Consolidating Weak Features in High-quality Mesh Simplification: A Unified Functional Approach

Core Concepts

A unified functional that simultaneously considers accuracy, triangle quality, and feature alignment, effectively consolidating both strong and weak features during mesh simplification.

Abstract

The paper proposes a smooth functional that unifies the requirements of accuracy, triangle quality, and feature alignment for mesh simplification. 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 normal anisotropy term inherits the spirit of Quadric Error Metrics (QEM) but operates in a continuous setting, while the CVT term encourages even point distribution, allowing various surface metrics. The authors introduce a decaying weight to automatically balance the two terms, effectively reducing the conflict between them and enabling the alignment of both strong and weak features. The authors also develop a simple yet effective technique to facilitate the computation of the Restricted Voronoi Diagram (RVD) on thin-plate models, which can encounter challenges with the "one site, one region" property. Experimental results on 100 CAD models from the ABC dataset and 21 organic models demonstrate that the proposed approach outperforms state-of-the-art mesh simplification methods in preserving accuracy, triangle quality, and feature alignment, particularly for weak features.

Stats

The total surface area is normalized to 1.0. The normal anisotropy term (𝐸NA) can be reduced to nearly 0 for most CAD models, while the CVT term (𝐸CVT) does not vary significantly. The optimization generally requires tens of iterations for CAD models, and up to 100 iterations for organic models to prevent the deterioration of triangle quality.

Quotes

"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." "We introduce a decaying weight to automatically balance the two terms, effectively reducing the conflict between them and enabling the alignment of both strong and weak features." "We also develop a simple yet effective technique to facilitate the computation of the Restricted Voronoi Diagram (RVD) on thin-plate models, which can encounter challenges with the 'one site, one region' property."

Key Insights Distilled From

CWF: Consolidating Weak Features in High-quality Mesh Simplification

by Rui Xu,Longd... at arxiv.org 04-25-2024

https://arxiv.org/pdf/2404.15661.pdf

Deeper Inquiries

How can the proposed approach be extended to handle non-manifold or open boundary meshes?

The proposed approach can be extended to handle non-manifold or open boundary meshes by incorporating specific techniques tailored to address these challenges. For non-manifold meshes, the algorithm can be modified to detect and handle vertices, edges, or faces that do not adhere to the manifold property. This may involve implementing additional checks during the surface decomposition phase to ensure that each region maintains manifoldness. For open boundary meshes, the algorithm can be enhanced to identify and process boundary vertices separately, ensuring that the mesh simplification process does not introduce non-manifold elements at the boundaries. By incorporating these modifications, the algorithm can effectively handle non-manifold and open boundary meshes while maintaining high-quality triangulations.

What are the potential applications of the consolidated weak feature preservation in downstream tasks, such as shape understanding or analysis?

The consolidated weak feature preservation achieved by the proposed approach has several potential applications in downstream tasks related to shape understanding and analysis. Shape Recognition: Preserving weak features during mesh simplification can enhance shape recognition algorithms by maintaining crucial visual cues that aid in identifying and distinguishing shapes accurately. CAD Modeling: In Computer-Aided Design (CAD) applications, the ability to consolidate weak features ensures that intricate design details are retained, leading to more accurate representations of complex geometries. Simulation and Analysis: For simulations and analysis tasks, preserving weak features can improve the accuracy of numerical computations by ensuring that subtle variations in the geometry are maintained, leading to more reliable results. Virtual Reality and Augmented Reality: In VR and AR applications, consolidated weak feature preservation can enhance the visual fidelity of virtual environments, providing a more realistic and immersive user experience. Medical Imaging: In medical imaging applications, maintaining weak features in anatomical models can aid in the accurate representation of patient-specific structures, leading to improved diagnostic capabilities and treatment planning. Overall, the consolidated weak feature preservation can benefit a wide range of tasks in shape understanding and analysis by ensuring that important geometric details are retained throughout the mesh simplification process.

Can the decaying weight mechanism be further optimized to achieve a more seamless balance between the normal anisotropy and CVT energy terms?

The decaying weight mechanism can be further optimized to achieve a more seamless balance between the normal anisotropy and CVT energy terms by fine-tuning the decay rate and incorporating adaptive strategies based on the characteristics of the input mesh. Adaptive Decay Rate: Instead of using a fixed decay rate, the algorithm can dynamically adjust the decay rate based on the convergence behavior of the optimization process. By monitoring the changes in the normal anisotropy and CVT energy terms during optimization, the algorithm can adaptively modify the decay rate to maintain a balanced influence of both terms. Dynamic Weight Adjustment: Introducing a dynamic weight adjustment mechanism that considers the relative magnitudes of the normal anisotropy and CVT energy terms can help in achieving a more seamless balance. By continuously evaluating the contributions of each term and adjusting the weights accordingly, the algorithm can ensure that neither term dominates the optimization process. Multi-Objective Optimization: Transforming the optimization problem into a multi-objective optimization task can enable the algorithm to simultaneously optimize for accuracy, triangle quality, and feature alignment. By incorporating multiple objectives and defining appropriate weightings for each objective, the algorithm can achieve a more holistic balance between the normal anisotropy and CVT energy terms. By implementing these optimizations, the decaying weight mechanism can be enhanced to achieve a more seamless and effective balance between the normal anisotropy and CVT energy terms, leading to improved mesh simplification results.