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We introduce novel algorithms for the joint recovery of an ensemble of signals that live on a smooth manifold from their under sampled measurements. Unlike current methods that are designed to recover a single signal assuming perfect knowledge of the manifold model, the proposed algorithms exploit similarity between the signals without prior knowledge of the underlying manifold structure. Our first algorithm is a two-step scheme, where the Laplacian of the graph associated with the manifold is estimated from similar under sampled measurements made on the signals; this Laplacian is used to formulate the problem as a penalized optimization scheme, where smoothness of the signals on the manifold is chosen as the penalty. The second algorithm is an iterative scheme that alternates between computation of the Laplacian and the signals. Validation of the proposed algorithms using simulations and experimental MRI data demonstrate their utility in accelerating free breathing cardiac MRI.

Joint recovery of under sampled signals on a manifold: Application to free breathing cardiac MRI