Exact quantification by the nonparametric fast Padé transform using only shape estimation of high-order derivatives of envelopes
Dževad Belkić, Karen L. Belkic · Journal of Mathematical Chemistry · 2017
This study deals exclusively with total shape spectra (envelopes). Yet, the goal is to exactly reconstruct all the components of the given envelope by relying only upon non-parametric signal processors (shape estimators). To this end, the so-called derivative envelope spectra are investigated. A derivative spectrum is the result of the application of the differentiation transform $$\left( {\hbox {d}/\hbox {d} u } \right) ^{m}$$ d / d ν m to the given conventional spectrum. Here, non-negative integer m is the order of differentiation and $$ u $$ ν is the real linear sweep frequency. For the customary envelope (the $$\hbox {zero}{\mathrm{th}}$$ zero th -order derivative, $$m=0)$$ m = 0 ) , we use the non-parametric fast Padé transform to generate the derivative fast Padé transform (dFPT). Explicit computations are carried out by successively increasing the differentiation order m from low through intermediate to high values of derivatives of complex envelopes. The dFPT can disentangle the spectrally crowded regions by splitting apart any multiplet of closely packed peaks. Hidden resonances, even those that are very weak, can be not only visualized, but also exactly quantified by the dFPT, despite performing shape estimations alone. Most importantly, while the envelopes in the derivative fast Fourier transform exhibit huge noise amplification with increasing m, the same-order of the differentiation transform in the dFPT acts as an effective noise suppressor. The results of the dFPT are illustrated for the envelopes with overlapping peaks stemming from synthesized noise-free and noise-contaminated time signals associated with encoding by in vitro proton magnetic resonance spectroscopy (MRS) of breast cancer tissue. This new methodology is anticipated to significantly enhance resolution as well as signal-to-noise ratio and the overall performance of single-voxel MRS in clinical diagnostics. It is also expected to be of special benefit for volumetric coverage of the scanned tissue by magnetic resonance spectroscopic imaging.