Code and data supplement for "Sparsity/Undersampling Tradeoffs in Anisotropic Undersampling, with Applications in MR Imaging/Spectroscopy"

The data and code provided here are supplementary material for the Information and Inference paper “Sparsity/Undersampling Tradeoffs in Anisotropic Undersampling, with Applications in MR Imaging/Spectroscopy" by H. Monajemi, and D.L. Donoho. Please read README file for reproducing the results of the article.

Abstract of the article:
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 dense undersampling.
Extensive empirical work shows that our formulas accurately describe observed finite-N behavior,
while the usual asymptotic predictions 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 spectrum decreases with
an increasing number of exhaustively-sampled dimensions.