MATTopo: Topology-preserving Medial Axis Transform with Restricted Power Diagram (2403.18761v4)

Abstract: We present a novel topology-preserving 3D medial axis computation framework based on volumetric restricted power diagram (RPD), while preserving the medial features and geometric convergence simultaneously, for both 3D CAD and organic shapes. The volumetric RPD discretizes the input 3D volume into sub-regions given a set of medial spheres. With this intermediate structure, we convert the homotopy equivalency between the generated medial mesh and the input 3D shape into a localized contractibility checking for each restricted element (power cell, power face, power edge), by checking their connected components and Euler characteristics. We further propose a fractional Euler characteristic algorithm for efficient GPU-based computation of Euler characteristic for each restricted element on the fly while computing the volumetric RPD. Compared with existing voxel-based or point-cloud-based methods, our approach is the first to adaptively and directly revise the medial mesh without globally modifying the dependent structure, such as voxel size or sampling density, while preserving its topology and medial features. In comparison with the feature preservation method MATFP, our method provides geometrically comparable results with fewer spheres and more robustly captures the topology of the input 3D shape.

• The paper introduces MATTopo, which uses Restricted Power Diagrams to ensure topological preservation in medial axis computation for both CAD and organic shapes.
• The method employs a sphere-shrinking initialization and iterative refinement to maintain geometric accuracy and capture medial features.
• Quantitative results show MATTopo reduces the medial sphere count tenfold while minimizing Hausdorff distance errors compared to previous techniques.

Analyzing "MATTopo: Topology-preserving Medial Axis Transform with Restricted Power Diagram"

The paper "MATTopo: Topology-preserving Medial Axis Transform with Restricted Power Diagram", authored by Ningna Wang et al., presents an innovative approach aimed at computing the medial axis of three-dimensional shapes with an emphasis on preserving topological properties. This work hinges on the integration of volumetric Restricted Power Diagrams (RPD) to ensure topology preservation alongside medial feature retention and geometric convergence.

Core Contributions and Methodology

The authors propose a computational framework that smartly leverages the volumetric RPD to derive medial axes for both CAD and organic shapes while ensuring the preservation of topology, medial features, and geometric accuracy. The framework prioritizes the homotopy equivalence between the generated medial mesh and the input 3D shape, offering a significant improvement over previous methods which either failed to preserve topology or handled medial features inadequately.

Primary Steps in the MATTopo Framework:

1. Initialization: Begin with a medial mesh derived from an initial set of medial spheres, computed using a sphere-shrinking algorithm.
2. Topology Preservation: Use the Nerve Theorem to ensure that each restricted element of the RPD maintains its contractibility, implying homotopy equivalence.
3. Feature Preservation: Ensure both external and internal medial features are captured accurately by inserting appropriate medial spheres.
4. Geometric Accuracy: Refine the medial mesh iteratively to maintain a user-defined geometric error threshold with respect to the original shape.

Innovations in Computational Techniques

The authors incorporate a novel Fractional Euler Characteristic strategy, which allows the computation of Euler characteristics of restricted power elements during the parallel RPD computation on a GPU. This approach ensures that topological guarantees are maintained efficiently. Furthermore, the adaptation of local RPD revisions markedly improves computational efficiency, making the framework scalable to detailed 3D shapes.

Comparison with Previous Methods

The MATTopo method distinguishes itself primarily in its balanced handling of topology and feature preservation. Previous methods like MATFP and PC often required trade-offs between these aspects. The MATTopo framework, however, manages to achieve homotopy equivalency without needing a markedly high sampling density or voxel resolution, reducing computational redundancy and providing comparable or superior surface reconstruction quality.

Numerical Results

Quantitative results demonstrate that the MATTopo method can generate medial meshes that are topologically correct with fewer medial spheres—approximately ten times fewer than those generated by MATFP—while maintaining alignment with the original shape's geometry. Notably, the reconstructed shapes from the medial axis present minimal Hausdorff distance errors, showcasing the method's practical feasibility in producing geometrically accurate results.

Implications and Speculations on Future Directions

The author's approach adds to the robustness of medial axis computation by ensuring topological integrity without compromising on speed or accuracy. This methodology opens avenues for advancing applications in computer graphics, CAD modeling, and possibly in domains needing rigorous shape analysis like biomedical imaging.

In future advancements, there is potential for integrating machine learning techniques to predict and analyze complex medial features, potentially accelerating calculations further and providing real-time topology-preserving computations in evolving digital environments. Additionally, streamlining the GPU-based RPD computations could facilitate handling larger models or more intricate shapes seamlessly.

Overall, this research represents an essential step in medial axis computation, paving the way for topologically robust shape analysis methodologies suitable for a plethora of applied contexts.