Geodesic distance on non-singular coherency matrix space in polarization optics

Area of Science:

• Optics and Photonics
• Data Science
• Mathematical Physics

Background:

• Polarization analysis is crucial in imaging polarimetry.
• Current methods often use Euclidean distance, which may not be optimal for polarization data.
• Non-singular coherency matrices describe polarization states.

Purpose of the Study:

• To define and investigate a geodesic distance for the polarization space of non-singular coherency matrices.
• To compare the properties of this geodesic distance with the Euclidean distance for interpixel comparisons.
• To evaluate the performance of a geodesic K-means clustering algorithm in imaging polarimetry.

Main Methods:

• Definition of a geodesic distance on the manifold of Hermitian positive definite matrices (HPD(2)).
• Mathematical formulation of the distance and mean value in this metric space.
• Application of a geodesic K-means clustering algorithm to simulated and real polarimetric imaging data.

Main Results:

• The geodesic distance is directly related to the Jones calculus.
• The geodesic distance offers advantages over the Euclidean distance for interpixel comparisons in imaging polarimetry.
• The geodesic K-means algorithm demonstrates improved performance compared to classical methods.

Conclusions:

• The proposed geodesic distance provides a more appropriate metric for polarization space analysis.
• This approach enhances the accuracy and effectiveness of clustering algorithms in imaging polarimetry.
• The geodesic method offers a significant advancement for analyzing polarimetric data.