Deformable templates using large deformation kinematics

TLDR

Large‑magnitude deformations require kinematic nonlinearities, and unlike quadratic penalty methods, stress relaxation over time permits such deformations. The study proposes an automatic method to accommodate local shape variation when aligning a template image with a topologically similar target image. The method applies a smooth vector‑field transformation constrained by a Stokesian limit, solves nonlinear PDEs for optimal mapping, and automatically regrids when the finite lattice becomes singular. Application to intersubject registration of neuroanatomical structures demonstrates the method’s ability to account for local anatomical variability.

Abstract

A general automatic approach is presented for accommodating local shape variation when mapping a two-dimensional (2-D) or three-dimensional (3-D) template image into alignment with a topologically similar target image. Local shape variability is accommodated by applying a vector-field transformation to the underlying material coordinate system of the template while constraining the transformation to be smooth (globally positive definite Jacobian). Smoothness is guaranteed without specifically penalizing large-magnitude deformations of small subvolumes by constraining the transformation on the basis of a Stokesian limit of the fluid-dynamical Navier-Stokes equations. This differs fundamentally from quadratic penalty methods, such as those based on linearized elasticity or thin-plate splines, in that stress restraining the motion relaxes over time allowing large-magnitude deformations. Kinematic nonlinearities are inherently necessary to maintain continuity of structures during large-magnitude deformations, and are included in all results. After initial global registration, final mappings are obtained by numerically solving a set of nonlinear partial differential equations associated with the constrained optimization problem. Automatic regridding is performed by propagating templates as the nonlinear transformations evaluated on a finite lattice become singular. Application of the method to intersubject registration of neuroanatomical structures illustrates the ability to account for local anatomical variability.