PoNQ: a Neural QEM-based Mesh Representation

TL;DR

This work introduces a novel learnable mesh representation through a set of local 3D sample Points and their associated Normals and Quadric error metrics (QEM) w.r.t. the underlying shape, which is denote PoNQ, and demonstrates the efficacy of PoNQ through a learning-based mesh prediction from SDF grids.

Abstract

Although polygon meshes have been a standard representation in geometry processing, their irregular and combinatorial nature hinders their suitability for learning-based applications. In this work, we introduce a novel learnable mesh representation through a set of local 3D sample Points and their associated Normals and Quadric error metrics (QEM) w.r.t. the underlying shape, which we denote PoNQ. A global mesh is directly derived from PoNQ by efficiently leveraging the knowledge of the local quadric errors. Besides marking the first use of QEM within a neural shape representation, our contribution guarantees both topological and geometrical properties by ensuring that a PoNQ mesh does not self-intersect and is always the boundary of a volume. Notably, our representation does not rely on a regular grid, is supervised directly by the target surface alone, and also handles open surfaces with boundaries and/or sharp features. We demonstrate the efficacy of PoNQ through a learning-based mesh prediction from SDF grids and show that our method surpasses recent state-of-the-art techniques in terms of both surface and edge-based metrics.

Figures (17)

• Figure 1: PoNQ representation. Quadric error metric (QEM) matrices are fitted and visualized on a given shape (left). Note that their aspect ratios (green to blue) capture the underlying surface information: pancakes for flat regions, cigars for sharp edges and balls for corners. Our mesh extraction outputs a watertight (or, optionally, open, see bottom) and non-self-intersecting mesh that preserves salient features of the input shape (center) and is more concise and faithful than current implicit approaches (right).
• Figure 2: 2D illustration of PoNQ: (a) a sampled ground-truth shape $S$ (blue dots) is represented by PoNQ as points $\mathbf{p}_i$ (whose Voronoi diagram (dotted lines) partitions the input samples), along with normals $\mathbf{n}_i$ and quadrics $\mathbf{Q}_i$ encoding the shape within each Voronoi cell. (b) The PoNQ mesh (black solid lines) is the boundary of the union of labeled tetrahedra from the Delaunay triangulation of the QEM-optimal vertices, providing a better fit than simply interpolating the points (black dotted lines).
• Figure 3: Overview of our learning pipeline with PoNQ.
• Figure 4: Optimization-based results $(32^3)$.
• Figure 5: Learning results (top: $32^3$; bottom: $64^3$) on ABC.