A Compressed Sensing Framework for Magnetic Resonance Fingerprinting

Abstract

Abstract. Inspired by the recently proposed Magnetic Resonance Fingerprinting (MRF) technique, we develop a principled compressed sensing framework for quantitative MRI. The three key components are: a random pulse excitation sequence following the MRF technique; a random EPI subsampling strategy and an iterative projection algorithm that imposes consistency with the Bloch equations. We show that theoretically, as long as the excitation sequence possesses an appropriate form of persistent excitation, we are able to accurately recover the proton density, T1, T2 and off-resonance maps simultaneously from a limited number of samples. These results are further supported through extensive simulations using a brain phantom.Key words. Compressed sensing, MRI, Bloch equations, manifolds, Johnston-Linderstrauss embedding 1. Introduction. Inspired by the recently proposed procedure of Magnetic Resonance Fingerprinting (MRF), which gives a new technique for quantitative MRI, we investigate this idea from a compressed sensing perspective. While MRF itself, was inspired by the recent growth of compressed sensing (CS) techniques in MRI [29], the exact link to CS was not made explicit, and the paper does not consider a full CS formulation. Indeed the role of sparsity, random excitation and sampling are not clarified. The goal of this current paper is to make the links with CS explicit, shed light on the appropriate acquisition and reconstruction procedures and hence to develop a full compressed sensing strategy for quantitative MRI.In particular, we identify separate roles for the pulse excitation and the subsampling of k-space. We identify the Bloch response manifold as the appropriate low dimensional signal model on which the CS acquisition is performed, and interpret the "model-based" dictionary of [29] as a natural discretization of this response manifold We also discuss what is necessary in order to have an appropriate CS-type acquisition scheme.Having identified the underlying signal model we next turn to the reconstruction process. In [29] this was performed through pattern matching using a matched filter based on the model-based dictionary. However, this does not offer the opportunity for exact reconstruction, even if the signal is hypothesised to be 1-sparse in this dictionary due to the undersampling of k-space. This suggests that we should look to a model based CS framework that directly supports such manifold models [4]. Recent algorithmic work in this direction has been presented by Iwen and Maggioni [23], however, their approach is not practical in the present context as the computational cost of their scheme grows exponentially with the dimension of the manifold. Instead, we leverage recent results from [11] and develop a recovery algorithm based on the Projected Landweber Algorithm (PLA). This method also has the appealing interpretation of an iterated refinement of the original MRF scheme.The remainder of the paper is set out as follows. We begin by giving a brief overview of MRI acquisition. Then we discuss ...

Low rank magnetic resonance fingerprinting

“…We compare our results to the methods developed by Davies et al 22 and Zhao 29 , and show that FLOR provides quantitative parameter maps with higher accuracy or correspondence to literature compared to those methods.…”

“…Our final algorithm is detailed in Algorithm 4 and referred to as magnetic resonance Fingerprint with LOw Rank (FLOR), where the parameter λ is chosen experimentally. Note that by setting λ = 0, enforcing R to have one-sparse rows and eliminating the acceleration step, FLOR reduces to BLIP 22 . Figure 4 shows the reconstruction error of FLOR as the number of iterations varies with and without the acceleration step.…”

“…Approaches based on exploiting the sparsity of the signal in some transform domain in a compressed sensing (CS) 20,21 framework are examined by Davies et al 22 and Wang et al 23 . Zhao et al 24 formulated MRF as a maximum likelihood (ML) problem, and developed an iterative reconstruction approach for each time point.…”

Low rank alternating direction method of multipliers reconstruction for MR fingerprinting

“…ADMM, like other iterative reconstructions proposed for MRF, constrains the reconstruction directly to atoms of the dictionary (12)(13)(14)20). However, these approaches are nonconvex problems, which are prone to convergence issues.…”

“…A finer grid allowsD to be updated even after small changes inx, potentially improving the convergence. However, the present implementation penalizes the distance to the dictionary, rather than strictly tyingx to the dictionary, as done in the BLIP algorithm (12). Therefore, the step size in the dictionary is not expected to be crucial for convergence and a detailed analysis is omitted here.…”