Generalization of the Poincare sphere to process 2D displacement signals

Traditionally the multiple phase method has been considered as an essential tool for phase information recovery. The in-quadrature phase method that theoretically is an alternative pathway to achieve the same goal failed in actual applications. The authors in a previous paper dealing with 1D signals have shown that properly implemented the in-quadrature method yields phase values with the same accuracy than the multiple phase method. The present paper extends the methodology developed in 1D to 2D. This extension is not a straight forward process and requires the introduction of a number of additional concepts and developments. The concept of monogenic function provides the necessary tools required for the extension process. The monogenic function has a graphic representation through the Poincare sphere familiar in the field of Photoelasticity and through the developments introduced in this paper connected to the analysis of displacement fringe patterns. The paper is illustrated with examples of application that show that multiple phases method and the in-quadrature are two aspects of the same basic theoretical model.