Fast Algorithm for Sparse Signal Approximation using Multiple Additive Dictionaries

There are several models for sparse approximation: one where a signal is a sparse linear combination of vectors over a redundant dictionary and a second model in which a collection of signals is a simultaneous sparse linear combination over a single dictionary. In this work, interpolate between these two models to synthesize a single signal of interest from K highly incoherent dictionaries while enforcing simultaneous sparsity on the K resulting coefficient vectors. We define this as the parallel approximation problem, which arises quite naturally in many applications such as MRI parallel excitation using multiple transmission coils. We present an efficient algorithm to solve the parallel approximation problem called Parallel Orthogonal Matching Pursuit (POMP). We prove its correctness in a general setting and then discuss adaptations needed to make it suitable for use in an MRI parallel excitation setting. We then discuss parallel excitation in more detail and demonstrate how POMP solves the problem as accurately, but much faster, than previously proposed convex optimization methods.