Joint Shape Reconstruction and Registration via a Shared Hybrid Diffeomorphic Flow.

Deep implicit functions (DIFs) effectively represent shapes by using a neural network to map 3D spatial coordinates to scalar values that encode the shape's geometry, but it is difficult to establish correspondences between shapes directly, limiting their use in medical image registration. The recently presented deformation field-based methods achieve implicit templates learning via template field learning with DIFs and deformation field learning, establishing shape correspondence through deformation fields. Although these approaches enable joint learning of shape representation and shape correspondence, the decoupled optimization for template field and deformation field, caused by the absence of deformation annotations lead to a relatively accurate template field but an underoptimized deformation field. In this paper, we propose a novel implicit template learning framework via a shared hybrid diffeomorphic flow (SHDF), which enables shared optimization for deformation and template, contributing to better deformations and shape representation. Specifically, we formulate the signed distance function (SDF, a type of DIFs) as a one-dimensional (1D) integral, unifying dimensions to match the form used in solving ordinary differential equation (ODE) for deformation field learning. Then, SDF in 1D integral form is integrated seamlessly into the deformation field learning. Using a recurrent learning strategy, we frame shape representations and deformations as solving different initial value problems of the same ODE. We also introduce a global smoothness regularization to handle local optima due to limited outside-of-shape data. Experiments on medical datasets show that SHDF outperforms state-of-the-art methods in shape representation and registration.