  Tagged magnetic resonance imaging (tMRI) has been employed for decades to
measure the motion of tissue undergoing deformation. However,
registration-based motion estimation from tMRI is difficult due to the periodic
patterns in these images, particularly when the motion is large. With a larger
motion the registration approach gets trapped in a local optima, leading to
motion estimation errors. We introduce a novel "momenta, shooting, and
correction" framework for Lagrangian motion estimation in the presence of
repetitive patterns and large motion. This framework, grounded in Lie algebra
and Lie group principles, accumulates momenta in the tangent vector space and
employs exponential mapping in the diffeomorphic space for rapid approximation
towards true optima, circumventing local optima. A subsequent correction step
ensures convergence to true optima. The results on a 2D synthetic dataset and a
real 3D tMRI dataset demonstrate our method's efficiency in estimating
accurate, dense, and diffeomorphic 2D/3D motion fields amidst large motion and
repetitive patterns.
