Random matrices in Magnetic Resonance signal processing

Many important problems in the processing of Magnetic Resonance data can be reduced to well known ill-posed inverse problems. MR relaxometry, spectroscopy and MR image formation problems can be formulated respectively as the numerical inversion of Laplace transform, the modal analysis problem and the truncated trigonometric moment problem. A unified framework to efficiently solve these problems is provided by considering random Pade’ approximants to the Z-transform s(z) of the measured signal. It turns out that the singularities of s(z) on the complex plane are the key quantities to make inference. Their location can be estimated from the logarithmic potential of the condensed density of the Pade’ poles or, equivalently, of the eigenvalues of a random pencil of Hankel matrices. Several alternatives to compute these quantities are discussed and compared by numerical examples.