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

摘要

We study anisotropic undersampling schemes like those used inmulti-dimensional NMR spectroscopy and MR imaging, which sample exhaustively incertain time dimensions and randomly in others. Our analysis shows that anisotropic undersampling schemes are equivalent tocertain block-diagonal measurement systems. We develop novel exact formulas forthe sparsity/undersampling tradeoffs in such measurement systems. Our formulaspredict finite-$N$ phase transition behavior differing substantially from thewell known asymptotic phase transitions for classical Gaussian undersampling.Extensive empirical work shows that our formulas accurately describe observedfinite-$N$ behavior, while the usual formulas based on universality aresubstantially inaccurate. We also vary the anisotropy, keeping the total number of samples fixed, andfor each variation we determine the precise sparsity/undersampling tradeoff(phase transition). We show that, other things being equal, the ability torecover a sparse object decreases with an increasing number ofexhaustively-sampled dimensions.