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# Underdetermined Polarimetric Measurements for Mueller Extrapolations

preprint

posted on 2023-01-10, 03:23 authored by Quinn Jarecki, Meredith KupinskiPolarized light-matter interactions are mathematically described by the Mueller matrix (MM)-valued polarized bidirectional reflectance distribution function (pBRDF). A pBRDF is parameterized by 16 degrees of freedom that depend upon scattering geometry. A triple degenerate (TD) MM assumption reduces the degrees of freedom to eight: one for reflectance, six for non-depolarizing properties, and one for depolarization. When the non-depolarizing dominant process is known or assumed (e.g. Fresnel reflection), the degrees of freedom are further reduced to two. For a given material, if the TD model is appropriate and the dominant non-depolarizing process is known, then these two degrees of freedom can be estimated from as few as two polarimetric measurements. Thus, the MM can be extrapolated from a reduced number of measurements. The primary contribution of this work is the development and demonstration of a linear estimator for a MM's dominant eigenvalue (i.e. single depolarization parameter) which requires fewer measurements than a full MM reconstruction. MM extrapolations from single snapshot acquisitions with a Sony Triton 5.0MP Polarization Camera are performed at 30 acquisition geometries and two wavelengths on an ensemble of LEGO bricks treated to have varying surface roughness. These extrapolated MMs are compared to MMs reconstructed from a complete dual rotating retarder (DRR) Mueller imaging polarimeter. The flux error mean and mode are 11.06% and 1.03%, respectively, despite a 10x reduction in the number of polarimetric measurements.