Sparsity/Undersampling Tradeoffs in Anisotropic Undersampling, with Applications in MR Imaging/Spectroscopy

We study anisotropic undersampling schemes like those used in multi-dimensional NMR spectroscopy and MR 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. 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. 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.