Quasi-Medial Distance Field (Q-MDF): A Robust Method for Approximating and Discretizing Neural Medial Axes

TL;DR

This work addresses robust extraction of the medial axis from imperfect 3D data by learning a neural implicit representation called Q-MDF, defined as the difference between the medial field and the absolute signed distance field, i.e., $f_q(p)=f_m(p)-|f_{sdf}(p)|$. This representation enables a differentiable, compact embedding of the medial axis, from which a medial membrane is extracted via a modified double-covering procedure that collapses to a zero-volume medial surface; the radius information is retrieved from the SDF. The approach demonstrates superior robustness and accuracy across challenging meshes, sparse point clouds, and real scans, outperforming traditional discretization-based methods and prior learning-based schemes in producing topologically coherent and smooth medial structures. The work further enhances fidelity near sharp features through MF-guided priors and discusses practical avenues for efficiency and topology improvement, highlighting its potential impact on digital geometry processing, shape analysis, and CAD-like modeling.

Abstract

The medial axis, a lower-dimensional descriptor that captures the extrinsic structure of a shape, plays an important role in digital geometry processing. Despite its importance, computing the medial axis transform robustly from diverse inputs, especially point clouds with defects, remains a challenging problem. In this paper, we propose a new implicit method that deviates from traditional explicit medial axis computation. Our key technical insight is that the difference between the signed distance field (SDF) and the medial field (MF) of a solid shape relates to the unsigned distance field (UDF) of the shape's medial axis. This observation allows us to formulate medial axis extraction as an implicit reconstruction problem. By employing a modified double covering strategy, we recover the medial axis as the zero level-set of the UDF. Extensive experiments demonstrate that our method achieves higher accuracy and robustness in learning compact medial axis transforms from challenging meshes and point clouds, outperforming existing approaches.

Figures (20)

• Figure 1: 2D illustration of the medial axis and medial axis transform. The medial axis, a lower-dimensional and inherently unstable representation of the original shape, is highly sensitive to boundary perturbations that can significantly alter its topology.
• Figure 2: Distance fields and medial fields. Given a 2D triangle, (a) visualizes the signed distance field, and (b) the medial field. In (c), the medial distance field (MDF), which is defined as the UDF of the medial axis. (d) illustrates the quasi-medial distance field (Q-MDF), computed by subtracting the SDF from the MF, which effectively approximates the medial distance field.
• Figure 3: Pipeline of our medial axis computation method. The input consists of either a point cloud or sampled points from a given mesh. After joint training of the signed distance field and medial field, we compute the implicit medial axis representation, termed Q-MDF, by taking their difference. A double covering envelope of the medial axis is then extracted and collapsed to form a zero-volume medial membrane. Optionally, sharp features can be enhanced by detecting feature points from both the input point cloud and the medial membrane, followed by tr-training the network with this additional feature constraint.
• Figure 4: 2D illustration of feature point localization. (a) Sample gradients at four neighboring points. (b) Cluster them into two directions representing bilateral gradients; the marching direction is determined by their sum at the boundary point. (c)-(d) March along this direction while gradually decreasing the step size $d_i$ until it approaches zero. (e) Identify the $k$ nearest surface points to the shifted boundary point. (f) Apply a quadratic error metric to determine the feature point position for this corner.
• Figure 5: SDF, MF, Q-MDF values queried along the $x$-axis of a cone, showing that the local minimum of the deep Q-MDF occurs at the central axis ($x=0$).

Theorems & Definitions (2)

• Definition 1
• Definition 2