Fast Low Rank Column-Wise Compressive Sensing For Accelerated Dynamic MRI

Abstract

In recent work we developed a fast and sample-efficient gradient descent (GD) solution to the following "Low Rank column-wise Compressive Sensing (LRcCS)": recover an n × q, rank-r matrix X <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> from measurements ${{\mathbf{y}}_k} = {{\mathbf{A}}_k}{\mathbf{x}}_k^{\ast}$, $k = 1,2, \ldots ,q$ when each y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> is an m-length vector with m < n, and the rank r ≪ min(n,q). Accelerated dynamic MRI is a key application where this problem occurs. In this work, we show the power of our approach (and of its modification for the MRI setting) for four very different highly undersampled dynamic MRI applications. Without any application-specific parameter tuning, in most settings, our approach outperforms the state-of-the-art MRI methods, while also being significantly faster in all settings.