Postprocessing and sparse blind source separation of positive and partially overlapped data

We study sparse blind source separation (BSS) for a class of positive and partially overlapped signals. The signals are only allowed to have nonoverlapping at certain locations, while they could overlap with each other elsewhere. For nonnegative data, a novel approach has been proposed by Naanaa and Nuzillard (NN) assuming that nonoverlapping exists for each source signal at some location of acquisition variable. However, the NN method introduces errors (spurious peaks) in the output when their nonoverlapping condition is not satisfied. To resolve this problem and improve robustness of separation, postprocessing techniques are developed in two aspects. One is to detect coherent and uncertain components from NN outputs by using multiple mixture data, then removing the uncertain portion to enhance signals. The other is to find better estimation of mixing matrix by leveraging reliable source peak structures in NN output. Numerical results on examples including NMR spectra of a 13C-1-acetylated carbohydrate with overlapping proton spin multiplets show satisfactory performance of the postprocessed sparse BSS and offer promise to resolve complex spectra without using multidimensional NMR methods.