Tagged magnetic resonance imaging (tMRI) is a valuable tool for visualizing and quantifying tissue deformation in vivo. Its use is often hampered, however, by tag fading, long computation times, and the challenge of ensuring diffeomorphic, incompressible motion fields. In this paper, we describe a novel integration of the harmonic phase (HARP) approach to tMRI analysis with an unsupervised deep learning-based registration framework to estimate 2D and 3D motion fields that are diffeomorphic and nearly incompressible. The resulting method, called deep sinusoidally transformed HARP, or DSHARP, enables end-to-end network training by implementing a transformation of the harmonic phase to remove phase-wrapping discontinuities. It produces diffeomorphic motion by estimating a stationary velocity field from which motion is computed using the scaling and squaring technique. Finally, it encourages incompressibility using a novel Jacobian determinant loss term during network training. We evaluated DSHARP on 2D and 3D phantom data with simulated incompressible motions, real 3D human tongue data acquired during speech from both healthy and glossectomy subjects, and cardiac tagged MRI from the public STACOM 2011 benchmark. Our approach outperforms HARP, SinMod, SyN, PVIRA, VoxelMorph, and DeepTag in tracking accuracy, computation speed, and preservation of incompressibility.
Bian et al. (2026) studied this question.