2017/02/09 by Hatef Monajemi, David L. Donoho, Monajemi, Hatef +1
Chemistry · Computer Science · Engineering · Mathematics · Medicine · Physics and Astronomy · #Advanced MRI Techniques and Applications #Advanced NMR Techniques and Applications #Algorithm #Anisotropy #Artificial intelligence #Computer science #Diagonal #FOS: Computer and information sciences #Gaussian #Geometry #Information Theory (cs.IT) #Mathematics #NMR spectroscopy and applications #Optics #Physics #Quantum mechanics #Sparse and Compressive Sensing Techniques #Statistical physics #Undersampling #cs.IT #math.IT
paper · pdf · doi:10.48550/arxiv.1702.03062
openalex publication_date 2017/02/09 · arxiv created 2018/03/16 · arxiv updated 2018/03/20 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We study anisotropic undersampling schemes like those used in\nmulti-dimensional NMR spectroscopy and MR imaging, which sample exhaustively in\ncertain time dimensions and randomly in others.\n Our analysis shows that anisotropic undersampling schemes are equivalent to\ncertain block-diagonal measurement systems. We develop novel exact formulas for\nthe sparsity/undersampling tradeoffs in such measurement systems. Our formulas\npredict finite-N phase transition behavior differing substantially from the\nwell known asymptotic phase transitions for classical Gaussian undersampling.\nExtensive empirical work shows that our formulas accurately describe observed\nfinite-N behavior, while the usual formulas based on universality are\nsubstantially inaccurate.\n We also vary the anisotropy, keeping the total number of samples fixed, and\nfor each variation we determine the precise sparsity/undersampling tradeoff\n(phase transition). We show that, other things being equal, the ability to\nrecover a sparse object decreases with an increasing number of\nexhaustively-sampled dimensions.\n