  Deep Implicit Functions (DIFs) represent 3D geometry with continuous signed
distance functions learned through deep neural nets. Recently DIFs-based
methods have been proposed to handle shape reconstruction and dense point
correspondences simultaneously, capturing semantic relationships across shapes
of the same class by learning a DIFs-modeled shape template. These methods
provide great flexibility and accuracy in reconstructing 3D shapes and
inferring correspondences. However, the point correspondences built from these
methods do not intrinsically preserve the topology of the shapes, unlike
mesh-based template matching methods. This limits their applications on 3D
geometries where underlying topological structures exist and matter, such as
anatomical structures in medical images. In this paper, we propose a new model
called Neural Diffeomorphic Flow (NDF) to learn deep implicit shape templates,
representing shapes as conditional diffeomorphic deformations of templates,
intrinsically preserving shape topologies. The diffeomorphic deformation is
realized by an auto-decoder consisting of Neural Ordinary Differential Equation
(NODE) blocks that progressively map shapes to implicit templates. We conduct
extensive experiments on several medical image organ segmentation datasets to
evaluate the effectiveness of NDF on reconstructing and aligning shapes. NDF
achieves consistently state-of-the-art organ shape reconstruction and
registration results in both accuracy and quality. The source code is publicly
available at https://github.com/Siwensun/Neural_Diffeomorphic_Flow--NDF.
