Lorentzian Peak Sharpening and Sparse Blind Source Separation for NMR Spectroscopy

In this paper, we introduce a preprocessing technique for blind source separation (BSS) of nonnegative and overlapped data. For Nuclear Magnetic Resonance spectroscopy (NMR), the classical method of Naanaa and Nuzillard (NN) requires the condition that source signals to be non-overlapping at certain locations while they are allowed to overlap with each other elsewhere. NN's method works well with data signals that possess stand alone peaks (SAP). The SAP does not hold completely for realistic NMR spectra however. Violation of SAP often introduces errors or artifacts in the NN's separation results. To address this issue, a preprocessing technique is developed here based on Lorentzian peak shapes and weighted peak sharpening. The idea is to superimpose the original peak signal with its weighted negative second order derivative. The resulting sharpened (narrower and taller) peaks enable NN's method to work with a more relaxed SAP condition, the so called dominant peaks condition (DPS), and deliver improved results. To achieve an optimal sharpening while preserving the data nonnegativity, we prove the existence of an upper bound of the weight parameter and propose a selection criterion. Numerical experiments on NMR spectroscopy data show satisfactory performance of our proposed method.