decaying weight represents a nuanced approach to term balancing in optimization processes for mesh simplification, which might influence future research in similar areas. Practically, this method offers a reliable tool for industries relying on 3D model manipulation, such as animation, CAD, and virtual reality, providing an efficient means to handle complex models without sacrificing detail or quality.

Future Directions

Potential future developments could explore the automatic adjustment of the balancing weights based on the input model's characteristics or the integration of machine learning techniques to predict optimal parameters dynamically. Additionally, extending this framework to accommodate non-Euclidean geometries more robustly could broaden its utility.

In conclusion, this paper presents a significant step forward in mesh simplification, particularly in the challenging area of weak feature consolidation, without compromising the overall mesh quality.