Cover tree compressed sensing for fast mr fingerprint recovery

We adopt a data structure in the form of cover trees and iteratively apply approximate nearest neighbour (ANN) searches for fast compressed sensing reconstruction of signals living on discrete smooth manifolds. Leveraging on the recent stability results for the inexact Iterative Projected Gradient (IPG) algorithm and by using the cover tree's ANN searches, we decrease the projection cost of the IPG algorithm to be logarithmically growing with data population for low dimensional smooth manifolds. We apply our results to quantitative MRI compressed sensing and in particular within the Magnetic Resonance Fingerprinting (MRF) framework. For a similar (or sometimes better) reconstruction accuracy, we report 2-3 orders of magnitude reduction in computations compared to the standard iterative method, which uses brute-force searches.