Sparsity/undersampling tradeoffs in anisotropic undersampling, with applications in MR imaging/spectroscopy

Abstract

Abstract We study anisotropic undersampling schemes like those used in multi-dimensional magnetic resonance (MR) spectroscopy and imaging, which sample exhaustively in certain time dimensions and randomly in others. Our analysis shows that anisotropic undersampling schemes are equivalent to certain block-diagonal measurement systems. We develop novel exact formulas for the sparsity/undersampling tradeoffs in such measurement systems, assuming uniform sparsity fractions in each column. Our formulas predict finite-$N$ phase transition behavior differing substantially from the well-known asymptotic phase transitions for classical Gaussian undersampling. Extensive empirical work shows that our formulas accurately describe observed finite-$N$ behavior, while the usual formulas based on universality are substantially inaccurate at the moderate $N$ involved in realistic applications. We also vary the anisotropy, keeping the total number of samples fixed, and for each variation we determine the precise sparsity/undersampling tradeoff (phase transition). We show that, other things being equal, the ability to recover a sparse object decreases with an increasing number of exhaustively sampled dimensions.