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.
View on arXiv