Learning Geodesics of Geometric Shape Deformations From Images
This addresses the challenge of quantifying deformable shapes in images for medical imaging and shape analysis, though it appears incremental by building on existing registration networks.
The paper tackles the problem of learning geodesic flows of deformation fields from images, introducing geodesic deformable networks (GDN) to predict geodesics for quantifying and comparing deformable shapes, and demonstrates effectiveness on 2D synthetic and 3D brain MRI data with improved regularizability and generalizability.
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).