  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.
