[Paper Review] Computing Medial Axis Transform with Feature Preservation via Restricted Power Diagram

Ningna Wang, Bin Wang+2 more|arXiv (Cornell University)|Oct 25, 2022

3D Shape Modeling and Analysis4 citations

TL;DR

This paper proposes a novel restricted power diagram (RPD)-based framework for computing the 3D medial axis transform that preserves both external features (e.g., sharp edges and corners) and internal features (e.g., seams and junctions). By leveraging the dual structure of RPD and surface-restricted power cells (RPCs), the method detects insufficient sampling, ensures correct connectivity, and enables feature-preserving sampling and refinement of medial spheres, resulting in high-quality medial meshes with topologically and geometrically accurate features.

ABSTRACT

We propose a novel framework for computing the medial axis transform of 3D shapes while preserving their medial features via restricted power diagram (RPD). Medial features, including external features such as the sharp edges and corners of the input mesh surface and internal features such as the seams and junctions of medial axis, are important shape descriptors both topologically and geometrically. However, existing medial axis approximation methods fail to capture and preserve them due to the fundamentally under-sampling in the vicinity of medial features, and the difficulty to build their correct connections. In this paper we use the RPD of medial spheres and its affiliated structures to help solve these challenges. The dual structure of RPD provides the connectivity of medial spheres. The surface restricted power cell (RPC) of each medial sphere provides the tangential surface regions that these spheres have contact with. The connected components (CC) of surface RPC give us the classification of each sphere, to be on a medial sheet, a seam, or a junction. They allow us to detect insufficient sphere sampling around medial features and develop necessary conditions to preserve them. Using this RPD-based framework, we are able to construct high quality medial meshes with features preserved. Compared with existing sampling-based or voxel-based methods, our method is the first one that can preserve not only external features but also internal features of medial axes.

Motivation & Objective

• To address the failure of existing medial axis approximation methods in preserving both external and internal medial features in 3D shapes.
• To develop a sampling and refinement strategy that guarantees correct tangential properties of medial spheres at features such as seams and junctions.
• To identify insufficient sampling around medial features using connected components (CCs) of surface RPCs.
• To establish a topologically correct and geometrically accurate medial mesh via RPD-based connectivity and refinement.
• To enable feature-preserving medial mesh generation without relying on excessive global sampling or voxelization.

Proposed method

• The method uses restricted power diagrams (RPD) of medial spheres to encode the topological and geometric structure of the medial axis.
• Surface-restricted power cells (RPCs) of each medial sphere are computed to determine the connected tangential surface regions the sphere contacts.
• Connected components (CCs) of each RPC classify the sphere as being on a medial sheet, seam, or junction, enabling feature detection and sampling guidance.
• A geometry-guided thinning algorithm uses RPD to connect medial spheres via medial triangles, ensuring correct topology and geometry.
• Sphere refinement is guided by RPC-based updates to achieve accurate position and radius that match the true medial features.
• The framework detects broken features by analyzing CCs and enforces necessary conditions to preserve them during sampling.

Experimental results

Research questions

• RQ1 How can medial spheres be sampled and updated to preserve their tangential contact properties at external and internal features?
• RQ2 How can insufficient sampling around medial features be detected and corrected?
• RQ3 How can the medial axis be reconstructed with correct topology and geometry, especially for hidden internal features like seams and junctions?
• RQ4 Can a single framework preserve both external and internal features simultaneously, avoiding the need for excessive sampling or global refinement?
• RQ5 What is the role of RPC-connected components in identifying and preserving medial features?

Key findings

• The proposed method generates medial meshes with significantly fewer medial spheres (e.g., 8k for the bear model) compared to existing methods (e.g., 119k for SAT), while maintaining high reconstruction accuracy.
• The method preserves both external features (e.g., sharp corners and edges) and internal features (e.g., junctions and seams), which previous sampling-based and voxel-based methods fail to do.
• Experimental results show that the method achieves lower Hausdorff error in surface reconstruction compared to PC, SAT, and VC, especially on CAD models with sharp features.
• The use of RPC-connected components enables reliable detection of feature breakdowns and guides targeted sampling to preserve medial features.
• The method outperforms existing approaches in preserving the thinness and structural fidelity of the medial axis, particularly in complex CAD models with multiple corners and junctions.
• Despite strong empirical results, the method does not yet guarantee topological equivalence in all cases, with 19 out of 73 test models showing incorrect Euler characteristics in the supplementary material.