Abstract
Nonnegative Matrix Factorization (NMF) is a well-known method for Blind Source Separation (BSS). Recently, BSS for polarized signals in spectropolarimetric data, containing both polarization and spectral information, was introduced. This information was encoded in 4-dimensional Stokes vectors represented by quaternion numbers. In the proposed Quaternion NMF (QNMF), the common challenge of determining the (usually) unknown number of quaternion signals remained unaddressed. Estimating the number of signals (aka model determination) is important, since an underestimation of this number results in poor source separation and omission of signals, while overestimation leads to extraction of noisy signals without physical meaning. Here, we introduce a method for determining the number of polarized signals in spectropolarimetric data, named QNMF $k$ . QNMF $k$ integrates: (a) Quaternion Alternating Direction Method of Multipliers (QADMM) implemented for QNMF, (b) random resampling of the initial quaternion data, and (c) custom clustering of sets of QADMM solutions with same number of sources, $k$ , needed to estimate the stability of the solutions. The appropriate latent dimension is determined based on the stability of the solutions. We demonstrate that, without any prior information, QNMF $k$ accurately extracts the correct number of signals used to generate synthetic quaternion datasets and a benchmark spectropolarimetric data.
Highlights
Polarization is a property of transverse waves that describes oscillations perpendicular to the direction of propagation of the wave
Quaternion NMF (QNMF) WITH AUTOMATIC MODEL DETERMINATION The Quaternion Alternating Direction Method of Multipliers (QADMM) algorithm is adequate for extraction of latent polarized sources when their true number k is known, this information is not typically known in practice
To determine this number, we introduce here, QNMFk, a method that integrates: (a) QADMM minimization, (b) random resampling of the initial data, and (c) custom clustering
Summary
Polarization is a property of transverse waves that describes oscillations perpendicular to the direction of propagation of the wave. The polarized wave includes two real-valued signals (u(t), v(t)) which can be represented as a complex function φ(t) ∈ C2; where, φ(t) = u(t) + iv(t) [4]. The complex function, φ(t), describes an elliptical trajectory and the corresponding wave field is called elliptically polarized. When the direction of the arrival of such wave field is known, a standard description of the polarization ellipse can be done by the Stokes parameters, S0, S1, S2, and S3, which are measurable quantities [5]. S0 is the total intensity of the wave while S1, S2, S3 describe the polarization. The first constraint (i) S0 ≥ 0 indicates that there cannot be a negative intensity wave, while the second constraint (ii) S02 ≥ S12 + S22 + S32, imposes that the total wave intensity is greater than or equal to the polarized components magnitude
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