  Dense vertex-to-vertex correspondence between 3D faces is a fundamental and
challenging issue for 3D&2D face analysis. While the sparse landmarks have
anatomically ground-truth correspondence, the dense vertex correspondences on
most facial regions are unknown. In this view, the current literatures commonly
result in reasonable but diverse solutions, which deviate from the optimum to
the 3D face dense registration problem. In this paper, we revisit dense
registration by a dimension-degraded problem, i.e. proportional segmentation of
a line, and employ an iterative dividing and diffusing method to reach the
final solution uniquely. This method is then extended to 3D surface by
formulating a local registration problem for dividing and a linear least-square
problem for diffusing, with constraints on fixed features. On this basis, we
further propose a multi-resolution algorithm to accelerate the computational
process. The proposed method is linked to a novel local scaling metric, where
we illustrate the physical meaning as smooth rearrangement for local cells of
3D facial shapes. Extensive experiments on public datasets demonstrate the
effectiveness of the proposed method in various aspects. Generally, the
proposed method leads to coherent local registrations and elegant mesh grid
routines for fine-grained 3D face dense registrations, which benefits many
downstream applications significantly. It can also be applied to dense
correspondence for other format of data which are not limited to face. The core
code will be publicly available at
https://github.com/NaughtyZZ/3D_face_dense_registration.
