Exact quantification of time signals in Padé-based magnetic resonance spectroscopy

This study reports on the fast Padé transform (FPT) for parametric signal processing of realistically synthesized free induction decay curves whose main spectral features are similar to those encoded clinically from a healthy human brain by means of magnetic resonance spectroscopy (MRS). Here, for the purpose of diagnostics, it is of paramount importance to be able to perform accurate and robust quantification of the investigated time signals. This amounts to solving the challenging harmonic inversion problem as a spectral decomposition of the given time signal by means of reconstruction of the unknown total number of resonances, their complex frequencies and amplitudes yielding the peak positions, widths, heights and phases. On theoretical grounds, the FPT solves exactly this mathematically ill-conditioned inverse problem for any noiseless synthesized time signal comprised of an arbitrarily large (finite or infinite) number of damped complex exponentials with stationary and non-stationary polynomial-type amplitudes leading to Lorentzian (non-degenerate) and non-Lorentzian (degenerate) spectra. Convergent validation for this fact is given via the proof-of-principle which is thoroughly demonstrated by the exact numerical solution of a typical quantification problem from MRS. The presently designed study is a paradigm shift for signal processing in MRS with particular relevance to clinical oncology, due to the unprecedented capability of the fast Padé transform to unequivocally resolve and quantify isolated, tightly overlapped and nearly coincident resonances.