Kernel Bi-Linear Modeling for Reconstructing Data on Manifolds: The Dynamic-MRI Case

This paper establishes a kernel-based framework for reconstructing data on manifolds, tailored to fit the dynamic-(d)MRI-data recovery problem. The proposed methodology exploits simple tangent-space geometries of manifolds in reproducing kernel Hilbert spaces, and follows classical kernel-approximation arguments to form the data-recovery task as a bilinear inverse problem. Departing from mainstream approaches, the proposed methodology uses no training data, employs no graph Laplacian matrix to penalize the optimization task, uses no costly (kernel) preimaging step to map feature points back to the input space, and utilizes complex-valued kernel functions to account for k-space data. The framework is validated on synthetically generated dMRI data, where comparisons against state-of-the-art schemes highlight the rich potential of the proposed approach in data-recovery problems.