Extrapolating Mueller matrices from linear Stokes images

Abstract

An object’s polarimetric bidirectional reflection distribution function (pBRDF) is fully parameterized by the 16 degrees of freedom of a Mueller matrix (MM) at each scattering geometry. A common pBRDF approximation to reduce the degrees of freedom is as a weighted sum of a Fresnel reflection term and an ideal depolarizer term. The weights on these terms represent fractional specular and diffuse reflection and are typically fit independently. Any MM for which the smallest three eigenvalues of the Cloude MM decomposition are identical,¹ can be rewritten as a convex sum of a dominant non-depolarizing MM and an ideal depolarizer.^{2, 3} Therefore, the fractional contribution of each term in this pBRDF model is a single depolarization parameter which corresponds to the largest eigenvalue.² The reduced degrees of freedom for pBRDFs described by this single depolarization parameter create an opportunity to utilize partial polarimetry. The primary contribution of this work is a linear estimator for a MM’s dominant eigenvalue which requires fewer measurements than a full MM reconstruction. Despite reducing the number of simulated measurements by a factor of 10, partial-polarimetry and full Mueller polarimetry eigenvalue estimates are comparable. Root-mean-squared error (RMSE) averaged over acquisition geometry for eigenvalues of a white and a gray balance card were 0.027 and 0.025 respectively for 4 polarimetric measurements, and 0.019 and 0.032 respectively for 40 polarimetric measurements. MM extrapolations from measurements with a commercial off-the-shelf linear Stokes camera are performed at 25 acquisition geometries on an ensemble of LEGO bricks treated to have varying surface roughness. Averaged over the acquisition geometries, the partial-polarimetry extrapolated MMs achieve a 7.3% minimum and 15.1% maximum flux discrepancy from full-polarimetry reconstructed MMs over the varying surface textures. This work demonstrates the first approach, known to the authors, for extrapolating depolarizing MMs.