MATStruct: High-Quality Medial Mesh Computation via Structure-aware Variational Optimization

TL;DR

This framework is the first to integrate structural awareness into the optimization process, yielding medial meshes with explicit structural decomposition, topological correctness, and geometric fidelity.

Abstract

We propose a novel optimization framework for computing the medial axis transform that simultaneously preserves the medial structure and ensures high medial mesh quality. The medial structure, consisting of interconnected sheets, seams, and junctions, provides a natural volumetric decomposition of a 3D shape. Our method introduces a structure-aware, particle-based optimization pipeline guided by the restricted power diagram (RPD), which partitions the input volume into convex cells whose dual encodes the connectivity of the medial mesh. Structure-awareness is enforced through a spherical quadratic error metric (SQEM) projection that constrains the movement of medial spheres, while a Gaussian kernel energy encourages an even spatial distribution. Compared to feature-preserving methods such as MATFP and MATTopo, our approach produces cleaner and more accurate medial structures with significantly improved mesh quality. In contrast to voxel-based, point-cloud-based, and variational methods, our framework is the first to integrate structural awareness into the optimization process, yielding medial meshes with superior geometric fidelity, topological correctness, and explicit structural decomposition.

Figures (10)

• Figure 1: The medial mesh generated for a hollow cylinder (rendered transparently) using VC yan2018voxel, MATFP 2022MATFP, and our method. Medial seams are shown as red lines, with corresponding spheres highlighted. (a) VC produces excessive medial sheets, resulting in redundant and spurious seams. (b) MATFP generates densely clustered medial spheres around seams, leading to numerical instability and erroneous seam connections (see Sec. \ref{['sec:pre_challenge']} for details). (c) Our method produces a high-quality 3D medial mesh with clear structural decomposition. Note that we show zoom-ins at two different scales to better reveal the redundant sheets in VC.
• Figure 2: Left: The medial axis of a cube illustrating the medial structure described in Sec. \ref{['sec:pre_structure']}. Right: 2D illustration of classification (a), overcrowding issues (b) described in Sec. \ref{['sec:pre_challenge']} using three spheres with centers $\boldsymbol \theta_i$, $\boldsymbol \theta_j$, and $\boldsymbol \theta_k$, where two of them slightly deviate from the medial axis (shown as a red dotted line).
• Figure 3: Illustration of our RPD-based spherical quadratic error metric described in Sec. \ref{['sec:method_qem']}.
• Figure 4: Illustration of our gradient projection strategy described in Sec. \ref{['sec:particle_grad_proj']}.
• Figure 5: Left: the sphere-shrinking algorithm ma20123shrink used in Sec. \ref{['sec:tech_sph_proj']}. Right: Illustration of pruning invalid medial connections described in Sec. \ref{['sec:feature_post_processing']}.