NeurEPDiff: Neural Operators to Predict Geodesics in Deformation Spaces
This work addresses the computational bottleneck in diffeomorphic image registration for medical imaging and synthetic data, offering a faster and more generalizable solution, though it is incremental as it builds on existing geodesic shooting methods.
This paper tackles the problem of efficiently predicting geodesics in deformation spaces for image registration by introducing NeurEPDiff, a neural operator that learns the evolving trajectory of geodesic deformations, resulting in significantly reduced computational cost and demonstrating effectiveness on 2D synthetic and 3D brain MRI datasets with improved registration accuracy and efficiency compared to state-of-the-art methods.
This paper presents NeurEPDiff, a novel network to fast predict the geodesics in deformation spaces generated by a well known Euler-Poincaré differential equation (EPDiff). To achieve this, we develop a neural operator that for the first time learns the evolving trajectory of geodesic deformations parameterized in the tangent space of diffeomorphisms(a.k.a velocity fields). In contrast to previous methods that purely fit the training images, our proposed NeurEPDiff learns a nonlinear mapping function between the time-dependent velocity fields. A composition of integral operators and smooth activation functions is formulated in each layer of NeurEPDiff to effectively approximate such mappings. The fact that NeurEPDiff is able to rapidly provide the numerical solution of EPDiff (given any initial condition) results in a significantly reduced computational cost of geodesic shooting of diffeomorphisms in a high-dimensional image space. Additionally, the properties of discretiztion/resolution-invariant of NeurEPDiff make its performance generalizable to multiple image resolutions after being trained offline. We demonstrate the effectiveness of NeurEPDiff in registering two image datasets: 2D synthetic data and 3D brain resonance imaging (MRI). The registration accuracy and computational efficiency are compared with the state-of-the-art diffeomophic registration algorithms with geodesic shooting.