Nonnegative matrix factorization (NMF) has been used in blind fluorescence unmixing for multispectral in-vivo fluorescence imaging, which decomposes a mixed source data into a set of constituent fluorescence spectra and corresponding concentrations. However, most classical NMF algorithms have ill convergence problems and they always fail to unmix multiple fluorescent targets from background autofluorescence for the sparse?acquisition?of multispectral fluorescence imaging, which introduces incomplete measurements and severe discontinuities in multispectral fluorescence emissions?across the multiple spectral bands. Observing the spatial distinction between the diffusive autofluorescence and the sparse fluorescent targets, we propose to separate the mixed sparse multispectral data into equality constrained two-hierarchical updating within NMF framework by dividing the concentration matrix of entire endmembers into two hierarchies: the fluorescence targets and the background autofluorescence. Specifically, when updating concentrations of multiple fluorescent targets in the two-hierarchical NMF, we assume that the concentration of autofluorescence is fixed and known, and vice versa. Furthermore, a sparsity constraint is imposed on the concentration matrix components of fluorescence targets only. Synthetic data sets, in vivo fluorescence imaging data are employed to demonstrate and validate the performance of our approach. The proposed algorithm can achieve more satisfying results of spectral unmixing and autofluorescence removal compared to other state-of-the-art methods, especially for the sparse multispectral fluorescence imaging. The proposed algorithm can successfully tackle the sparse acquisition and ill-posed problems in the NMF-based fluorescence unmixing through equality constraint along with partial sparsity constraint during two-hierarchical NMF optimization, at which fixing sparsity constrained target fluorescence can make the update of autofluorescence as accurate as possible and vice versa.
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