  This article introduces quaternion non-negative matrix factorization (QNMF),
which generalizes the usual non-negative matrix factorization (NMF) to the case
of polarized signals. Polarization information is represented by Stokes
parameters, a set of 4 energetic parameters widely used in polarimetric
imaging. QNMF relies on two key ingredients: (i) the algebraic representation
of Stokes parameters thanks to quaternions and (ii) the exploitation of
physical constraints on Stokes parameters. These constraints generalize
non-negativity to the case of polarized signals, encoding positive
semi-definiteness of the covariance matrix associated which each source.
Uniqueness conditions for QNMF are presented. Remarkably, they encompass known
sufficient uniqueness conditions from NMF. Meanwhile, QNMF further relaxes NMF
uniqueness conditions requiring sources to exhibit a certain zero-pattern, by
leveraging the complete polarization information. We introduce a simple yet
efficient algorithm called quaternion alternating least squares (QALS) to solve
the QNMF problem in practice. Closed-form quaternion updates are obtained using
the recently introduced generalized HR calculus. Numerical experiments on
synthetic data demonstrate the relevance of the approach. QNMF defines a
promising generic low-rank approximation tool to handle polarization, notably
for blind source separation problems arising in imaging applications.
