  This work introduces an on-the-fly (i.e., online) linear unmixing method
which is able to sequentially analyze spectral data acquired on a
spectrum-by-spectrum basis. After deriving a sequential counterpart of the
conventional linear mixing model, the proposed approach recasts the linear
unmixing problem into a linear state-space estimation framework. Under Gaussian
noise and state models, the estimation of the pure spectra can be efficiently
conducted by resorting to Kalman filtering. Interestingly, it is shown that
this Kalman filter can operate in a lower-dimensional subspace while ensuring
the nonnegativity constraint inherent to pure spectra. This dimensionality
reduction allows significantly lightening the computational burden, while
leveraging recent advances related to the representation of essential spectral
information. The proposed method is evaluated through extensive numerical
experiments conducted on synthetic and real Raman data sets. The results show
that this Kalman filter-based method offers a convenient trade-off between
unmixing accuracy and computational efficiency, which is crucial for operating
in an on-the-fly setting. To the best of the authors' knowledge, this is the
first operational method which is able to solve the spectral unmixing problem
efficiently in a dynamic fashion. It also constitutes a valuable building block
for benefiting from acquisition and processing frameworks recently proposed in
the microscopy literature, which are motivated by practical issues such as
reducing acquisition time and avoiding potential damages being inflicted to
photosensitive samples.
