Two-hierarchical nonnegative matrix factorization distinguishing the fluorescent targets from autofluorescence for fluorescence imaging

Abstract

Background: 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.

Methods: 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.

Results: 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.

Conclusions: 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.

Fig. 1

Synthetic data. The three rows are concentration vectors, fluorescence spectra and data spatial distributions. The first three columns correspond to the three different endmembers. a AF488. b AF594, c AF. d Simulated phantom acquired at 555 nm

Fig. 2

Ground truth synthetic data when SNR=30 and the different unmixed spectra of AF488, AF594 and AF obtained with different algorithms. a Ground truth of fluorescence spectra for two simulated fluorescent targets and one simulated AF. The different unmixed spectra of AF488, AF594 and AF obtained with all algorithms: b MCR-ALS, c NMF$l_1$, d NMFsc, e sNMF, f thNMF

Fig. 3

Ground truth synthetic data when SNR = 30 and the different unmixed concentrations obtained with different algorithms. The three rows correspond to two target fluorescence endmembers: AF488 and AF594, and the background AF. a Ground truth of fluorescence concentrations for simulated AF488, AF594 and AF. The different unmixed abundances of AF488, AF594 and AF obtained with: b MCR-ALS, c NMF$l_1$, d NMFsc, e sNMF, f thNMF

Fig. 4

Evaluation for unmixing results of synthetic data. a $\overline{SAD}$ and b $\overline{RMSE}$ as functions of SNRs

Fig. 5

Unmixing results for nude mouse experiment. a-1–a-5 Raw fluorescence images acquired at the 525, 542, 579, 624 and 716 nm emission filters after subcutaneous injection of AF488 and AF594 dyes into nude mouse. The AF488 in the first three images is excited at 474 nm, the AF594 in the last two images is excited at 565 nm. The unmixed AF488, AF594, and AF obtained with: b-1–b-3 MCR-ALS, c-1–c-3 NMF$l_1$, d-1–d-3 NMFsc, e-1–e-3 sNMF, (f-1–(f-3) thNMF

Fig. 6

Unmixing results for BALB/c mouse experiment 1. a-1–a-5. Raw fluorescence images acquired at the 525, 542, 579, 624 and 716 nm emission filters after subcutaneous injection of AF488 and AF594 dyes into BALB/c mouse. The AF488 in the first three images is excited at 474 nm, the AF594 in the last two images is excited at 565 nm. The unmixed AF488, AF594, and AF obtained with: b-1–b-3 MCR-ALS, c-1–c-3 NMF$l_1$, d-1–d-3 NMFsc, e-1–e-3 sNMF, f-1–f-3 thNMF

Fig. 7

Unmixing results for BALB/c mouse experiment 2. a-1–a-4 Raw fluorescence images acquired at the 525, 542, 579 and 624 nm emission filters after subcutaneous injection of AF488 and AF555 dyes into BALB/c mouse. The AF488 in the first three images is excited at 474 nm and AF555 in the last two images at 500 nm. The unmixed AF488, AF555, and AF obtained with: b-1–b-3 MCR-ALS, c-1–c-3 NMF$l_1$, d-1–d-3 sNMF, e-1–e-3 thNMF