Sparsity/Undersampling Tradeoffs in Anisotropic Undersampling, with Applications in MR Imaging/Spectroscopy
| Autor: | David L. Donoho, Hatef Monajemi |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Statistics and Probability
FOS: Computer and information sciences Phase transition Computer science Computer Science - Information Theory Gaussian 02 engineering and technology 01 natural sciences 010104 statistics & probability symbols.namesake Multiple time dimensions 0202 electrical engineering electronic engineering information engineering 0101 mathematics Anisotropy Spectroscopy Numerical Analysis Applied Mathematics System of measurement Information Theory (cs.IT) 020206 networking & telecommunications Universality (dynamical systems) Computational Theory and Mathematics Undersampling symbols Algorithm Analysis |
| DOI: | 10.48550/arxiv.1702.03062 |
| Popis: | We study anisotropic undersampling schemes like those used in multi-dimensional magnetic resonance (MR) spectroscopy and 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, assuming uniform sparsity fractions in each column. 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 at the moderate $N$ involved in realistic applications. 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. |
| Databáze: | OpenAIRE |
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