ENF-SSM: Amortizing Anatomical Statistical Shape Modeling with Elastic Neural Flow

Riemannian statistical shape analysis repeatedly computes deformation paths to estimate templates, evolve initial fields, and transfer deformations. We introduce ENF-SSM, which amortizes these computations through a shared endpoint-conditioned neural path. Building on the Spectral Meets Spatial backbone, we use an elastic regularizer derived from established bending, scaling, and shearing decompositions, together with spatial smoothness. The learned path supports a finite-step logarithm surrogate, frozen-network endpoint inversion, recursive template estimation, and deformation-based path regression. We derive the conditional bounds that separate time-discretization error, reference-path error, inverse conditioning, and endpoint attachment. These results identify the assumptions needed for operator reuse. Existing experiments on pancreas, hippocampus, and heart meshes report competitive reconstruction and lower online runtimes than the evaluated iterative implementations. They provide evidence for the computational utility of the shared path and its potential values in anatomical shape analysis.