  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.
