  Purpose: Training auto-encoders on simulated signal evolution and inserting
the decoder into the forward model improves reconstructions through more
compact, Bloch-equation-based representations of signal in comparison to linear
subspaces.
  Methods: Building on model-based nonlinear and linear subspace techniques
that enable reconstruction of signal dynamics, we train auto-encoders on
dictionaries of simulated signal evolution to learn more compact, non-linear,
latent representations. The proposed Latent Signal Model framework inserts the
decoder portion of the auto-encoder into the forward model and directly
reconstructs the latent representation. Latent Signal Models essentially serve
as a proxy for fast and feasible differentiation through the Bloch-equations
used to simulate signal. This work performs experiments in the context of
T2-shuffling, gradient echo EPTI, and MPRAGE-shuffling. We compare how
efficiently auto-encoders represent signal evolution in comparison to linear
subspaces. Simulation and in-vivo experiments then evaluate if reducing degrees
of freedom by inserting the decoder into the forward model improves
reconstructions in comparison to subspace constraints.
  Results: An auto-encoder with one real latent variable represents FSE, EPTI,
and MPRAGE signal evolution as well as linear subspaces characterized by four
basis vectors. In simulated/in-vivo T2-shuffling and in-vivo EPTI experiments,
the proposed framework achieves consistent quantitative NRMSE and qualitative
improvement over linear approaches. From qualitative evaluation, the proposed
approach yields images with reduced blurring and noise amplification in MPRAGE
shuffling experiments.
  Conclusion: Directly solving for non-linear latent representations of signal
evolution improves time-resolved MRI reconstructions through reduced degrees of
freedom.
