10. Extension of the Monogenic Phasor Method to Extract Displacements and Their Derivatives from 3-D Fringe Patterns

Determination of displacement field and its derivatives from fringe patterns entails four steps: (1) information inscription; (2) data recovery; (3) data processing; (4) data analysis. Phase retrieval and processing are very important in fringe analysis. In [1], these steps were discussed for 1D signals, introducing a 2D abstract space as phase evaluation requires a vectorial function. The Hilbert transform allows to obtain signal in-quadrature defining local phase. A 3D abstract space must be generated to handle the analysis of 2D signals and simultaneously extend Hilbert transform to 2D. The theory of a monogenic function introduced in [2] is elaborated here: the 3D vector in a Cartesian complex space is graphically represented by a Poincare sphere. This provides a generalization of the Hilbert transform to a 2D version of the Riesz transform, a modified version of that described in [2]. Theoretical derivations are supported by actual application of theory and corresponding algorithms to 2D fringe patterns and by comparing obtained results with literature.