Abstract

This paper presents a novel method, named geodesic deformable networks (GDN), that for the first time enables the learning of geodesic flows of deformation fields derived from images. In particular, the capability of our proposed GDN being able to predict geodesics is important for quantifying and comparing deformable shape presented in images. The geodesic deformations, also known as optimal transformations that align pairwise images, are often parameterized by a time sequence of smooth vector fields governed by nonlinear differential equations. A bountiful literature has been focusing on learning the initial conditions (e.g., initial velocity fields) based on registration networks. However, the definition of geodesics central to deformation-based shape analysis is blind to the networks. To address this problem, we carefully develop an efficient neural operator to treat the geodesics as unknown mapping functions learned from the latent deformation spaces. A composition of integral operators and smooth activation functions is then formulated to effectively approximate such mappings. In contrast to previous works, our GDN jointly optimizes a newly defined geodesic loss, which adds additional benefits to promote the network regularizability and generalizability. We demonstrate the effectiveness of GDN on both 2D synthetic data and 3D real brain magnetic resonance imaging (MRI).