Interpretation of measured 3x3 partial depolarizing Mueller matrices

Mueller polarimetry is a powerful technique with broad applications in astronomy, remote sensing, advanced material analysis, and biomedical imaging. However, instrumental constraints frequently restrict the measurement to an incomplete Mueller matrix limited to its upper-left 3x3 submatrix. Simply padding the missing entries with zeros to form a 4x4 matrix can produce physically inconsistent results, even for nondepolarizing systems. To address this issue, we present a systematic procedure to complete 3x3 measured Mueller matrices into physically consistent 4x4 matrices. The method relies on the covariance matrix formalism and selects, among the infinitely many admissible completions, the one with maximal polarimetric purity. This criterion ensures that the synthesized matrix corresponds to the least random (most deterministic) model compatible with the measurement. The procedure is fully general and can be applied to any 3x3 partial Mueller polarimetric data, providing a reliable and physically grounded reconstruction tool for polarimetric imaging and materials characterization.