  This paper studies a problem of learning surface mesh via implicit functions
in an emerging field of deep learning surface reconstruction, where implicit
functions are popularly implemented as multi-layer perceptrons (MLPs) with
rectified linear units (ReLU). To achieve meshing from learned implicit
functions, existing methods adopt the de-facto standard algorithm of marching
cubes; while promising, they suffer from loss of precision learned in the MLPs,
due to the discretization nature of marching cubes. Motivated by the knowledge
that a ReLU based MLP partitions its input space into a number of linear
regions, we identify from these regions analytic cells and analytic faces that
are associated with zero-level isosurface of the implicit function, and
characterize the theoretical conditions under which the identified analytic
faces are guaranteed to connect and form a closed, piecewise planar surface.
Based on our theorem, we propose a naturally parallelizable algorithm of
analytic marching, which marches among analytic cells to exactly recover the
mesh captured by a learned MLP. Experiments on deep learning mesh
reconstruction verify the advantages of our algorithm over existing ones.
