  Reconstruction of object or scene surfaces has tremendous applications in
computer vision, computer graphics, and robotics. In this paper, we study a
fundamental problem in this context about recovering a surface mesh from an
implicit field function whose zero-level set captures the underlying surface.
To achieve the goal, existing methods rely on traditional meshing algorithms;
while promising, they suffer from loss of precision learned in the implicit
surface networks, due to the use of discrete space sampling in marching cubes.
Given that an MLP with activations of Rectified Linear Unit (ReLU) partitions
its input space into a number of linear regions, we are motivated to connect
this local linearity with a same property owned by the desired result of
polygon mesh. More specifically, we identify from the linear regions,
partitioned by an MLP based implicit function, the analytic cells and analytic
faces that are associated with the function's zero-level isosurface. We prove
that under mild conditions, the identified analytic faces are guaranteed to
connect and form a closed, piecewise planar surface. Based on the theorem, we
propose an algorithm of analytic marching, which marches among analytic cells
to exactly recover the mesh captured by an implicit surface network. We also
show that our theory and algorithm are equally applicable to advanced MLPs with
shortcut connections and max pooling. Given the parallel nature of analytic
marching, we contribute AnalyticMesh, a software package that supports
efficient meshing of implicit surface networks via CUDA parallel computing, and
mesh simplification for efficient downstream processing. We apply our method to
different settings of generative shape modeling using implicit surface
networks. Extensive experiments demonstrate our advantages over existing
methods in terms of both meshing accuracy and efficiency.
