Learning Geodesics of Geometric Shape Deformations From Images

TL;DR

Geodesic deformations provide a principled basis for comparing deformable shapes in images, but existing deep registration methods learn only initial velocity fields and ignore geodesic structure. The authors propose GDN, a Geodesic Neural Operator that learns geodesic mappings in latent deformation spaces and enforces a geodesic loss, enabling direct prediction of geodesic flows without solving differential equations at inference. They model the geodesic evolution with a Fourier-based neural operator, demonstrate close alignment with numerical EPDiff solutions on 2D data and 3D brain MRIs, and show improvements in regularization, generalization, and inference speed compared with baselines. The work lays groundwork for interpretable deformation analysis and potential geodesic regression in population studies.

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).

Figures (9)

• Figure 1: An overview of our proposed network GDN.
• Figure 2: Visulization of predicted geodesics from GDN vs. real numerical solutions zhang2015fast and predictions from NeurEPDiff wu2023neurepdiff. Left to right: source and target images, predicted geodesic deformations along time $t$. Top to bottom: deformed images, transformation fields, and velocity fields.
• Figure 3: Top to bottom: Comparison of predicted geodesics by GDN/NeurEPDiff vs. numerical integration of EPDiff. Left to right: MSE between deformed images, transformations, and velocities along the geodesic path.
• Figure 4: A comparison of predicted transformation grids and their associated determinant of Jacobian maps on in-distribution (ID) testing data from our method GDN vs. baselines. From top to bottom: source and target image pairs, resulting deformed images, predicted transformation grids, and determinant of Jacobian maps.
• Figure 5: A comparison of predicted transformation grids and their associated determinant of Jacobian maps on out-of-distribution (OOD) testing data from our method GDN vs. baselines. Top to bottom: source and target images, deformed source images, predicted deformation fields, and determinant of Jacobian maps.