Validation of reconstructed component spectra from non-parametric derivative envelopes: comparison with component lineshapes from parametric derivative estimations with the solved quantification problem

Of late, we have put forward a new branch called high-order derivative signal processing. This investigative strategy is universally relevant for all spectroscopies, the progress of which ultimately depends on resolution improvement and noise suppression. The high-order, non-parametric derivative fast Padé transform (dFPT) simultaneously solves these two problems of utmost importance. The present work goes one critical step further than our previous two studies on this particular topic, by setting up the goal of validating the non-parametric dFPT by its parametric counterpart. This is done by comparing the full lineshapes of derivative envelopes from the non-parametric dFPT with the corresponding derivative component spectra from the parametric dFPT. The non-parametric dFPT, as a shape estimator, never solves the quantification problem (or equivalently, the spectral analysis problem via e.g. an eigenvalue problem, rooting characteristic/secular equations, etc.). The parametric dFPT first solves the quantification problem from which the lineshapes of components and envelopes are plotted. Thus, if the derivative component spectra from the parametric dFPT could be fully reconstructed by the derivative envelopes from the non-parametric dFPT, the goal of achieving quantification would be done by derivative lineshape processing alone. This would amount to providing stand-alone quantification without actually solving the quantification problem (and, of course, without fitting, either). The present study accomplishes this goal, with an important application to data encountered in magnetic resonance spectroscopy for clinical diagnostics of breast cancer.